[Top][All Lists]
[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
[Emacs-diffs] Changes to calc.texi
|
From: |
Glenn Morris |
|
Subject: |
[Emacs-diffs] Changes to calc.texi |
|
Date: |
Thu, 06 Sep 2007 04:58:47 +0000 |
CVSROOT: /sources/emacs
Module name: emacs
Changes by: Glenn Morris <gm> 07/09/06 04:58:46
Index: calc.texi
===================================================================
RCS file: calc.texi
diff -N calc.texi
--- /dev/null 1 Jan 1970 00:00:00 -0000
+++ calc.texi 6 Sep 2007 04:58:46 -0000 1.1
@@ -0,0 +1,36190 @@
+\input texinfo @c -*-texinfo-*-
address@hidden %**start of header (This is for running Texinfo on a region.)
address@hidden smallbook
address@hidden ../info/calc
address@hidden [title]
address@hidden GNU Emacs Calc 2.1 Manual
address@hidden odd
address@hidden %**end of header (This is for running Texinfo on a region.)
+
address@hidden The following macros are used for conditional output for single
lines.
address@hidden @texline foo
address@hidden `foo' will appear only in TeX output
address@hidden @infoline foo
address@hidden `foo' will appear only in non-TeX output
+
address@hidden @expr{expr} will typeset an expression;
address@hidden $x$ in TeX, @samp{x} otherwise.
+
address@hidden
address@hidden texline
address@hidden macro
address@hidden infoline=comment
address@hidden expr=math
address@hidden tfn=code
address@hidden mathit=expr
address@hidden cpi{}
address@hidden@pi{}}
address@hidden macro
address@hidden cpiover{den}
address@hidden@pi/\den\}
address@hidden macro
address@hidden iftex
+
address@hidden
address@hidden texline=comment
address@hidden infoline{stuff}
+\stuff\
address@hidden macro
address@hidden expr=samp
address@hidden tfn=t
address@hidden mathit=i
address@hidden cpi{}
address@hidden
address@hidden macro
address@hidden cpiover{den}
address@hidden/\den\}
address@hidden macro
address@hidden ifnottex
+
+
address@hidden
+% Suggested by Karl Berry <karl@@freefriends.org>
+\gdef\!{\mskip-\thinmuskip}
address@hidden tex
+
address@hidden Fix some other things specifically for this manual.
address@hidden
address@hidden
address@hidden@:=`@: @c Make Calc fractions come out right in math mode
address@hidden
+\gdef\coloneq{\mathrel{\mathord:\mathord=}}
+
+\gdef\beforedisplay{\vskip-10pt}
+\gdef\afterdisplay{\vskip-5pt}
+\gdef\beforedisplayh{\vskip-25pt}
+\gdef\afterdisplayh{\vskip-10pt}
address@hidden tex
address@hidden@kyvpos @kyvpos=0pt
address@hidden@kyhpos @kyhpos=0pt
address@hidden@calcclubpenalty @calcclubpenalty=1000
address@hidden
address@hidden@calcpageno
address@hidden@calcoldeverypar @address@hidden
address@hidden@address@hidden@calcoldeverypar}
address@hidden@address@hidden@address@hidden@fi
address@hidden@address@hidden@address@hidden@fi
address@hidden@\=0 address@hidden
address@hidden
address@hidden @address@hidden@active
address@hidden ignore
address@hidden iftex
+
address@hidden
+This file documents Calc, the GNU Emacs calculator.
+
+Copyright @copyright{} 1990, 1991, 2001, 2002, 2003, 2004,
+2005, 2006, 2007 Free Software Foundation, Inc.
+
address@hidden
+Permission is granted to copy, distribute and/or modify this document
+under the terms of the GNU Free Documentation License, Version 1.2 or
+any later version published by the Free Software Foundation; with the
+Invariant Sections being just ``GNU GENERAL PUBLIC LICENSE'', with the
+Front-Cover texts being ``A GNU Manual,'' and with the Back-Cover
+Texts as in (a) below. A copy of the license is included in the section
+entitled ``GNU Free Documentation License.''
+
+(a) The FSF's Back-Cover Text is: ``You have freedom to copy and modify
+this GNU Manual, like GNU software. Copies published by the Free
+Software Foundation raise funds for GNU development.''
address@hidden quotation
address@hidden copying
+
address@hidden Emacs
address@hidden
+* Calc: (calc). Advanced desk calculator and mathematical tool.
address@hidden direntry
+
address@hidden
address@hidden 6
address@hidden @titlefont{Calc Manual}
address@hidden 4
address@hidden GNU Emacs Calc Version 2.1
address@hidden [volume]
address@hidden 5
address@hidden Dave Gillespie
address@hidden daveg@@synaptics.com
address@hidden
+
address@hidden 0pt plus 1filll
+Copyright @copyright{} 1990, 1991, 2001, 2002, 2003, 2004,
+ 2005, 2006, 2007 Free Software Foundation, Inc.
address@hidden
address@hidden titlepage
+
+
address@hidden
+
address@hidden [end]
+
address@hidden
+
address@hidden [begin]
address@hidden
address@hidden Top, Getting Started, (dir), (dir)
address@hidden The GNU Emacs Calculator
+
address@hidden
address@hidden is an advanced desk calculator and mathematical tool
+written by Dave Gillespie that runs as part of the GNU Emacs environment.
+
+This manual, also written (mostly) by Dave Gillespie, is divided into
+three major parts: ``Getting Started,'' the ``Calc Tutorial,'' and the
+``Calc Reference.'' The Tutorial introduces all the major aspects of
+Calculator use in an easy, hands-on way. The remainder of the manual is
+a complete reference to the features of the Calculator.
address@hidden ifnottex
+
address@hidden
+For help in the Emacs Info system (which you are using to read this
+file), type @kbd{?}. (You can also type @kbd{h} to run through a
+longer Info tutorial.)
address@hidden ifinfo
+
address@hidden
+* Getting Started:: General description and overview.
address@hidden
+* Interactive Tutorial::
address@hidden ifinfo
+* Tutorial:: A step-by-step introduction for beginners.
+
+* Introduction:: Introduction to the Calc reference manual.
+* Data Types:: Types of objects manipulated by Calc.
+* Stack and Trail:: Manipulating the stack and trail buffers.
+* Mode Settings:: Adjusting display format and other modes.
+* Arithmetic:: Basic arithmetic functions.
+* Scientific Functions:: Transcendentals and other scientific functions.
+* Matrix Functions:: Operations on vectors and matrices.
+* Algebra:: Manipulating expressions algebraically.
+* Units:: Operations on numbers with units.
+* Store and Recall:: Storing and recalling variables.
+* Graphics:: Commands for making graphs of data.
+* Kill and Yank:: Moving data into and out of Calc.
+* Keypad Mode:: Operating Calc from a keypad.
+* Embedded Mode:: Working with formulas embedded in a file.
+* Programming:: Calc as a programmable calculator.
+
+* Copying:: How you can copy and share Calc.
+* GNU Free Documentation License:: The license for this documentation.
+* Customizing Calc:: Customizing Calc.
+* Reporting Bugs:: How to report bugs and make suggestions.
+
+* Summary:: Summary of Calc commands and functions.
+
+* Key Index:: The standard Calc key sequences.
+* Command Index:: The interactive Calc commands.
+* Function Index:: Functions (in algebraic formulas).
+* Concept Index:: General concepts.
+* Variable Index:: Variables used by Calc (both user and internal).
+* Lisp Function Index:: Internal Lisp math functions.
address@hidden menu
+
address@hidden
address@hidden Getting Started, Interactive Tutorial, Top, Top
address@hidden ifinfo
address@hidden
address@hidden Getting Started, Tutorial, Top, Top
address@hidden ifnotinfo
address@hidden Getting Started
address@hidden
+This chapter provides a general overview of Calc, the GNU Emacs
+Calculator: What it is, how to start it and how to exit from it,
+and what are the various ways that it can be used.
+
address@hidden
+* What is Calc::
+* About This Manual::
+* Notations Used in This Manual::
+* Demonstration of Calc::
+* Using Calc::
+* History and Acknowledgements::
address@hidden menu
+
address@hidden What is Calc, About This Manual, Getting Started, Getting Started
address@hidden What is Calc?
+
address@hidden
address@hidden is an advanced calculator and mathematical tool that runs as
+part of the GNU Emacs environment. Very roughly based on the HP-28/48
+series of calculators, its many features include:
+
address@hidden @bullet
address@hidden
+Choice of algebraic or RPN (stack-based) entry of calculations.
+
address@hidden
+Arbitrary precision integers and floating-point numbers.
+
address@hidden
+Arithmetic on rational numbers, complex numbers (rectangular and polar),
+error forms with standard deviations, open and closed intervals, vectors
+and matrices, dates and times, infinities, sets, quantities with units,
+and algebraic formulas.
+
address@hidden
+Mathematical operations such as logarithms and trigonometric functions.
+
address@hidden
+Programmer's features (bitwise operations, non-decimal numbers).
+
address@hidden
+Financial functions such as future value and internal rate of return.
+
address@hidden
+Number theoretical features such as prime factorization and arithmetic
+modulo @var{m} for any @var{m}.
+
address@hidden
+Algebraic manipulation features, including symbolic calculus.
+
address@hidden
+Moving data to and from regular editing buffers.
+
address@hidden
+Embedded mode for manipulating Calc formulas and data directly
+inside any editing buffer.
+
address@hidden
+Graphics using GNUPLOT, a versatile (and free) plotting program.
+
address@hidden
+Easy programming using keyboard macros, algebraic formulas,
+algebraic rewrite rules, or extended Emacs Lisp.
address@hidden itemize
+
+Calc tries to include a little something for everyone; as a result it is
+large and might be intimidating to the first-time user. If you plan to
+use Calc only as a traditional desk calculator, all you really need to
+read is the ``Getting Started'' chapter of this manual and possibly the
+first few sections of the tutorial. As you become more comfortable with
+the program you can learn its additional features. Calc does not
+have the scope and depth of a fully-functional symbolic math package,
+but Calc has the advantages of convenience, portability, and freedom.
+
address@hidden About This Manual, Notations Used in This Manual, What is Calc,
Getting Started
address@hidden About This Manual
+
address@hidden
+This document serves as a complete description of the GNU Emacs
+Calculator. It works both as an introduction for novices, and as
+a reference for experienced users. While it helps to have some
+experience with GNU Emacs in order to get the most out of Calc,
+this manual ought to be readable even if you don't know or use Emacs
+regularly.
+
+The manual is divided into three major parts:@: the ``Getting
+Started'' chapter you are reading now, the Calc tutorial (chapter 2),
+and the Calc reference manual (the remaining chapters and appendices).
address@hidden [when-split]
address@hidden This manual has been printed in two volumes, the @dfn{Tutorial}
and the
address@hidden @dfn{Reference}. Both volumes include a copy of the ``Getting
Started''
address@hidden chapter.
+
+If you are in a hurry to use Calc, there is a brief ``demonstration''
+below which illustrates the major features of Calc in just a couple of
+pages. If you don't have time to go through the full tutorial, this
+will show you everything you need to know to begin.
address@hidden of Calc}.
+
+The tutorial chapter walks you through the various parts of Calc
+with lots of hands-on examples and explanations. If you are new
+to Calc and you have some time, try going through at least the
+beginning of the tutorial. The tutorial includes about 70 exercises
+with answers. These exercises give you some guided practice with
+Calc, as well as pointing out some interesting and unusual ways
+to use its features.
+
+The reference section discusses Calc in complete depth. You can read
+the reference from start to finish if you want to learn every aspect
+of Calc. Or, you can look in the table of contents or the Concept
+Index to find the parts of the manual that discuss the things you
+need to know.
+
address@hidden Marginal notes
+Every Calc keyboard command is listed in the Calc Summary, and also
+in the Key Index. Algebraic functions, @kbd{M-x} commands, and
+variables also have their own indices.
address@hidden Each
address@hidden In the printed manual, each
+paragraph that is referenced in the Key or Function Index is marked
+in the margin with its index entry.
+
address@hidden [fix-ref Help Commands]
+You can access this manual on-line at any time within Calc by
+pressing the @kbd{h i} key sequence. Outside of the Calc window,
+you can press @kbd{C-x * i} to read the manual on-line. Also, you
+can jump directly to the Tutorial by pressing @kbd{h t} or @kbd{C-x * t},
+or to the Summary by pressing @kbd{h s} or @kbd{C-x * s}. Within Calc,
+you can also go to the part of the manual describing any Calc key,
+function, or variable using @address@hidden k}}, @kbd{h f}, or @kbd{h v},
+respectively. @xref{Help Commands}.
+
address@hidden
+The Calc manual can be printed, but because the manual is so large, you
+should only make a printed copy if you really need it. To print the
+manual, you will need the @TeX{} typesetting program (this is a free
+program by Donald Knuth at Stanford University) as well as the
address@hidden program and @file{texinfo.tex} file, both of which can
+be obtained from the FSF as part of the @code{texinfo} package.
+To print the Calc manual in one huge tome, you will need the
+source code to this manual, @file{calc.texi}, available as part of the
+Emacs source. Once you have this file, type @kbd{texi2dvi calc.texi}.
+Alternatively, change to the @file{man} subdirectory of the Emacs
+source distribution, and type @kbd{make calc.dvi}. (Don't worry if you
+get some ``overfull box'' warnings while @TeX{} runs.)
+The result will be a device-independent output file called
address@hidden, which you must print in whatever way is right
+for your system. On many systems, the command is
+
address@hidden
+lpr -d calc.dvi
address@hidden example
+
address@hidden
+or
+
address@hidden
+dvips calc.dvi
address@hidden example
address@hidden ifnottex
address@hidden Printed copies of this manual are also available from the Free
Software
address@hidden Foundation.
+
address@hidden Notations Used in This Manual, Demonstration of Calc, About This
Manual, Getting Started
address@hidden Notations Used in This Manual
+
address@hidden
+This section describes the various notations that are used
+throughout the Calc manual.
+
+In keystroke sequences, uppercase letters mean you must hold down
+the shift key while typing the letter. Keys pressed with Control
+held down are shown as @kbd{C-x}. Keys pressed with Meta held down
+are shown as @kbd{M-x}. Other notations are @key{RET} for the
+Return key, @key{SPC} for the space bar, @key{TAB} for the Tab key,
address@hidden for the Delete key, and @key{LFD} for the Line-Feed key.
+The @key{DEL} key is called Backspace on some keyboards, it is
+whatever key you would use to correct a simple typing error when
+regularly using Emacs.
+
+(If you don't have the @key{LFD} or @key{TAB} keys on your keyboard,
+the @kbd{C-j} and @kbd{C-i} keys are equivalent to them, respectively.
+If you don't have a Meta key, look for Alt or Extend Char. You can
+also press @key{ESC} or @kbd{C-[} first to get the same effect, so
+that @kbd{M-x}, @address@hidden x}, and @kbd{C-[ x} are all equivalent.)
+
+Sometimes the @key{RET} key is not shown when it is ``obvious''
+that you must press @key{RET} to proceed. For example, the @key{RET}
+is usually omitted in key sequences like @kbd{M-x calc-keypad @key{RET}}.
+
+Commands are generally shown like this: @kbd{p} (@code{calc-precision})
+or @kbd{C-x * k} (@code{calc-keypad}). This means that the command is
+normally used by pressing the @kbd{p} key or @kbd{C-x * k} key sequence,
+but it also has the full-name equivalent shown, e.g., @kbd{M-x calc-precision}.
+
+Commands that correspond to functions in algebraic notation
+are written: @kbd{C} (@code{calc-cos}) address@hidden This means
+the @kbd{C} key is equivalent to @kbd{M-x calc-cos}, and that
+the corresponding function in an algebraic-style formula would
+be @samp{cos(@var{x})}.
+
+A few commands don't have key equivalents: @code{calc-sincos}
address@hidden
+
address@hidden Demonstration of Calc, Using Calc, Notations Used in This
Manual, Getting Started
address@hidden A Demonstration of Calc
+
address@hidden
address@hidden Demonstration of Calc
+This section will show some typical small problems being solved with
+Calc. The focus is more on demonstration than explanation, but
+everything you see here will be covered more thoroughly in the
+Tutorial.
+
+To begin, start Emacs if necessary (usually the command @code{emacs}
+does this), and type @kbd{C-x * c} to start the
+Calculator. (You can also use @kbd{M-x calc} if this doesn't work.
address@hidden Calc}, for various ways of starting the Calculator.)
+
+Be sure to type all the sample input exactly, especially noting the
+difference between lower-case and upper-case letters. Remember,
address@hidden, @key{TAB}, @key{DEL}, and @key{SPC} are the Return, Tab,
+Delete, and Space keys.
+
address@hidden calculation.} In RPN, you type the input number(s) first,
+then the command to operate on the numbers.
+
address@hidden
+Type @kbd{2 @key{RET} 3 + Q} to compute
address@hidden @math{\sqrt{2+3} = 2.2360679775}.
address@hidden the square root of 2+3, which is 2.2360679775.
+
address@hidden
+Type @kbd{P 2 ^} to compute
address@hidden @math{\pi^2 = 9.86960440109}.
address@hidden the value of `pi' squared, 9.86960440109.
+
address@hidden
+Type @key{TAB} to exchange the order of these two results.
+
address@hidden
+Type @kbd{- I H S} to subtract these results and compute the Inverse
+Hyperbolic sine of the difference, 2.72996136574.
+
address@hidden
+Type @key{DEL} to erase this result.
+
address@hidden calculation.} You can also enter calculations using
+conventional ``algebraic'' notation. To enter an algebraic formula,
+use the apostrophe key.
+
address@hidden
+Type @kbd{' sqrt(2+3) @key{RET}} to compute
address@hidden @math{\sqrt{2+3}}.
address@hidden the square root of 2+3.
+
address@hidden
+Type @kbd{' pi^2 @key{RET}} to enter
address@hidden @math{\pi^2}.
address@hidden `pi' squared.
+To evaluate this symbolic formula as a number, type @kbd{=}.
+
address@hidden
+Type @kbd{' arcsinh($ - $$) @key{RET}} to subtract the second-most-recent
+result from the most-recent and compute the Inverse Hyperbolic sine.
+
address@hidden mode.} If you are using the X window system, press
address@hidden@kbd{C-x * k}} to get Keypad mode. (If you don't use X, skip to
+the next section.)
+
address@hidden
+Click on the @key{2}, @key{ENTER}, @key{3}, @key{+}, and @key{SQRT}
+``buttons'' using your left mouse button.
+
address@hidden
+Click on @key{PI}, @key{2}, and @tfn{y^x}.
+
address@hidden
+Click on @key{INV}, then @key{ENTER} to swap the two results.
+
address@hidden
+Click on @key{-}, @key{INV}, @key{HYP}, and @key{SIN}.
+
address@hidden
+Click on @key{<-} to erase the result, then click @key{OFF} to turn
+the Keypad Calculator off.
+
address@hidden data.} Type @kbd{C-x * x} if necessary to exit Calc.
+Now select the following numbers as an Emacs region: ``Mark'' the
+front of the list by typing @address@hidden or @kbd{C-@@} there,
+then move to the other end of the list. (Either get this list from
+the on-line copy of this manual, accessed by @address@hidden * i}}, or just
+type these numbers into a scratch file.) Now type @kbd{C-x * g} to
+``grab'' these numbers into Calc.
+
address@hidden
address@hidden
+1.23 1.97
+1.6 2
+1.19 1.08
address@hidden group
address@hidden example
+
address@hidden
+The result @samp{[1.23, 1.97, 1.6, 2, 1.19, 1.08]} is a Calc ``vector.''
+Type @address@hidden R +}} to compute the sum of these numbers.
+
address@hidden
+Type @kbd{U} to Undo this command, then type @kbd{V R *} to compute
+the product of the numbers.
+
address@hidden
+You can also grab data as a rectangular matrix. Place the cursor on
+the upper-leftmost @samp{1} and set the mark, then move to just after
+the lower-right @samp{8} and press @kbd{C-x * r}.
+
address@hidden
+Type @kbd{v t} to transpose this
address@hidden @math{3\times2}
address@hidden 3x2
+matrix into a
address@hidden @math{2\times3}
address@hidden 2x3
+matrix. Type @address@hidden u}} to unpack the rows into two separate
+vectors. Now type @address@hidden R + @key{TAB} V R +}} to compute the sums
+of the two original columns. (There is also a special
+grab-and-sum-columns command, @kbd{C-x * :}.)
+
address@hidden conversion.} Units are entered algebraically.
+Type @address@hidden' 43 mi/hr @key{RET}}} to enter the quantity 43
miles-per-hour.
+Type @address@hidden c km/hr @key{RET}}}. Type @address@hidden c m/s
@key{RET}}}.
+
address@hidden arithmetic.} Type @kbd{t N} to get the current date and
+time. Type @kbd{90 +} to find the date 90 days from now. Type
address@hidden' <25 dec 87> @key{RET}} to enter a date, then @kbd{- 7 /} to see
how
+many weeks have passed since then.
+
address@hidden Algebraic entries can also include formulas
+or equations involving variables. Type @address@hidden' [x + y} = a, x y = 1]
@key{RET}}
+to enter a pair of equations involving three variables.
+(Note the leading apostrophe in this example; also, note that the space
+between @samp{x y} is required.) Type @address@hidden S x,y @key{RET}}} to
solve
+these equations for the variables @expr{x} and @expr{y}.
+
address@hidden
+Type @kbd{d B} to view the solutions in more readable notation.
+Type @address@hidden C}} to view them in C language notation, @kbd{d T}
+to view them in the notation for the @TeX{} typesetting system,
+and @kbd{d L} to view them in the notation for the address@hidden typesetting
+system. Type @kbd{d N} to return to normal notation.
+
address@hidden
+Type @kbd{7.5}, then @kbd{s l a @key{RET}} to let @expr{a = 7.5} in these
formulas.
+(That's a letter @kbd{l}, not a numeral @kbd{1}.)
+
address@hidden
address@hidden functions.} You can read about any command in the on-line
+manual. Type @kbd{C-x * c} to return to Calc after each of these
+commands: @kbd{h k t N} to read about the @kbd{t N} command,
address@hidden f sqrt @key{RET}} to read about the @code{sqrt} function, and
address@hidden s} to read the Calc summary.
address@hidden ifnotinfo
address@hidden
address@hidden functions.} You can read about any command in the on-line
+manual. Remember to type the letter @kbd{l}, then @kbd{C-x * c}, to
+return here after each of these commands: @address@hidden k t N}} to read
+about the @address@hidden N}} command, @kbd{h f sqrt @key{RET}} to read about
the
address@hidden function, and @kbd{h s} to read the Calc summary.
address@hidden ifinfo
+
+Press @key{DEL} repeatedly to remove any leftover results from the stack.
+To exit from Calc, press @kbd{q} or @kbd{C-x * c} again.
+
address@hidden Using Calc, History and Acknowledgements, Demonstration of Calc,
Getting Started
address@hidden Using Calc
+
address@hidden
+Calc has several user interfaces that are specialized for
+different kinds of tasks. As well as Calc's standard interface,
+there are Quick mode, Keypad mode, and Embedded mode.
+
address@hidden
+* Starting Calc::
+* The Standard Interface::
+* Quick Mode Overview::
+* Keypad Mode Overview::
+* Standalone Operation::
+* Embedded Mode Overview::
+* Other C-x * Commands::
address@hidden menu
+
address@hidden Starting Calc, The Standard Interface, Using Calc, Using Calc
address@hidden Starting Calc
+
address@hidden
+On most systems, you can type @kbd{C-x *} to start the Calculator.
+The key sequence @kbd{C-x *} is bound to the command @code{calc-dispatch},
+which can be rebound if convenient (@pxref{Customizing Calc}).
+
+When you press @kbd{C-x *}, Emacs waits for you to press a second key to
+complete the command. In this case, you will follow @kbd{C-x *} with a
+letter (upper- or lower-case, it doesn't matter for @kbd{C-x *}) that says
+which Calc interface you want to use.
+
+To get Calc's standard interface, type @kbd{C-x * c}. To get
+Keypad mode, type @kbd{C-x * k}. Type @kbd{C-x * ?} to get a brief
+list of the available options, and type a second @kbd{?} to get
+a complete list.
+
+To ease typing, @kbd{C-x * *} also works to start Calc. It starts the
+same interface (either @kbd{C-x * c} or @address@hidden * k}}) that you last
+used, selecting the @kbd{C-x * c} interface by default.
+
+If @kbd{C-x *} doesn't work for you, you can always type explicit
+commands like @kbd{M-x calc} (for the standard user interface) or
address@hidden@kbd{M-x calc-keypad}} (for Keypad mode). First type @kbd{M-x}
+(that's Meta with the letter @kbd{x}), then, at the prompt,
+type the full command (like @kbd{calc-keypad}) and press Return.
+
+The same commands (like @kbd{C-x * c} or @kbd{C-x * *}) that start
+the Calculator also turn it off if it is already on.
+
address@hidden The Standard Interface, Quick Mode Overview, Starting Calc,
Using Calc
address@hidden The Standard Calc Interface
+
address@hidden
address@hidden Standard user interface
+Calc's standard interface acts like a traditional RPN calculator,
+operated by the normal Emacs keyboard. When you type @kbd{C-x * c}
+to start the Calculator, the Emacs screen splits into two windows
+with the file you were editing on top and Calc on the bottom.
+
address@hidden
address@hidden
+
+...
+--**-Emacs: myfile (Fundamental)----All----------------------
+--- Emacs Calculator Mode --- |Emacs Calculator Trail
+2: 17.3 | 17.3
+1: -5 | 3
+ . | 2
+ | 4
+ | * 8
+ | ->-5
+ |
+--%%-Calc: 12 Deg (Calculator)----All----- --%%-Emacs: *Calc Trail*
address@hidden group
address@hidden smallexample
+
+In this figure, the mode-line for @file{myfile} has moved up and the
+``Calculator'' window has appeared below it. As you can see, Calc
+actually makes two windows side-by-side. The lefthand one is
+called the @dfn{stack window} and the righthand one is called the
address@hidden window.} The stack holds the numbers involved in the
+calculation you are currently performing. The trail holds a complete
+record of all calculations you have done. In a desk calculator with
+a printer, the trail corresponds to the paper tape that records what
+you do.
+
+In this case, the trail shows that four numbers (17.3, 3, 2, and 4)
+were first entered into the Calculator, then the 2 and 4 were
+multiplied to get 8, then the 3 and 8 were subtracted to get @mathit{-5}.
+(The @samp{>} symbol shows that this was the most recent calculation.)
+The net result is the two numbers 17.3 and @mathit{-5} sitting on the stack.
+
+Most Calculator commands deal explicitly with the stack only, but
+there is a set of commands that allow you to search back through
+the trail and retrieve any previous result.
+
+Calc commands use the digits, letters, and punctuation keys.
+Shifted (i.e., upper-case) letters are different from lowercase
+letters. Some letters are @dfn{prefix} keys that begin two-letter
+commands. For example, @kbd{e} means ``enter exponent'' and shifted
address@hidden means @expr{e^x}. With the @kbd{d} (``display modes'') prefix
+the letter ``e'' takes on very different meanings: @kbd{d e} means
+``engineering notation'' and @kbd{d E} means address@hidden language mode.''
+
+There is nothing stopping you from switching out of the Calc
+window and back into your editing window, say by using the Emacs
address@hidden@kbd{C-x o}} (@code{other-window}) command. When the cursor is
+inside a regular window, Emacs acts just like normal. When the
+cursor is in the Calc stack or trail windows, keys are interpreted
+as Calc commands.
+
+When you quit by pressing @kbd{C-x * c} a second time, the Calculator
+windows go away but the actual Stack and Trail are not gone, just
+hidden. When you press @kbd{C-x * c} once again you will get the
+same stack and trail contents you had when you last used the
+Calculator.
+
+The Calculator does not remember its state between Emacs sessions.
+Thus if you quit Emacs and start it again, @kbd{C-x * c} will give you
+a fresh stack and trail. There is a command (@kbd{m m}) that lets
+you save your favorite mode settings between sessions, though.
+One of the things it saves is which user interface (standard or
+Keypad) you last used; otherwise, a freshly started Emacs will
+always treat @kbd{C-x * *} the same as @kbd{C-x * c}.
+
+The @kbd{q} key is another equivalent way to turn the Calculator off.
+
+If you type @kbd{C-x * b} first and then @kbd{C-x * c}, you get a
+full-screen version of Calc (@code{full-calc}) in which the stack and
+trail windows are still side-by-side but are now as tall as the whole
+Emacs screen. When you press @kbd{q} or @kbd{C-x * c} again to quit,
+the file you were editing before reappears. The @kbd{C-x * b} key
+switches back and forth between ``big'' full-screen mode and the
+normal partial-screen mode.
+
+Finally, @kbd{C-x * o} (@code{calc-other-window}) is like @kbd{C-x * c}
+except that the Calc window is not selected. The buffer you were
+editing before remains selected instead. @kbd{C-x * o} is a handy
+way to switch out of Calc momentarily to edit your file; type
address@hidden * c} to switch back into Calc when you are done.
+
address@hidden Quick Mode Overview, Keypad Mode Overview, The Standard
Interface, Using Calc
address@hidden Quick Mode (Overview)
+
address@hidden
address@hidden mode} is a quick way to use Calc when you don't need the
+full complexity of the stack and trail. To use it, type @kbd{C-x * q}
+(@code{quick-calc}) in any regular editing buffer.
+
+Quick mode is very simple: It prompts you to type any formula in
+standard algebraic notation (like @samp{4 - 2/3}) and then displays
+the result at the bottom of the Emacs screen (@mathit{3.33333333333}
+in this case). You are then back in the same editing buffer you
+were in before, ready to continue editing or to type @kbd{C-x * q}
+again to do another quick calculation. The result of the calculation
+will also be in the Emacs ``kill ring'' so that a @kbd{C-y} command
+at this point will yank the result into your editing buffer.
+
+Calc mode settings affect Quick mode, too, though you will have to
+go into regular Calc (with @kbd{C-x * c}) to change the mode settings.
+
address@hidden [fix-ref Quick Calculator mode]
address@hidden Calculator}, for further information.
+
address@hidden Keypad Mode Overview, Standalone Operation, Quick Mode Overview,
Using Calc
address@hidden Keypad Mode (Overview)
+
address@hidden
address@hidden mode} is a mouse-based interface to the Calculator.
+It is designed for use with terminals that support a mouse. If you
+don't have a mouse, you will have to operate Keypad mode with your
+arrow keys (which is probably more trouble than it's worth).
+
+Type @kbd{C-x * k} to turn Keypad mode on or off. Once again you
+get two new windows, this time on the righthand side of the screen
+instead of at the bottom. The upper window is the familiar Calc
+Stack; the lower window is a picture of a typical calculator keypad.
+
address@hidden
+\dimen0=\pagetotal%
+\advance \dimen0 by 24\baselineskip%
+\ifdim \dimen0>\pagegoal \vfill\eject \fi%
+\medskip
address@hidden tex
address@hidden
address@hidden
+|--- Emacs Calculator Mode ---
+|2: 17.3
+|1: -5
+| .
+|--%%-Calc: 12 Deg (Calcul
+|----+-----Calc 2.1------+----1
+|FLR |CEIL|RND |TRNC|CLN2|FLT |
+|----+----+----+----+----+----|
+| LN |EXP | |ABS |IDIV|MOD |
+|----+----+----+----+----+----|
+|SIN |COS |TAN |SQRT|y^x |1/x |
+|----+----+----+----+----+----|
+| ENTER |+/- |EEX |UNDO| <- |
+|-----+---+-+--+--+-+---++----|
+| INV | 7 | 8 | 9 | / |
+|-----+-----+-----+-----+-----|
+| HYP | 4 | 5 | 6 | * |
+|-----+-----+-----+-----+-----|
+|EXEC | 1 | 2 | 3 | - |
+|-----+-----+-----+-----+-----|
+| OFF | 0 | . | PI | + |
+|-----+-----+-----+-----+-----+
address@hidden group
address@hidden smallexample
+
+Keypad mode is much easier for beginners to learn, because there
+is no need to memorize lots of obscure key sequences. But not all
+commands in regular Calc are available on the Keypad. You can
+always switch the cursor into the Calc stack window to use
+standard Calc commands if you need. Serious Calc users, though,
+often find they prefer the standard interface over Keypad mode.
+
+To operate the Calculator, just click on the ``buttons'' of the
+keypad using your left mouse button. To enter the two numbers
+shown here you would click @address@hidden 7 .@: 3 ENTER 5 +/- ENTER}}; to
+add them together you would then click @kbd{+} (to get 12.3 on
+the stack).
+
+If you click the right mouse button, the top three rows of the
+keypad change to show other sets of commands, such as advanced
+math functions, vector operations, and operations on binary
+numbers.
+
+Because Keypad mode doesn't use the regular keyboard, Calc leaves
+the cursor in your original editing buffer. You can type in
+this buffer in the usual way while also clicking on the Calculator
+keypad. One advantage of Keypad mode is that you don't need an
+explicit command to switch between editing and calculating.
+
+If you press @kbd{C-x * b} first, you get a full-screen Keypad mode
+(@code{full-calc-keypad}) with three windows: The keypad in the lower
+left, the stack in the lower right, and the trail on top.
+
address@hidden [fix-ref Keypad Mode]
address@hidden Mode}, for further information.
+
address@hidden Standalone Operation, Embedded Mode Overview, Keypad Mode
Overview, Using Calc
address@hidden Standalone Operation
+
address@hidden
address@hidden Standalone Operation
+If you are not in Emacs at the moment but you wish to use Calc,
+you must start Emacs first. If all you want is to run Calc, you
+can give the commands:
+
address@hidden
+emacs -f full-calc
address@hidden example
+
address@hidden
+or
+
address@hidden
+emacs -f full-calc-keypad
address@hidden example
+
address@hidden
+which run a full-screen Calculator (as if by @kbd{C-x * b C-x * c}) or
+a full-screen X-based Calculator (as if by @kbd{C-x * b C-x * k}).
+In standalone operation, quitting the Calculator (by pressing
address@hidden or clicking on the keypad @key{EXIT} button) quits Emacs
+itself.
+
address@hidden Embedded Mode Overview, Other C-x * Commands, Standalone
Operation, Using Calc
address@hidden Embedded Mode (Overview)
+
address@hidden
address@hidden mode} is a way to use Calc directly from inside an
+editing buffer. Suppose you have a formula written as part of a
+document like this:
+
address@hidden
address@hidden
+The derivative of
+
+ ln(ln(x))
+
+is
address@hidden group
address@hidden smallexample
+
address@hidden
+and you wish to have Calc compute and format the derivative for
+you and store this derivative in the buffer automatically. To
+do this with Embedded mode, first copy the formula down to where
+you want the result to be:
+
address@hidden
address@hidden
+The derivative of
+
+ ln(ln(x))
+
+is
+
+ ln(ln(x))
address@hidden group
address@hidden smallexample
+
+Now, move the cursor onto this new formula and press @kbd{C-x * e}.
+Calc will read the formula (using the surrounding blank lines to
+tell how much text to read), then push this formula (invisibly)
+onto the Calc stack. The cursor will stay on the formula in the
+editing buffer, but the buffer's mode line will change to look
+like the Calc mode line (with mode indicators like @samp{12 Deg}
+and so on). Even though you are still in your editing buffer,
+the keyboard now acts like the Calc keyboard, and any new result
+you get is copied from the stack back into the buffer. To take
+the derivative, you would type @kbd{a d x @key{RET}}.
+
address@hidden
address@hidden
+The derivative of
+
+ ln(ln(x))
+
+is
+
+1 / ln(x) x
address@hidden group
address@hidden smallexample
+
+To make this look nicer, you might want to press @kbd{d =} to center
+the formula, and even @kbd{d B} to use Big display mode.
+
address@hidden
address@hidden
+The derivative of
+
+ ln(ln(x))
+
+is
+% [calc-mode: justify: center]
+% [calc-mode: language: big]
+
+ 1
+ -------
+ ln(x) x
address@hidden group
address@hidden smallexample
+
+Calc has added annotations to the file to help it remember the modes
+that were used for this formula. They are formatted like comments
+in the @TeX{} typesetting language, just in case you are using @TeX{} or
address@hidden (In this example @TeX{} is not being used, so you might want
+to move these comments up to the top of the file or otherwise put them
+out of the way.)
+
+As an extra flourish, we can add an equation number using a
+righthand label: Type @kbd{d @} (1) @key{RET}}.
+
address@hidden
address@hidden
+% [calc-mode: justify: center]
+% [calc-mode: language: big]
+% [calc-mode: right-label: " (1)"]
+
+ 1
+ ------- (1)
+ ln(x) x
address@hidden group
address@hidden smallexample
+
+To leave Embedded mode, type @kbd{C-x * e} again. The mode line
+and keyboard will revert to the way they were before.
+
+The related command @kbd{C-x * w} operates on a single word, which
+generally means a single number, inside text. It uses any
+non-numeric characters rather than blank lines to delimit the
+formula it reads. Here's an example of its use:
+
address@hidden
+A slope of one-third corresponds to an angle of 1 degrees.
address@hidden smallexample
+
+Place the cursor on the @samp{1}, then type @kbd{C-x * w} to enable
+Embedded mode on that number. Now type @kbd{3 /} (to get one-third),
+and @kbd{I T} (the Inverse Tangent converts a slope into an angle),
+then @address@hidden * w}} again to exit Embedded mode.
+
address@hidden
+A slope of one-third corresponds to an angle of 18.4349488229 degrees.
address@hidden smallexample
+
address@hidden [fix-ref Embedded Mode]
address@hidden Mode}, for full details.
+
address@hidden Other C-x * Commands, , Embedded Mode Overview, Using Calc
address@hidden Other @kbd{C-x *} Commands
+
address@hidden
+Two more Calc-related commands are @kbd{C-x * g} and @kbd{C-x * r},
+which ``grab'' data from a selected region of a buffer into the
+Calculator. The region is defined in the usual Emacs way, by
+a ``mark'' placed at one end of the region, and the Emacs
+cursor or ``point'' placed at the other.
+
+The @kbd{C-x * g} command reads the region in the usual left-to-right,
+top-to-bottom order. The result is packaged into a Calc vector
+of numbers and placed on the stack. Calc (in its standard
+user interface) is then started. Type @kbd{v u} if you want
+to unpack this vector into separate numbers on the stack. Also,
address@hidden C-x * g} interprets the region as a single number or
+formula.
+
+The @kbd{C-x * r} command reads a rectangle, with the point and
+mark defining opposite corners of the rectangle. The result
+is a matrix of numbers on the Calculator stack.
+
+Complementary to these is @kbd{C-x * y}, which ``yanks'' the
+value at the top of the Calc stack back into an editing buffer.
+If you type @address@hidden * y}} while in such a buffer, the value is
+yanked at the current position. If you type @kbd{C-x * y} while
+in the Calc buffer, Calc makes an educated guess as to which
+editing buffer you want to use. The Calc window does not have
+to be visible in order to use this command, as long as there
+is something on the Calc stack.
+
+Here, for reference, is the complete list of @kbd{C-x *} commands.
+The shift, control, and meta keys are ignored for the keystroke
+following @kbd{C-x *}.
+
address@hidden
+Commands for turning Calc on and off:
+
address@hidden @kbd
address@hidden *
+Turn Calc on or off, employing the same user interface as last time.
+
address@hidden =, +, -, /, \, &, #
+Alternatives for @kbd{*}.
+
address@hidden C
+Turn Calc on or off using its standard bottom-of-the-screen
+interface. If Calc is already turned on but the cursor is not
+in the Calc window, move the cursor into the window.
+
address@hidden O
+Same as @kbd{C}, but don't select the new Calc window. If
+Calc is already turned on and the cursor is in the Calc window,
+move it out of that window.
+
address@hidden B
+Control whether @kbd{C-x * c} and @kbd{C-x * k} use the full screen.
+
address@hidden Q
+Use Quick mode for a single short calculation.
+
address@hidden K
+Turn Calc Keypad mode on or off.
+
address@hidden E
+Turn Calc Embedded mode on or off at the current formula.
+
address@hidden J
+Turn Calc Embedded mode on or off, select the interesting part.
+
address@hidden W
+Turn Calc Embedded mode on or off at the current word (number).
+
address@hidden Z
+Turn Calc on in a user-defined way, as defined by a @kbd{Z I} command.
+
address@hidden X
+Quit Calc; turn off standard, Keypad, or Embedded mode if on.
+(This is like @kbd{q} or @key{OFF} inside of Calc.)
address@hidden table
address@hidden
address@hidden 2
address@hidden iftex
+
address@hidden
+Commands for moving data into and out of the Calculator:
+
address@hidden @kbd
address@hidden G
+Grab the region into the Calculator as a vector.
+
address@hidden R
+Grab the rectangular region into the Calculator as a matrix.
+
address@hidden :
+Grab the rectangular region and compute the sums of its columns.
+
address@hidden _
+Grab the rectangular region and compute the sums of its rows.
+
address@hidden Y
+Yank a value from the Calculator into the current editing buffer.
address@hidden table
address@hidden
address@hidden 2
address@hidden iftex
+
address@hidden
+Commands for use with Embedded mode:
+
address@hidden @kbd
address@hidden A
+``Activate'' the current buffer. Locate all formulas that
+contain @samp{:=} or @samp{=>} symbols and record their locations
+so that they can be updated automatically as variables are changed.
+
address@hidden D
+Duplicate the current formula immediately below and select
+the duplicate.
+
address@hidden F
+Insert a new formula at the current point.
+
address@hidden N
+Move the cursor to the next active formula in the buffer.
+
address@hidden P
+Move the cursor to the previous active formula in the buffer.
+
address@hidden U
+Update (i.e., as if by the @kbd{=} key) the formula at the current point.
+
address@hidden `
+Edit (as if by @code{calc-edit}) the formula at the current point.
address@hidden table
address@hidden
address@hidden 2
address@hidden iftex
+
address@hidden
+Miscellaneous commands:
+
address@hidden @kbd
address@hidden I
+Run the Emacs Info system to read the Calc manual.
+(This is the same as @kbd{h i} inside of Calc.)
+
address@hidden T
+Run the Emacs Info system to read the Calc Tutorial.
+
address@hidden S
+Run the Emacs Info system to read the Calc Summary.
+
address@hidden L
+Load Calc entirely into memory. (Normally the various parts
+are loaded only as they are needed.)
+
address@hidden M
+Read a region of written keystroke names (like @kbd{C-n a b c @key{RET}})
+and record them as the current keyboard macro.
+
address@hidden 0
+(This is the ``zero'' digit key.) Reset the Calculator to
+its initial state: Empty stack, and initial mode settings.
address@hidden table
+
address@hidden History and Acknowledgements, , Using Calc, Getting Started
address@hidden History and Acknowledgements
+
address@hidden
+Calc was originally started as a two-week project to occupy a lull
+in the author's schedule. Basically, a friend asked if I remembered
+the value of
address@hidden @math{2^{32}}.
address@hidden @expr{2^32}.
+I didn't offhand, but I said, ``that's easy, just call up an
address@hidden'' @code{Xcalc} duly reported that the answer to our
+question was @samp{4.294967e+09}---with no way to see the full ten
+digits even though we knew they were there in the program's memory! I
+was so annoyed, I vowed to write a calculator of my own, once and for
+all.
+
+I chose Emacs Lisp, a) because I had always been curious about it
+and b) because, being only a text editor extension language after
+all, Emacs Lisp would surely reach its limits long before the project
+got too far out of hand.
+
+To make a long story short, Emacs Lisp turned out to be a distressingly
+solid implementation of Lisp, and the humble task of calculating
+turned out to be more open-ended than one might have expected.
+
+Emacs Lisp didn't have built-in floating point math (now it does), so
+this had to be
+simulated in software. In fact, Emacs integers will only comfortably
+fit six decimal digits or so---not enough for a decent calculator. So
+I had to write my own high-precision integer code as well, and once I had
+this I figured that arbitrary-size integers were just as easy as large
+integers. Arbitrary floating-point precision was the logical next step.
+Also, since the large integer arithmetic was there anyway it seemed only
+fair to give the user direct access to it, which in turn made it practical
+to support fractions as well as floats. All these features inspired me
+to look around for other data types that might be worth having.
+
+Around this time, my friend Rick Koshi showed me his nifty new HP-28
+calculator. It allowed the user to manipulate formulas as well as
+numerical quantities, and it could also operate on matrices. I
+decided that these would be good for Calc to have, too. And once
+things had gone this far, I figured I might as well take a look at
+serious algebra systems for further ideas. Since these systems did
+far more than I could ever hope to implement, I decided to focus on
+rewrite rules and other programming features so that users could
+implement what they needed for themselves.
+
+Rick complained that matrices were hard to read, so I put in code to
+format them in a 2D style. Once these routines were in place, Big mode
+was obligatory. Gee, what other language modes would be useful?
+
+Scott Hemphill and Allen Knutson, two friends with a strong mathematical
+bent, contributed ideas and algorithms for a number of Calc features
+including modulo forms, primality testing, and float-to-fraction conversion.
+
+Units were added at the eager insistence of Mass Sivilotti. Later,
+Ulrich Mueller at CERN and Przemek Klosowski at NIST provided invaluable
+expert assistance with the units table. As far as I can remember, the
+idea of using algebraic formulas and variables to represent units dates
+back to an ancient article in Byte magazine about muMath, an early
+algebra system for microcomputers.
+
+Many people have contributed to Calc by reporting bugs and suggesting
+features, large and small. A few deserve special mention: Tim Peters,
+who helped develop the ideas that led to the selection commands, rewrite
+rules, and many other algebra features;
address@hidden Fran\c{c}ois
address@hidden Francois
+Pinard, who contributed an early prototype of the Calc Summary appendix
+as well as providing valuable suggestions in many other areas of Calc;
+Carl Witty, whose eagle eyes discovered many typographical and factual
+errors in the Calc manual; Tim Kay, who drove the development of
+Embedded mode; Ove Ewerlid, who made many suggestions relating to the
+algebra commands and contributed some code for polynomial operations;
+Randal Schwartz, who suggested the @code{calc-eval} function; Robert
+J. Chassell, who suggested the Calc Tutorial and exercises; and Juha
+Sarlin, who first worked out how to split Calc into quickly-loading
+parts. Bob Weiner helped immensely with the Lucid Emacs port.
+
address@hidden Bibliography
address@hidden Knuth, Art of Computer Programming
address@hidden Numerical Recipes
address@hidden Should these be expanded into more complete references?
+Among the books used in the development of Calc were Knuth's @emph{Art
+of Computer Programming} (especially volume II, @emph{Seminumerical
+Algorithms}); @emph{Numerical Recipes} by Press, Flannery, Teukolsky,
+and Vetterling; Bevington's @emph{Data Reduction and Error Analysis
+for the Physical Sciences}; @emph{Concrete Mathematics} by Graham,
+Knuth, and Patashnik; Steele's @emph{Common Lisp, the Language}; the
address@hidden Standard Math Tables} (William H. Beyer, ed.); and
+Abramowitz and Stegun's venerable @emph{Handbook of Mathematical
+Functions}. Also, of course, Calc could not have been written without
+the excellent @emph{GNU Emacs Lisp Reference Manual}, by Bil Lewis and
+Dan LaLiberte.
+
+Final thanks go to Richard Stallman, without whose fine implementations
+of the Emacs editor, language, and environment, Calc would have been
+finished in two weeks.
+
address@hidden [tutorial]
+
address@hidden
address@hidden This node is accessed by the `C-x * t' command.
address@hidden Interactive Tutorial, Tutorial, Getting Started, Top
address@hidden Tutorial
+
address@hidden
+Some brief instructions on using the Emacs Info system for this tutorial:
+
+Press the space bar and Delete keys to go forward and backward in a
+section by screenfuls (or use the regular Emacs scrolling commands
+for this).
+
+Press @kbd{n} or @kbd{p} to go to the Next or Previous section.
+If the section has a @dfn{menu}, press a digit key like @kbd{1}
+or @kbd{2} to go to a sub-section from the menu. Press @kbd{u} to
+go back up from a sub-section to the menu it is part of.
+
+Exercises in the tutorial all have cross-references to the
+appropriate page of the ``answers'' section. Press @kbd{f}, then
+the exercise number, to see the answer to an exercise. After
+you have followed a cross-reference, you can press the letter
address@hidden to return to where you were before.
+
+You can press @kbd{?} at any time for a brief summary of Info commands.
+
+Press @kbd{1} now to enter the first section of the Tutorial.
+
address@hidden
+* Tutorial::
address@hidden menu
+
address@hidden Tutorial, Introduction, Interactive Tutorial, Top
address@hidden ifinfo
address@hidden
address@hidden Tutorial, Introduction, Getting Started, Top
address@hidden ifnotinfo
address@hidden Tutorial
+
address@hidden
+This chapter explains how to use Calc and its many features, in
+a step-by-step, tutorial way. You are encouraged to run Calc and
+work along with the examples as you read (@pxref{Starting Calc}).
+If you are already familiar with advanced calculators, you may wish
address@hidden [not-split]
+to skip on to the rest of this manual.
address@hidden [when-split]
address@hidden to skip on to volume II of this manual, the @dfn{Calc Reference}.
+
address@hidden [fix-ref Embedded Mode]
+This tutorial describes the standard user interface of Calc only.
+The Quick mode and Keypad mode interfaces are fairly
+self-explanatory. @xref{Embedded Mode}, for a description of
+the Embedded mode interface.
+
+The easiest way to read this tutorial on-line is to have two windows on
+your Emacs screen, one with Calc and one with the Info system. (If you
+have a printed copy of the manual you can use that instead.) Press
address@hidden * c} to turn Calc on or to switch into the Calc window, and
+press @kbd{C-x * i} to start the Info system or to switch into its window.
+
+This tutorial is designed to be done in sequence. But the rest of this
+manual does not assume you have gone through the tutorial. The tutorial
+does not cover everything in the Calculator, but it touches on most
+general areas.
+
address@hidden
+You may wish to print out a copy of the Calc Summary and keep notes on
+it as you learn Calc. @xref{About This Manual}, to see how to make a
+printed summary. @xref{Summary}.
address@hidden ifnottex
address@hidden
+The Calc Summary at the end of the reference manual includes some blank
+space for your own use. You may wish to keep notes there as you learn
+Calc.
address@hidden iftex
+
address@hidden
+* Basic Tutorial::
+* Arithmetic Tutorial::
+* Vector/Matrix Tutorial::
+* Types Tutorial::
+* Algebra Tutorial::
+* Programming Tutorial::
+
+* Answers to Exercises::
address@hidden menu
+
address@hidden Basic Tutorial, Arithmetic Tutorial, Tutorial, Tutorial
address@hidden Basic Tutorial
+
address@hidden
+In this section, we learn how RPN and algebraic-style calculations
+work, how to undo and redo an operation done by mistake, and how
+to control various modes of the Calculator.
+
address@hidden
+* RPN Tutorial:: Basic operations with the stack.
+* Algebraic Tutorial:: Algebraic entry; variables.
+* Undo Tutorial:: If you make a mistake: Undo and the trail.
+* Modes Tutorial:: Common mode-setting commands.
address@hidden menu
+
address@hidden RPN Tutorial, Algebraic Tutorial, Basic Tutorial, Basic Tutorial
address@hidden RPN Calculations and the Stack
+
address@hidden RPN notation
address@hidden
address@hidden
+Calc normally uses RPN notation. You may be familiar with the RPN
+system from Hewlett-Packard calculators, FORTH, or PostScript.
+(Reverse Polish Notation, RPN, is named after the Polish mathematician
+Jan Lukasiewicz.)
address@hidden ifnottex
address@hidden
+\noindent
+Calc normally uses RPN notation. You may be familiar with the RPN
+system from Hewlett-Packard calculators, FORTH, or PostScript.
+(Reverse Polish Notation, RPN, is named after the Polish mathematician
+Jan \L ukasiewicz.)
address@hidden tex
+
+The central component of an RPN calculator is the @dfn{stack}. A
+calculator stack is like a stack of dishes. New dishes (numbers) are
+added at the top of the stack, and numbers are normally only removed
+from the top of the stack.
+
address@hidden Operators
address@hidden Operands
+In an operation like @expr{2+3}, the 2 and 3 are called the @dfn{operands}
+and the @expr{+} is the @dfn{operator}. In an RPN calculator you always
+enter the operands first, then the operator. Each time you type a
+number, Calc adds or @dfn{pushes} it onto the top of the Stack.
+When you press an operator key like @kbd{+}, Calc @dfn{pops} the appropriate
+number of operands from the stack and pushes back the result.
+
+Thus we could add the numbers 2 and 3 in an RPN calculator by typing:
address@hidden @key{RET} 3 @key{RET} +}. (The @key{RET} key, Return,
corresponds to
+the @key{ENTER} key on traditional RPN calculators.) Try this now if
+you wish; type @kbd{C-x * c} to switch into the Calc window (you can type
address@hidden * c} again or @kbd{C-x * o} to switch back to the Tutorial
window).
+The first four keystrokes ``push'' the numbers 2 and 3 onto the stack.
+The @kbd{+} key ``pops'' the top two numbers from the stack, adds them,
+and pushes the result (5) back onto the stack. Here's how the stack
+will look at various points throughout the calculation:
+
address@hidden
address@hidden
+ . 1: 2 2: 2 1: 5 .
+ . 1: 3 .
+ .
+
+ C-x * c 2 @key{RET} 3 @key{RET} +
@key{DEL}
address@hidden group
address@hidden smallexample
+
+The @samp{.} symbol is a marker that represents the top of the stack.
+Note that the ``top'' of the stack is really shown at the bottom of
+the Stack window. This may seem backwards, but it turns out to be
+less distracting in regular use.
+
address@hidden Stack levels
address@hidden Levels of stack
+The numbers @samp{1:} and @samp{2:} on the left are @dfn{stack level
+numbers}. Old RPN calculators always had four stack levels called
address@hidden, @expr{y}, @expr{z}, and @expr{t}. Calc's stack can grow
+as large as you like, so it uses numbers instead of letters. Some
+stack-manipulation commands accept a numeric argument that says
+which stack level to work on. Normal commands like @kbd{+} always
+work on the top few levels of the stack.
+
address@hidden [fix-ref Truncating the Stack]
+The Stack buffer is just an Emacs buffer, and you can move around in
+it using the regular Emacs motion commands. But no matter where the
+cursor is, even if you have scrolled the @samp{.} marker out of
+view, most Calc commands always move the cursor back down to level 1
+before doing anything. It is possible to move the @samp{.} marker
+upwards through the stack, temporarily ``hiding'' some numbers from
+commands like @kbd{+}. This is called @dfn{stack truncation} and
+we will not cover it in this tutorial; @pxref{Truncating the Stack},
+if you are interested.
+
+You don't really need the second @key{RET} in @kbd{2 @key{RET} 3
address@hidden +}. That's because if you type any operator name or
+other non-numeric key when you are entering a number, the Calculator
+automatically enters that number and then does the requested command.
+Thus @kbd{2 @key{RET} 3 +} will work just as well.
+
+Examples in this tutorial will often omit @key{RET} even when the
+stack displays shown would only happen if you did press @key{RET}:
+
address@hidden
address@hidden
+1: 2 2: 2 1: 5
+ . 1: 3 .
+ .
+
+ 2 @key{RET} 3 +
address@hidden group
address@hidden smallexample
+
address@hidden
+Here, after pressing @kbd{3} the stack would really show @samp{1: 2}
+with @samp{Calc:@: 3} in the minibuffer. In these situations, you can
+press the optional @key{RET} to see the stack as the figure shows.
+
+(@bullet{}) @strong{Exercise 1.} (This tutorial will include exercises
+at various points. Try them if you wish. Answers to all the exercises
+are located at the end of the Tutorial chapter. Each exercise will
+include a cross-reference to its particular answer. If you are
+reading with the Emacs Info system, press @kbd{f} and the
+exercise number to go to the answer, then the letter @kbd{l} to
+return to where you were.)
+
address@hidden
+Here's the first exercise: What will the keystrokes @kbd{1 @key{RET} 2
address@hidden 3 @key{RET} 4 + * -} compute? (@samp{*} is the symbol for
+multiplication.) Figure it out by hand, then try it with Calc to see
+if you're right. @xref{RPN Answer 1, 1}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 2.} Compute
address@hidden @math{(2\times4) + (7\times9.4) + {5\over4}}
address@hidden @expr{2*4 + 7*9.5 + 5/4}
+using the stack. @xref{RPN Answer 2, 2}. (@bullet{})
+
+The @key{DEL} key is called Backspace on some keyboards. It is
+whatever key you would use to correct a simple typing error when
+regularly using Emacs. The @key{DEL} key pops and throws away the
+top value on the stack. (You can still get that value back from
+the Trail if you should need it later on.) There are many places
+in this tutorial where we assume you have used @key{DEL} to erase the
+results of the previous example at the beginning of a new example.
+In the few places where it is really important to use @key{DEL} to
+clear away old results, the text will remind you to do so.
+
+(It won't hurt to let things accumulate on the stack, except that
+whenever you give a display-mode-changing command Calc will have to
+spend a long time reformatting such a large stack.)
+
+Since the @kbd{-} key is also an operator (it subtracts the top two
+stack elements), how does one enter a negative number? Calc uses
+the @kbd{_} (underscore) key to act like the minus sign in a number.
+So, typing @kbd{-5 @key{RET}} won't work because the @kbd{-} key
+will try to do a subtraction, but @kbd{_5 @key{RET}} works just fine.
+
+You can also press @kbd{n}, which means ``change sign.'' It changes
+the number at the top of the stack (or the number being entered)
+from positive to negative or vice-versa: @kbd{5 n @key{RET}}.
+
address@hidden Duplicating a stack entry
+If you press @key{RET} when you're not entering a number, the effect
+is to duplicate the top number on the stack. Consider this calculation:
+
address@hidden
address@hidden
+1: 3 2: 3 1: 9 2: 9 1: 81
+ . 1: 3 . 1: 9 .
+ . .
+
+ 3 @key{RET} @key{RET} * @key{RET}
*
address@hidden group
address@hidden smallexample
+
address@hidden
+(Of course, an easier way to do this would be @kbd{3 @key{RET} 4 ^},
+to raise 3 to the fourth power.)
+
+The space-bar key (denoted @key{SPC} here) performs the same function
+as @key{RET}; you could replace all three occurrences of @key{RET} in
+the above example with @key{SPC} and the effect would be the same.
+
address@hidden Exchanging stack entries
+Another stack manipulation key is @key{TAB}. This exchanges the top
+two stack entries. Suppose you have computed @kbd{2 @key{RET} 3 +}
+to get 5, and then you realize what you really wanted to compute
+was @expr{20 / (2+3)}.
+
address@hidden
address@hidden
+1: 5 2: 5 2: 20 1: 4
+ . 1: 20 1: 5 .
+ . .
+
+ 2 @key{RET} 3 + 20 @key{TAB} /
address@hidden group
address@hidden smallexample
+
address@hidden
+Planning ahead, the calculation would have gone like this:
+
address@hidden
address@hidden
+1: 20 2: 20 3: 20 2: 20 1: 4
+ . 1: 2 2: 2 1: 5 .
+ . 1: 3 .
+ .
+
+ 20 @key{RET} 2 @key{RET} 3 + /
address@hidden group
address@hidden smallexample
+
+A related stack command is @address@hidden (hold @key{META} and type
address@hidden). It rotates the top three elements of the stack upward,
+bringing the object in level 3 to the top.
+
address@hidden
address@hidden
+1: 10 2: 10 3: 10 3: 20 3: 30
+ . 1: 20 2: 20 2: 30 2: 10
+ . 1: 30 1: 10 1: 20
+ . . .
+
+ 10 @key{RET} 20 @key{RET} 30 @key{RET}
address@hidden address@hidden
address@hidden group
address@hidden smallexample
+
+(@bullet{}) @strong{Exercise 3.} Suppose the numbers 10, 20, and 30 are
+on the stack. Figure out how to add one to the number in level 2
+without affecting the rest of the stack. Also figure out how to add
+one to the number in level 3. @xref{RPN Answer 3, 3}. (@bullet{})
+
+Operations like @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/}, and @kbd{^} pop two
+arguments from the stack and push a result. Operations like @kbd{n} and
address@hidden (square root) pop a single number and push the result. You can
+think of them as simply operating on the top element of the stack.
+
address@hidden
address@hidden
+1: 3 1: 9 2: 9 1: 25 1: 5
+ . . 1: 16 . .
+ .
+
+ 3 @key{RET} @key{RET} * 4 @key{RET} @key{RET} * +
Q
address@hidden group
address@hidden smallexample
+
address@hidden
+(Note that capital @kbd{Q} means to hold down the Shift key while
+typing @kbd{q}. Remember, plain unshifted @kbd{q} is the Quit command.)
+
address@hidden Pythagorean Theorem
+Here we've used the Pythagorean Theorem to determine the hypotenuse of a
+right triangle. Calc actually has a built-in command for that called
address@hidden h}, but let's suppose we can't remember the necessary keystrokes.
+We can still enter it by its full name using @kbd{M-x} notation:
+
address@hidden
address@hidden
+1: 3 2: 3 1: 5
+ . 1: 4 .
+ .
+
+ 3 @key{RET} 4 @key{RET} M-x calc-hypot
address@hidden group
address@hidden smallexample
+
+All Calculator commands begin with the word @samp{calc-}. Since it
+gets tiring to type this, Calc provides an @kbd{x} key which is just
+like the regular Emacs @kbd{M-x} key except that it types the @samp{calc-}
+prefix for you:
+
address@hidden
address@hidden
+1: 3 2: 3 1: 5
+ . 1: 4 .
+ .
+
+ 3 @key{RET} 4 @key{RET} x hypot
address@hidden group
address@hidden smallexample
+
+What happens if you take the square root of a negative number?
+
address@hidden
address@hidden
+1: 4 1: -4 1: (0, 2)
+ . . .
+
+ 4 @key{RET} n Q
address@hidden group
address@hidden smallexample
+
address@hidden
+The notation @expr{(a, b)} represents a complex number.
+Complex numbers are more traditionally written @expr{a + b i};
+Calc can display in this format, too, but for now we'll stick to the
address@hidden(a, b)} notation.
+
+If you don't know how complex numbers work, you can safely ignore this
+feature. Complex numbers only arise from operations that would be
+errors in a calculator that didn't have complex numbers. (For example,
+taking the square root or logarithm of a negative number produces a
+complex result.)
+
+Complex numbers are entered in the notation shown. The @kbd{(} and
address@hidden,} and @kbd{)} keys manipulate ``incomplete complex numbers.''
+
address@hidden
address@hidden
+1: ( ... 2: ( ... 1: (2, ... 1: (2, ... 1: (2, 3)
+ . 1: 2 . 3 .
+ . .
+
+ ( 2 , 3 )
address@hidden group
address@hidden smallexample
+
+You can perform calculations while entering parts of incomplete objects.
+However, an incomplete object cannot actually participate in a calculation:
+
address@hidden
address@hidden
+1: ( ... 2: ( ... 3: ( ... 1: ( ... 1: ( ...
+ . 1: 2 2: 2 5 5
+ . 1: 3 . .
+ .
+ (error)
+ ( 2 @key{RET} 3 + +
address@hidden group
address@hidden smallexample
+
address@hidden
+Adding 5 to an incomplete object makes no sense, so the last command
+produces an error message and leaves the stack the same.
+
+Incomplete objects can't participate in arithmetic, but they can be
+moved around by the regular stack commands.
+
address@hidden
address@hidden
+2: 2 3: 2 3: 3 1: ( ... 1: (2, 3)
+1: 3 2: 3 2: ( ... 2 .
+ . 1: ( ... 1: 2 3
+ . . .
+
+2 @key{RET} 3 @key{RET} ( address@hidden
address@hidden )
address@hidden group
address@hidden smallexample
+
address@hidden
+Note that the @kbd{,} (comma) key did not have to be used here.
+When you press @kbd{)} all the stack entries between the incomplete
+entry and the top are collected, so there's never really a reason
+to use the comma. It's up to you.
+
+(@bullet{}) @strong{Exercise 4.} To enter the complex number @expr{(2, 3)},
+your friend Joe typed @kbd{( 2 , @key{SPC} 3 )}. What happened?
+(Joe thought of a clever way to correct his mistake in only two
+keystrokes, but it didn't quite work. Try it to find out why.)
address@hidden Answer 4, 4}. (@bullet{})
+
+Vectors are entered the same way as complex numbers, but with square
+brackets in place of parentheses. We'll meet vectors again later in
+the tutorial.
+
+Any Emacs command can be given a @dfn{numeric prefix argument} by
+typing a series of @key{META}-digits beforehand. If @key{META} is
+awkward for you, you can instead type @kbd{C-u} followed by the
+necessary digits. Numeric prefix arguments can be negative, as in
address@hidden M-3 M-5} or @address@hidden - 3 5}}. Calc commands use numeric
+prefix arguments in a variety of ways. For example, a numeric prefix
+on the @kbd{+} operator adds any number of stack entries at once:
+
address@hidden
address@hidden
+1: 10 2: 10 3: 10 3: 10 1: 60
+ . 1: 20 2: 20 2: 20 .
+ . 1: 30 1: 30
+ . .
+
+ 10 @key{RET} 20 @key{RET} 30 @key{RET} C-u 3
+
address@hidden group
address@hidden smallexample
+
+For stack manipulation commands like @key{RET}, a positive numeric
+prefix argument operates on the top @var{n} stack entries at once. A
+negative argument operates on the entry in level @var{n} only. An
+argument of zero operates on the entire stack. In this example, we copy
+the second-to-top element of the stack:
+
address@hidden
address@hidden
+1: 10 2: 10 3: 10 3: 10 4: 10
+ . 1: 20 2: 20 2: 20 3: 20
+ . 1: 30 1: 30 2: 30
+ . . 1: 20
+ .
+
+ 10 @key{RET} 20 @key{RET} 30 @key{RET} C-u -2
@key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden Clearing the stack
address@hidden Emptying the stack
+Another common idiom is @kbd{M-0 @key{DEL}}, which clears the stack.
+(The @kbd{M-0} numeric prefix tells @key{DEL} to operate on the
+entire stack.)
+
address@hidden Algebraic Tutorial, Undo Tutorial, RPN Tutorial, Basic Tutorial
address@hidden Algebraic-Style Calculations
+
address@hidden
+If you are not used to RPN notation, you may prefer to operate the
+Calculator in Algebraic mode, which is closer to the way
+non-RPN calculators work. In Algebraic mode, you enter formulas
+in traditional @expr{2+3} notation.
+
address@hidden:} Note that @samp{/} has lower precedence than
address@hidden, so that @samp{a/b*c} is interpreted as @samp{a/(b*c)}. See
+below for details.
+
+You don't really need any special ``mode'' to enter algebraic formulas.
+You can enter a formula at any time by pressing the apostrophe (@kbd{'})
+key. Answer the prompt with the desired formula, then press @key{RET}.
+The formula is evaluated and the result is pushed onto the RPN stack.
+If you don't want to think in RPN at all, you can enter your whole
+computation as a formula, read the result from the stack, then press
address@hidden to delete it from the stack.
+
+Try pressing the apostrophe key, then @kbd{2+3+4}, then @key{RET}.
+The result should be the number 9.
+
+Algebraic formulas use the operators @samp{+}, @samp{-}, @samp{*},
address@hidden/}, and @samp{^}. You can use parentheses to make the order
+of evaluation clear. In the absence of parentheses, @samp{^} is
+evaluated first, then @samp{*}, then @samp{/}, then finally
address@hidden and @samp{-}. For example, the expression
+
address@hidden
+2 + 3*4*5 / 6*7^8 - 9
address@hidden example
+
address@hidden
+is equivalent to
+
address@hidden
+2 + ((3*4*5) / (6*(7^8)) - 9
address@hidden example
+
address@hidden
+or, in large mathematical notation,
+
address@hidden
address@hidden
address@hidden
+ 3 * 4 * 5
+2 + --------- - 9
+ 8
+ 6 * 7
address@hidden group
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ 2 + { 3 \times 4 \times 5 \over 6 \times 7^8 } - 9 $$
+\afterdisplay
address@hidden tex
+
address@hidden
+The result of this expression will be the number @mathit{-6.99999826533}.
+
+Calc's order of evaluation is the same as for most computer languages,
+except that @samp{*} binds more strongly than @samp{/}, as the above
+example shows. As in normal mathematical notation, the @samp{*} symbol
+can often be omitted: @samp{2 a} is the same as @samp{2*a}.
+
+Operators at the same level are evaluated from left to right, except
+that @samp{^} is evaluated from right to left. Thus, @samp{2-3-4} is
+equivalent to @samp{(2-3)-4} or @mathit{-5}, whereas @samp{2^3^4} is equivalent
+to @samp{2^(3^4)} (a very large integer; try it!).
+
+If you tire of typing the apostrophe all the time, there is
+Algebraic mode, where Calc automatically senses
+when you are about to type an algebraic expression. To enter this
+mode, press the two letters @address@hidden a}}. (An @samp{Alg} indicator
+should appear in the Calc window's mode line.)
+
+Press @kbd{m a}, then @kbd{2+3+4} with no apostrophe, then @key{RET}.
+
+In Algebraic mode, when you press any key that would normally begin
+entering a number (such as a digit, a decimal point, or the @kbd{_}
+key), or if you press @kbd{(} or @kbd{[}, Calc automatically begins
+an algebraic entry.
+
+Functions which do not have operator symbols like @samp{+} and @samp{*}
+must be entered in formulas using function-call notation. For example,
+the function name corresponding to the square-root key @kbd{Q} is
address@hidden To compute a square root in a formula, you would use
+the notation @samp{sqrt(@var{x})}.
+
+Press the apostrophe, then type @kbd{sqrt(5*2) - 3}. The result should
+be @expr{0.16227766017}.
+
+Note that if the formula begins with a function name, you need to use
+the apostrophe even if you are in Algebraic mode. If you type @kbd{arcsin}
+out of the blue, the @kbd{a r} will be taken as an Algebraic Rewrite
+command, and the @kbd{csin} will be taken as the name of the rewrite
+rule to use!
+
+Some people prefer to enter complex numbers and vectors in algebraic
+form because they find RPN entry with incomplete objects to be too
+distracting, even though they otherwise use Calc as an RPN calculator.
+
+Still in Algebraic mode, type:
+
address@hidden
address@hidden
+1: (2, 3) 2: (2, 3) 1: (8, -1) 2: (8, -1) 1: (9, -1)
+ . 1: (1, -2) . 1: 1 .
+ . .
+
+ (2,3) @key{RET} (1,-2) @key{RET} * 1 @key{RET}
+
address@hidden group
address@hidden smallexample
+
+Algebraic mode allows us to enter complex numbers without pressing
+an apostrophe first, but it also means we need to press @key{RET}
+after every entry, even for a simple number like @expr{1}.
+
+(You can type @kbd{C-u m a} to enable a special Incomplete Algebraic
+mode in which the @kbd{(} and @kbd{[} keys use algebraic entry even
+though regular numeric keys still use RPN numeric entry. There is also
+Total Algebraic mode, started by typing @kbd{m t}, in which all
+normal keys begin algebraic entry. You must then use the @key{META} key
+to type Calc commands: @kbd{M-m t} to get back out of Total Algebraic
+mode, @kbd{M-q} to quit, etc.)
+
+If you're still in Algebraic mode, press @kbd{m a} again to turn it off.
+
+Actual non-RPN calculators use a mixture of algebraic and RPN styles.
+In general, operators of two numbers (like @kbd{+} and @kbd{*})
+use algebraic form, but operators of one number (like @kbd{n} and @kbd{Q})
+use RPN form. Also, a non-RPN calculator allows you to see the
+intermediate results of a calculation as you go along. You can
+accomplish this in Calc by performing your calculation as a series
+of algebraic entries, using the @kbd{$} sign to tie them together.
+In an algebraic formula, @kbd{$} represents the number on the top
+of the stack. Here, we perform the calculation
address@hidden @math{\sqrt{2\times4+1}},
address@hidden @expr{sqrt(2*4+1)},
+which on a traditional calculator would be done by pressing
address@hidden * 4 + 1 =} and then the square-root key.
+
address@hidden
address@hidden
+1: 8 1: 9 1: 3
+ . . .
+
+ ' 2*4 @key{RET} $+1 @key{RET} Q
address@hidden group
address@hidden smallexample
+
address@hidden
+Notice that we didn't need to press an apostrophe for the @kbd{$+1},
+because the dollar sign always begins an algebraic entry.
+
+(@bullet{}) @strong{Exercise 1.} How could you get the same effect as
+pressing @kbd{Q} but using an algebraic entry instead? How about
+if the @kbd{Q} key on your keyboard were broken?
address@hidden Answer 1, 1}. (@bullet{})
+
+The notations @kbd{$$}, @kbd{$$$}, and so on stand for higher stack
+entries. For example, @kbd{' $$+$ @key{RET}} is just like typing @kbd{+}.
+
+Algebraic formulas can include @dfn{variables}. To store in a
+variable, press @kbd{s s}, then type the variable name, then press
address@hidden (There are actually two flavors of store command:
address@hidden s} stores a number in a variable but also leaves the number
+on the stack, while @address@hidden t}} removes a number from the stack and
+stores it in the variable.) A variable name should consist of one
+or more letters or digits, beginning with a letter.
+
address@hidden
address@hidden
+1: 17 . 1: a + a^2 1: 306
+ . . .
+
+ 17 s t a @key{RET} ' a+a^2 @key{RET} =
address@hidden group
address@hidden smallexample
+
address@hidden
+The @kbd{=} key @dfn{evaluates} a formula by replacing all its
+variables by the values that were stored in them.
+
+For RPN calculations, you can recall a variable's value on the
+stack either by entering its name as a formula and pressing @kbd{=},
+or by using the @kbd{s r} command.
+
address@hidden
address@hidden
+1: 17 2: 17 3: 17 2: 17 1: 306
+ . 1: 17 2: 17 1: 289 .
+ . 1: 2 .
+ .
+
+ s r a @key{RET} ' a @key{RET} = 2 ^ +
address@hidden group
address@hidden smallexample
+
+If you press a single digit for a variable name (as in @kbd{s t 3}, you
+get one of ten @dfn{quick variables} @code{q0} through @code{q9}.
+They are ``quick'' simply because you don't have to type the letter
address@hidden or the @key{RET} after their names. In fact, you can type
+simply @kbd{s 3} as a shorthand for @kbd{s s 3}, and likewise for
address@hidden 3} and @address@hidden 3}}.
+
+Any variables in an algebraic formula for which you have not stored
+values are left alone, even when you evaluate the formula.
+
address@hidden
address@hidden
+1: 2 a + 2 b 1: 34 + 2 b
+ . .
+
+ ' 2a+2b @key{RET} =
address@hidden group
address@hidden smallexample
+
+Calls to function names which are undefined in Calc are also left
+alone, as are calls for which the value is undefined.
+
address@hidden
address@hidden
+1: 2 + log10(0) + log10(x) + log10(5, 6) + foo(3)
+ .
+
+ ' log10(100) + log10(0) + log10(x) + log10(5,6) + foo(3) @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+In this example, the first call to @code{log10} works, but the other
+calls are not evaluated. In the second call, the logarithm is
+undefined for that value of the argument; in the third, the argument
+is symbolic, and in the fourth, there are too many arguments. In the
+fifth case, there is no function called @code{foo}. You will see a
+``Wrong number of arguments'' message referring to @samp{log10(5,6)}.
+Press the @kbd{w} (``why'') key to see any other messages that may
+have arisen from the last calculation. In this case you will get
+``logarithm of zero,'' then ``number expected: @code{x}''. Calc
+automatically displays the first message only if the message is
+sufficiently important; for example, Calc considers ``wrong number
+of arguments'' and ``logarithm of zero'' to be important enough to
+report automatically, while a message like ``number expected: @code{x}''
+will only show up if you explicitly press the @kbd{w} key.
+
+(@bullet{}) @strong{Exercise 2.} Joe entered the formula @samp{2 x y},
+stored 5 in @code{x}, pressed @kbd{=}, and got the expected result,
address@hidden y}. He then tried the same for the formula @samp{2 x (1+y)},
+expecting @samp{10 (1+y)}, but it didn't work. Why not?
address@hidden Answer 2, 2}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 3.} What result would you expect
address@hidden @key{RET} 0 /} to give? What if you then type @kbd{0 *}?
address@hidden Answer 3, 3}. (@bullet{})
+
+One interesting way to work with variables is to use the
address@hidden (@samp{=>}) operator. It works like this:
+Enter a formula algebraically in the usual way, but follow
+the formula with an @samp{=>} symbol. (There is also an @kbd{s =}
+command which builds an @samp{=>} formula using the stack.) On
+the stack, you will see two copies of the formula with an @samp{=>}
+between them. The lefthand formula is exactly like you typed it;
+the righthand formula has been evaluated as if by typing @kbd{=}.
+
address@hidden
address@hidden
+2: 2 + 3 => 5 2: 2 + 3 => 5
+1: 2 a + 2 b => 34 + 2 b 1: 2 a + 2 b => 20 + 2 b
+ . .
+
+' 2+3 => @key{RET} ' 2a+2b @key{RET} s = 10 s t a @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+Notice that the instant we stored a new value in @code{a}, all
address@hidden>} operators already on the stack that referred to @expr{a}
+were updated to use the new value. With @samp{=>}, you can push a
+set of formulas on the stack, then change the variables experimentally
+to see the effects on the formulas' values.
+
+You can also ``unstore'' a variable when you are through with it:
+
address@hidden
address@hidden
+2: 2 + 5 => 5
+1: 2 a + 2 b => 2 a + 2 b
+ .
+
+ s u a @key{RET}
address@hidden group
address@hidden smallexample
+
+We will encounter formulas involving variables and functions again
+when we discuss the algebra and calculus features of the Calculator.
+
address@hidden Undo Tutorial, Modes Tutorial, Algebraic Tutorial, Basic Tutorial
address@hidden Undo and Redo
+
address@hidden
+If you make a mistake, you can usually correct it by pressing address@hidden,
+the ``undo'' command. First, clear the stack (@kbd{M-0 @key{DEL}}) and exit
+and restart Calc (@kbd{C-x * * C-x * *}) to make sure things start off
+with a clean slate. Now:
+
address@hidden
address@hidden
+1: 2 2: 2 1: 8 2: 2 1: 6
+ . 1: 3 . 1: 3 .
+ . .
+
+ 2 @key{RET} 3 ^ U *
address@hidden group
address@hidden smallexample
+
+You can undo any number of times. Calc keeps a complete record of
+all you have done since you last opened the Calc window. After the
+above example, you could type:
+
address@hidden
address@hidden
+1: 6 2: 2 1: 2 . .
+ . 1: 3 .
+ .
+ (error)
+ U U U U
address@hidden group
address@hidden smallexample
+
+You can also type @kbd{D} to ``redo'' a command that you have undone
+mistakenly.
+
address@hidden
address@hidden
+ . 1: 2 2: 2 1: 6 1: 6
+ . 1: 3 . .
+ .
+ (error)
+ D D D D
address@hidden group
address@hidden smallexample
+
address@hidden
+It was not possible to redo past the @expr{6}, since that was placed there
+by something other than an undo command.
+
address@hidden Time travel
+You can think of undo and redo as a sort of ``time machine.'' Press
address@hidden to go backward in time, @kbd{D} to go forward. If you go
+backward and do something (like @kbd{*}) then, as any science fiction
+reader knows, you have changed your future and you cannot go forward
+again. Thus, the inability to redo past the @expr{6} even though there
+was an earlier undo command.
+
+You can always recall an earlier result using the Trail. We've ignored
+the trail so far, but it has been faithfully recording everything we
+did since we loaded the Calculator. If the Trail is not displayed,
+press @kbd{t d} now to turn it on.
+
+Let's try grabbing an earlier result. The @expr{8} we computed was
+undone by a @kbd{U} command, and was lost even to Redo when we pressed
address@hidden, but it's still there in the trail. There should be a little
address@hidden>} arrow (the @dfn{trail pointer}) resting on the last trail
+entry. If there isn't, press @kbd{t ]} to reset the trail pointer.
+Now, press @address@hidden p}} to move the arrow onto the line containing
address@hidden, and press @address@hidden y}} to ``yank'' that number back onto
the
+stack.
+
+If you press @kbd{t ]} again, you will see that even our Yank command
+went into the trail.
+
+Let's go further back in time. Earlier in the tutorial we computed
+a huge integer using the formula @samp{2^3^4}. We don't remember
+what it was, but the first digits were ``241''. Press @kbd{t r}
+(which stands for trail-search-reverse), then type @kbd{241}.
+The trail cursor will jump back to the next previous occurrence of
+the string ``241'' in the trail. This is just a regular Emacs
+incremental search; you can now press @kbd{C-s} or @kbd{C-r} to
+continue the search forwards or backwards as you like.
+
+To finish the search, press @key{RET}. This halts the incremental
+search and leaves the trail pointer at the thing we found. Now we
+can type @kbd{t y} to yank that number onto the stack. If we hadn't
+remembered the ``241'', we could simply have searched for @kbd{2^3^4},
+then pressed @address@hidden t n} to halt and then move to the next item.
+
+You may have noticed that all the trail-related commands begin with
+the letter @kbd{t}. (The store-and-recall commands, on the other hand,
+all began with @kbd{s}.) Calc has so many commands that there aren't
+enough keys for all of them, so various commands are grouped into
+two-letter sequences where the first letter is called the @dfn{prefix}
+key. If you type a prefix key by accident, you can press @kbd{C-g}
+to cancel it. (In fact, you can press @kbd{C-g} to cancel almost
+anything in Emacs.) To get help on a prefix key, press that key
+followed by @kbd{?}. Some prefixes have several lines of help,
+so you need to press @kbd{?} repeatedly to see them all.
+You can also type @kbd{h h} to see all the help at once.
+
+Try pressing @kbd{t ?} now. You will see a line of the form,
+
address@hidden
+trail/time: Display; Fwd, Back; Next, Prev, Here, [, ]; Yank: [MORE] t-
address@hidden smallexample
+
address@hidden
+The word ``trail'' indicates that the @kbd{t} prefix key contains
+trail-related commands. Each entry on the line shows one command,
+with a single capital letter showing which letter you press to get
+that command. We have used @kbd{t n}, @kbd{t p}, @kbd{t ]}, and
address@hidden y} so far. The @samp{[MORE]} means you can press @kbd{?}
+again to see more @kbd{t}-prefix commands. Notice that the commands
+are roughly divided (by semicolons) into related groups.
+
+When you are in the help display for a prefix key, the prefix is
+still active. If you press another key, like @kbd{y} for example,
+it will be interpreted as a @kbd{t y} command. If all you wanted
+was to look at the help messages, press @kbd{C-g} afterwards to cancel
+the prefix.
+
+One more way to correct an error is by editing the stack entries.
+The actual Stack buffer is marked read-only and must not be edited
+directly, but you can press @kbd{`} (the backquote or accent grave)
+to edit a stack entry.
+
+Try entering @samp{3.141439} now. If this is supposed to represent
address@hidden, it's got several errors. Press @kbd{`} to edit this number.
+Now use the normal Emacs cursor motion and editing keys to change
+the second 4 to a 5, and to transpose the 3 and the 9. When you
+press @key{RET}, the number on the stack will be replaced by your
+new number. This works for formulas, vectors, and all other types
+of values you can put on the stack. The @kbd{`} key also works
+during entry of a number or algebraic formula.
+
address@hidden Modes Tutorial, , Undo Tutorial, Basic Tutorial
address@hidden Mode-Setting Commands
+
address@hidden
+Calc has many types of @dfn{modes} that affect the way it interprets
+your commands or the way it displays data. We have already seen one
+mode, namely Algebraic mode. There are many others, too; we'll
+try some of the most common ones here.
+
+Perhaps the most fundamental mode in Calc is the current @dfn{precision}.
+Notice the @samp{12} on the Calc window's mode line:
+
address@hidden
+--%%-Calc: 12 Deg (Calculator)----All------
address@hidden smallexample
+
address@hidden
+Most of the symbols there are Emacs things you don't need to worry
+about, but the @samp{12} and the @samp{Deg} are mode indicators.
+The @samp{12} means that calculations should always be carried to
+12 significant figures. That is why, when we type @kbd{1 @key{RET} 7 /},
+we get @expr{0.142857142857} with exactly 12 digits, not counting
+leading and trailing zeros.
+
+You can set the precision to anything you like by pressing @kbd{p},
+then entering a suitable number. Try pressing @kbd{p 30 @key{RET}},
+then doing @kbd{1 @key{RET} 7 /} again:
+
address@hidden
address@hidden
+1: 0.142857142857
+2: 0.142857142857142857142857142857
+ .
address@hidden group
address@hidden smallexample
+
+Although the precision can be set arbitrarily high, Calc always
+has to have @emph{some} value for the current precision. After
+all, the true value @expr{1/7} is an infinitely repeating decimal;
+Calc has to stop somewhere.
+
+Of course, calculations are slower the more digits you request.
+Press @address@hidden 12}} now to set the precision back down to the default.
+
+Calculations always use the current precision. For example, even
+though we have a 30-digit value for @expr{1/7} on the stack, if
+we use it in a calculation in 12-digit mode it will be rounded
+down to 12 digits before it is used. Try it; press @key{RET} to
+duplicate the number, then @address@hidden +}}. Notice that the @key{RET}
+key didn't round the number, because it doesn't do any calculation.
+But the instant we pressed @kbd{+}, the number was rounded down.
+
address@hidden
address@hidden
+1: 0.142857142857
+2: 0.142857142857142857142857142857
+3: 1.14285714286
+ .
address@hidden group
address@hidden smallexample
+
address@hidden
+In fact, since we added a digit on the left, we had to lose one
+digit on the right from even the 12-digit value of @expr{1/7}.
+
+How did we get more than 12 digits when we computed @samp{2^3^4}? The
+answer is that Calc makes a distinction between @dfn{integers} and
address@hidden numbers, or @dfn{floats}. An integer is a number
+that does not contain a decimal point. There is no such thing as an
+``infinitely repeating fraction integer,'' so Calc doesn't have to limit
+itself. If you asked for @samp{2^10000} (don't try this!), you would
+have to wait a long time but you would eventually get an exact answer.
+If you ask for @samp{2.^10000}, you will quickly get an answer which is
+correct only to 12 places. The decimal point tells Calc that it should
+use floating-point arithmetic to get the answer, not exact integer
+arithmetic.
+
+You can use the @kbd{F} (@code{calc-floor}) command to convert a
+floating-point value to an integer, and @kbd{c f} (@code{calc-float})
+to convert an integer to floating-point form.
+
+Let's try entering that last calculation:
+
address@hidden
address@hidden
+1: 2. 2: 2. 1: 1.99506311689e3010
+ . 1: 10000 .
+ .
+
+ 2.0 @key{RET} 10000 @key{RET} ^
address@hidden group
address@hidden smallexample
+
address@hidden
address@hidden Scientific notation, entry of
+Notice the letter @samp{e} in there. It represents ``times ten to the
+power of,'' and is used by Calc automatically whenever writing the
+number out fully would introduce more extra zeros than you probably
+want to see. You can enter numbers in this notation, too.
+
address@hidden
address@hidden
+1: 2. 2: 2. 1: 1.99506311678e3010
+ . 1: 10000. .
+ .
+
+ 2.0 @key{RET} 1e4 @key{RET} ^
address@hidden group
address@hidden smallexample
+
address@hidden Round-off errors
address@hidden
+Hey, the answer is different! Look closely at the middle columns
+of the two examples. In the first, the stack contained the
+exact integer @expr{10000}, but in the second it contained
+a floating-point value with a decimal point. When you raise a
+number to an integer power, Calc uses repeated squaring and
+multiplication to get the answer. When you use a floating-point
+power, Calc uses logarithms and exponentials. As you can see,
+a slight error crept in during one of these methods. Which
+one should we trust? Let's raise the precision a bit and find
+out:
+
address@hidden
address@hidden
+ . 1: 2. 2: 2. 1: 1.995063116880828e3010
+ . 1: 10000. .
+ .
+
+ p 16 @key{RET} 2. @key{RET} 1e4 ^ p 12
@key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
address@hidden Guard digits
+Presumably, it doesn't matter whether we do this higher-precision
+calculation using an integer or floating-point power, since we
+have added enough ``guard digits'' to trust the first 12 digits
+no matter what. And the verdict address@hidden Integer powers were more
+accurate; in fact, the result was only off by one unit in the
+last place.
+
address@hidden Guard digits
+Calc does many of its internal calculations to a slightly higher
+precision, but it doesn't always bump the precision up enough.
+In each case, Calc added about two digits of precision during
+its calculation and then rounded back down to 12 digits
+afterward. In one case, it was enough; in the other, it
+wasn't. If you really need @var{x} digits of precision, it
+never hurts to do the calculation with a few extra guard digits.
+
+What if we want guard digits but don't want to look at them?
+We can set the @dfn{float format}. Calc supports four major
+formats for floating-point numbers, called @dfn{normal},
address@hidden, @dfn{scientific notation}, and @dfn{engineering
+notation}. You get them by pressing @address@hidden n}}, @kbd{d f},
address@hidden s}, and @kbd{d e}, respectively. In each case, you can
+supply a numeric prefix argument which says how many digits
+should be displayed. As an example, let's put a few numbers
+onto the stack and try some different display modes. First,
+use @kbd{M-0 @key{DEL}} to clear the stack, then enter the four
+numbers shown here:
+
address@hidden
address@hidden
+4: 12345 4: 12345 4: 12345 4: 12345 4: 12345
+3: 12345. 3: 12300. 3: 1.2345e4 3: 1.23e4 3: 12345.000
+2: 123.45 2: 123. 2: 1.2345e2 2: 1.23e2 2: 123.450
+1: 12.345 1: 12.3 1: 1.2345e1 1: 1.23e1 1: 12.345
+ . . . . .
+
+ d n M-3 d n d s M-3 d s M-3 d f
address@hidden group
address@hidden smallexample
+
address@hidden
+Notice that when we typed @kbd{M-3 d n}, the numbers were rounded down
+to three significant digits, but then when we typed @kbd{d s} all
+five significant figures reappeared. The float format does not
+affect how numbers are stored, it only affects how they are
+displayed. Only the current precision governs the actual rounding
+of numbers in the Calculator's memory.
+
+Engineering notation, not shown here, is like scientific notation
+except the exponent (the power-of-ten part) is always adjusted to be
+a multiple of three (as in ``kilo,'' ``micro,'' etc.). As a result
+there will be one, two, or three digits before the decimal point.
+
+Whenever you change a display-related mode, Calc redraws everything
+in the stack. This may be slow if there are many things on the stack,
+so Calc allows you to type address@hidden before any mode command to
+prevent it from updating the stack. Anything Calc displays after the
+mode-changing command will appear in the new format.
+
address@hidden
address@hidden
+4: 12345 4: 12345 4: 12345 4: 12345 4: 12345
+3: 12345.000 3: 12345.000 3: 12345.000 3: 1.2345e4 3: 12345.
+2: 123.450 2: 123.450 2: 1.2345e1 2: 1.2345e1 2: 123.45
+1: 12.345 1: 1.2345e1 1: 1.2345e2 1: 1.2345e2 1: 12.345
+ . . . . .
+
+ H d s @key{DEL} U @key{TAB} d @key{SPC}
d n
address@hidden group
address@hidden smallexample
+
address@hidden
+Here the @kbd{H d s} command changes to scientific notation but without
+updating the screen. Deleting the top stack entry and undoing it back
+causes it to show up in the new format; swapping the top two stack
+entries reformats both entries. The @kbd{d @key{SPC}} command refreshes the
+whole stack. The @kbd{d n} command changes back to the normal float
+format; since it doesn't have an @kbd{H} prefix, it also updates all
+the stack entries to be in @kbd{d n} format.
+
+Notice that the integer @expr{12345} was not affected by any
+of the float formats. Integers are integers, and are always
+displayed exactly.
+
address@hidden Large numbers, readability
+Large integers have their own problems. Let's look back at
+the result of @kbd{2^3^4}.
+
address@hidden
+2417851639229258349412352
address@hidden example
+
address@hidden
+Quick---how many digits does this have? Try typing @kbd{d g}:
+
address@hidden
+2,417,851,639,229,258,349,412,352
address@hidden example
+
address@hidden
+Now how many digits does this have? It's much easier to tell!
+We can actually group digits into clumps of any size. Some
+people prefer @kbd{M-5 d g}:
+
address@hidden
+24178,51639,22925,83494,12352
address@hidden example
+
+Let's see what happens to floating-point numbers when they are grouped.
+First, type @kbd{p 25 @key{RET}} to make sure we have enough precision
+to get ourselves into trouble. Now, type @kbd{1e13 /}:
+
address@hidden
+24,17851,63922.9258349412352
address@hidden example
+
address@hidden
+The integer part is grouped but the fractional part isn't. Now try
address@hidden M-5 d g} (that's meta-minus-sign, meta-five):
+
address@hidden
+24,17851,63922.92583,49412,352
address@hidden example
+
+If you find it hard to tell the decimal point from the commas, try
+changing the grouping character to a space with @kbd{d , @key{SPC}}:
+
address@hidden
+24 17851 63922.92583 49412 352
address@hidden example
+
+Type @kbd{d , ,} to restore the normal grouping character, then
address@hidden g} again to turn grouping off. Also, press @kbd{p 12} to
+restore the default precision.
+
+Press @kbd{U} enough times to get the original big integer back.
+(Notice that @kbd{U} does not undo each mode-setting command; if
+you want to undo a mode-setting command, you have to do it yourself.)
+Now, type @kbd{d r 16 @key{RET}}:
+
address@hidden
+16#200000000000000000000
address@hidden example
+
address@hidden
+The number is now displayed in @dfn{hexadecimal}, or ``base-16'' form.
+Suddenly it looks pretty simple; this should be no surprise, since we
+got this number by computing a power of two, and 16 is a power of 2.
+In fact, we can use @address@hidden r 2 @key{RET}}} to see it in actual binary
+form:
+
address@hidden
+2#1000000000000000000000000000000000000000000000000000000 @dots{}
address@hidden example
+
address@hidden
+We don't have enough space here to show all the zeros! They won't
+fit on a typical screen, either, so you will have to use horizontal
+scrolling to see them all. Press @kbd{<} and @kbd{>} to scroll the
+stack window left and right by half its width. Another way to view
+something large is to press @kbd{`} (back-quote) to edit the top of
+stack in a separate window. (Press @kbd{C-c C-c} when you are done.)
+
+You can enter non-decimal numbers using the @kbd{#} symbol, too.
+Let's see what the hexadecimal number @samp{5FE} looks like in
+binary. Type @kbd{16#5FE} (the letters can be typed in upper or
+lower case; they will always appear in upper case). It will also
+help to turn grouping on with @kbd{d g}:
+
address@hidden
+2#101,1111,1110
address@hidden example
+
+Notice that @kbd{d g} groups by fours by default if the display radix
+is binary or hexadecimal, but by threes if it is decimal, octal, or any
+other radix.
+
+Now let's see that number in decimal; type @kbd{d r 10}:
+
address@hidden
+1,534
address@hidden example
+
+Numbers are not @emph{stored} with any particular radix attached. They're
+just numbers; they can be entered in any radix, and are always displayed
+in whatever radix you've chosen with @kbd{d r}. The current radix applies
+to integers, fractions, and floats.
+
address@hidden Roundoff errors, in non-decimal numbers
+(@bullet{}) @strong{Exercise 1.} Your friend Joe tried to enter one-third
+as @samp{3#0.1} in @kbd{d r 3} mode with a precision of 12. He got
address@hidden (with 25 2's) in the display. When he multiplied
+that by three, he got @samp{3#0.222222...} instead of the expected
address@hidden Next, Joe entered @samp{3#0.2} and, to his great relief,
+saw @samp{3#0.2} on the screen. But when he typed @kbd{2 /}, he got
address@hidden (some zeros omitted). What's going on here?
address@hidden Answer 1, 1}. (@bullet{})
+
address@hidden Scientific notation, in non-decimal numbers
+(@bullet{}) @strong{Exercise 2.} Scientific notation works in non-decimal
+modes in the natural way (the exponent is a power of the radix instead of
+a power of ten, although the exponent itself is always written in decimal).
+Thus @samp{8#1.23e3 = 8#1230.0}. Suppose we have the hexadecimal number
address@hidden times 16 to the 15th power: We write @samp{16#f.e8fe15}.
+What is wrong with this picture? What could we write instead that would
+work better? @xref{Modes Answer 2, 2}. (@bullet{})
+
+The @kbd{m} prefix key has another set of modes, relating to the way
+Calc interprets your inputs and does computations. Whereas @kbd{d}-prefix
+modes generally affect the way things look, @kbd{m}-prefix modes affect
+the way they are actually computed.
+
+The most popular @kbd{m}-prefix mode is the @dfn{angular mode}. Notice
+the @samp{Deg} indicator in the mode line. This means that if you use
+a command that interprets a number as an angle, it will assume the
+angle is measured in degrees. For example,
+
address@hidden
address@hidden
+1: 45 1: 0.707106781187 1: 0.500000000001 1: 0.5
+ . . . .
+
+ 45 S 2 ^ c 1
address@hidden group
address@hidden smallexample
+
address@hidden
+The address@hidden command computes the sine of an angle. The sine
+of 45 degrees is
address@hidden @math{\sqrt{2}/2};
address@hidden @expr{sqrt(2)/2};
+squaring this yields @expr{2/4 = 0.5}. However, there has been a slight
+roundoff error because the representation of
address@hidden @math{\sqrt{2}/2}
address@hidden @expr{sqrt(2)/2}
+wasn't exact. The @kbd{c 1} command is a handy way to clean up numbers
+in this case; it temporarily reduces the precision by one digit while it
+re-rounds the number on the top of the stack.
+
address@hidden Roundoff errors, examples
+(@bullet{}) @strong{Exercise 3.} Your friend Joe computed the sine
+of 45 degrees as shown above, then, hoping to avoid an inexact
+result, he increased the precision to 16 digits before squaring.
+What happened? @xref{Modes Answer 3, 3}. (@bullet{})
+
+To do this calculation in radians, we would type @kbd{m r} first.
+(The indicator changes to @samp{Rad}.) 45 degrees corresponds to
address@hidden radians. To get @cpi{}, press the @kbd{P} key. (Once
+again, this is a shifted capital @kbd{P}. Remember, unshifted
address@hidden sets the precision.)
+
address@hidden
address@hidden
+1: 3.14159265359 1: 0.785398163398 1: 0.707106781187
+ . . .
+
+ P 4 / m r S
address@hidden group
address@hidden smallexample
+
+Likewise, inverse trigonometric functions generate results in
+either radians or degrees, depending on the current angular mode.
+
address@hidden
address@hidden
+1: 0.707106781187 1: 0.785398163398 1: 45.
+ . . .
+
+ .5 Q m r I S m d U I S
address@hidden group
address@hidden smallexample
+
address@hidden
+Here we compute the Inverse Sine of
address@hidden @math{\sqrt{0.5}},
address@hidden @expr{sqrt(0.5)},
+first in radians, then in degrees.
+
+Use @kbd{c d} and @kbd{c r} to convert a number from radians to degrees
+and vice-versa.
+
address@hidden
address@hidden
+1: 45 1: 0.785398163397 1: 45.
+ . . .
+
+ 45 c r c d
address@hidden group
address@hidden smallexample
+
+Another interesting mode is @dfn{Fraction mode}. Normally,
+dividing two integers produces a floating-point result if the
+quotient can't be expressed as an exact integer. Fraction mode
+causes integer division to produce a fraction, i.e., a rational
+number, instead.
+
address@hidden
address@hidden
+2: 12 1: 1.33333333333 1: 4:3
+1: 9 . .
+ .
+
+ 12 @key{RET} 9 / m f U / m f
address@hidden group
address@hidden smallexample
+
address@hidden
+In the first case, we get an approximate floating-point result.
+In the second case, we get an exact fractional result (four-thirds).
+
+You can enter a fraction at any time using @kbd{:} notation.
+(Calc uses @kbd{:} instead of @kbd{/} as the fraction separator
+because @kbd{/} is already used to divide the top two stack
+elements.) Calculations involving fractions will always
+produce exact fractional results; Fraction mode only says
+what to do when dividing two integers.
+
address@hidden Fractions vs. floats
address@hidden Floats vs. fractions
+(@bullet{}) @strong{Exercise 4.} If fractional arithmetic is exact,
+why would you ever use floating-point numbers instead?
address@hidden Answer 4, 4}. (@bullet{})
+
+Typing @kbd{m f} doesn't change any existing values in the stack.
+In the above example, we had to Undo the division and do it over
+again when we changed to Fraction mode. But if you use the
+evaluates-to operator you can get commands like @kbd{m f} to
+recompute for you.
+
address@hidden
address@hidden
+1: 12 / 9 => 1.33333333333 1: 12 / 9 => 1.333 1: 12 / 9 => 4:3
+ . . .
+
+ ' 12/9 => @key{RET} p 4 @key{RET} m f
address@hidden group
address@hidden smallexample
+
address@hidden
+In this example, the righthand side of the @samp{=>} operator
+on the stack is recomputed when we change the precision, then
+again when we change to Fraction mode. All @samp{=>} expressions
+on the stack are recomputed every time you change any mode that
+might affect their values.
+
address@hidden Arithmetic Tutorial, Vector/Matrix Tutorial, Basic Tutorial,
Tutorial
address@hidden Arithmetic Tutorial
+
address@hidden
+In this section, we explore the arithmetic and scientific functions
+available in the Calculator.
+
+The standard arithmetic commands are @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/},
+and @kbd{^}. Each normally takes two numbers from the top of the stack
+and pushes back a result. The @kbd{n} and @kbd{&} keys perform
+change-sign and reciprocal operations, respectively.
+
address@hidden
address@hidden
+1: 5 1: 0.2 1: 5. 1: -5. 1: 5.
+ . . . . .
+
+ 5 & & n n
address@hidden group
address@hidden smallexample
+
address@hidden Binary operators
+You can apply a ``binary operator'' like @kbd{+} across any number of
+stack entries by giving it a numeric prefix. You can also apply it
+pairwise to several stack elements along with the top one if you use
+a negative prefix.
+
address@hidden
address@hidden
+3: 2 1: 9 3: 2 4: 2 3: 12
+2: 3 . 2: 3 3: 3 2: 13
+1: 4 1: 4 2: 4 1: 14
+ . . 1: 10 .
+ .
+
+2 @key{RET} 3 @key{RET} 4 M-3 + U 10 M--
M-3 +
address@hidden group
address@hidden smallexample
+
address@hidden Unary operators
+You can apply a ``unary operator'' like @kbd{&} to the top @var{n}
+stack entries with a numeric prefix, too.
+
address@hidden
address@hidden
+3: 2 3: 0.5 3: 0.5
+2: 3 2: 0.333333333333 2: 3.
+1: 4 1: 0.25 1: 4.
+ . . .
+
+2 @key{RET} 3 @key{RET} 4 M-3 & M-2 &
address@hidden group
address@hidden smallexample
+
+Notice that the results here are left in floating-point form.
+We can convert them back to integers by pressing @kbd{F}, the
+``floor'' function. This function rounds down to the next lower
+integer. There is also @kbd{R}, which rounds to the nearest
+integer.
+
address@hidden
address@hidden
+7: 2. 7: 2 7: 2
+6: 2.4 6: 2 6: 2
+5: 2.5 5: 2 5: 3
+4: 2.6 4: 2 4: 3
+3: -2. 3: -2 3: -2
+2: -2.4 2: -3 2: -2
+1: -2.6 1: -3 1: -3
+ . . .
+
+ M-7 F U M-7 R
address@hidden group
address@hidden smallexample
+
+Since dividing-and-flooring (i.e., ``integer quotient'') is such a
+common operation, Calc provides a special command for that purpose, the
+backslash @kbd{\}. Another common arithmetic operator is @kbd{%}, which
+computes the remainder that would arise from a @kbd{\} operation, i.e.,
+the ``modulo'' of two numbers. For example,
+
address@hidden
address@hidden
+2: 1234 1: 12 2: 1234 1: 34
+1: 100 . 1: 100 .
+ . .
+
+1234 @key{RET} 100 \ U %
address@hidden group
address@hidden smallexample
+
+These commands actually work for any real numbers, not just integers.
+
address@hidden
address@hidden
+2: 3.1415 1: 3 2: 3.1415 1: 0.1415
+1: 1 . 1: 1 .
+ . .
+
+3.1415 @key{RET} 1 \ U %
address@hidden group
address@hidden smallexample
+
+(@bullet{}) @strong{Exercise 1.} The @kbd{\} command would appear to be a
+frill, since you could always do the same thing with @kbd{/ F}. Think
+of a situation where this is not address@hidden/ F} would be inadequate.
+Now think of a way you could get around the problem if Calc didn't
+provide a @kbd{\} command. @xref{Arithmetic Answer 1, 1}. (@bullet{})
+
+We've already seen the @kbd{Q} (square root) and @kbd{S} (sine)
+commands. Other commands along those lines are @kbd{C} (cosine),
address@hidden (tangent), @kbd{E} (@expr{e^x}) and @kbd{L} (natural
+logarithm). These can be modified by the @kbd{I} (inverse) and
address@hidden (hyperbolic) prefix keys.
+
+Let's compute the sine and cosine of an angle, and verify the
+identity
address@hidden @math{\sin^2x + \cos^2x = 1}.
address@hidden @expr{sin(x)^2 + cos(x)^2 = 1}.
+We'll arbitrarily pick @mathit{-64} degrees as a good value for @expr{x}.
+With the angular mode set to degrees (type @address@hidden d}}), do:
+
address@hidden
address@hidden
+2: -64 2: -64 2: -0.89879 2: -0.89879 1: 1.
+1: -64 1: -0.89879 1: -64 1: 0.43837 .
+ . . . .
+
+ 64 n @key{RET} @key{RET} S @key{TAB} C
f h
address@hidden group
address@hidden smallexample
+
address@hidden
+(For brevity, we're showing only five digits of the results here.
+You can of course do these calculations to any precision you like.)
+
+Remember, @kbd{f h} is the @code{calc-hypot}, or square-root of sum
+of squares, command.
+
+Another identity is
address@hidden @math{\displaystyle\tan x = {\sin x \over \cos x}}.
address@hidden @expr{tan(x) = sin(x) / cos(x)}.
address@hidden
address@hidden
+
+2: -0.89879 1: -2.0503 1: -64.
+1: 0.43837 . .
+ .
+
+ U / I T
address@hidden group
address@hidden smallexample
+
+A physical interpretation of this calculation is that if you move
address@hidden units downward and @expr{0.43837} units to the right,
+your direction of motion is @mathit{-64} degrees from horizontal. Suppose
+we move in the opposite direction, up and to the left:
+
address@hidden
address@hidden
+2: -0.89879 2: 0.89879 1: -2.0503 1: -64.
+1: 0.43837 1: -0.43837 . .
+ . .
+
+ U U M-2 n / I T
address@hidden group
address@hidden smallexample
+
address@hidden
+How can the angle be the same? The answer is that the @kbd{/} operation
+loses information about the signs of its inputs. Because the quotient
+is negative, we know exactly one of the inputs was negative, but we
+can't tell which one. There is an @kbd{f T} address@hidden function which
+computes the inverse tangent of the quotient of a pair of numbers.
+Since you feed it the two original numbers, it has enough information
+to give you a full 360-degree answer.
+
address@hidden
address@hidden
+2: 0.89879 1: 116. 3: 116. 2: 116. 1: 180.
+1: -0.43837 . 2: -0.89879 1: -64. .
+ . 1: 0.43837 .
+ .
+
+ U U f T address@hidden M-2 n f T -
address@hidden group
address@hidden smallexample
+
address@hidden
+The resulting angles differ by 180 degrees; in other words, they
+point in opposite directions, just as we would expect.
+
+The @address@hidden we used in the third step is the
+``last-arguments'' command. It is sort of like Undo, except that it
+restores the arguments of the last command to the stack without removing
+the command's result. It is useful in situations like this one,
+where we need to do several operations on the same inputs. We could
+have accomplished the same thing by using @kbd{M-2 @key{RET}} to duplicate
+the top two stack elements right after the @kbd{U U}, then a pair of
address@hidden@key{TAB}} commands to cycle the 116 up around the duplicates.
+
+A similar identity is supposed to hold for hyperbolic sines and cosines,
+except that it is the @emph{difference}
address@hidden @math{\cosh^2x - \sinh^2x}
address@hidden @expr{cosh(x)^2 - sinh(x)^2}
+that always equals one. Let's try to verify this identity.
+
address@hidden
address@hidden
+2: -64 2: -64 2: -64 2: 9.7192e54 2: 9.7192e54
+1: -64 1: -3.1175e27 1: 9.7192e54 1: -64 1: 9.7192e54
+ . . . . .
+
+ 64 n @key{RET} @key{RET} H C 2 ^ @key{TAB}
H S 2 ^
address@hidden group
address@hidden smallexample
+
address@hidden
address@hidden Roundoff errors, examples
+Something's obviously wrong, because when we subtract these numbers
+the answer will clearly be zero! But if you think about it, if these
+numbers @emph{did} differ by one, it would be in the 55th decimal
+place. The difference we seek has been lost entirely to roundoff
+error.
+
+We could verify this hypothesis by doing the actual calculation with,
+say, 60 decimal places of precision. This will be slow, but not
+enormously so. Try it if you wish; sure enough, the answer is
+0.99999, reasonably close to 1.
+
+Of course, a more reasonable way to verify the identity is to use
+a more reasonable value for @expr{x}!
+
address@hidden Common logarithm
+Some Calculator commands use the Hyperbolic prefix for other purposes.
+The logarithm and exponential functions, for example, work to the base
address@hidden normally but use base-10 instead if you use the Hyperbolic
+prefix.
+
address@hidden
address@hidden
+1: 1000 1: 6.9077 1: 1000 1: 3
+ . . . .
+
+ 1000 L U H L
address@hidden group
address@hidden smallexample
+
address@hidden
+First, we mistakenly compute a natural logarithm. Then we undo
+and compute a common logarithm instead.
+
+The @kbd{B} key computes a general address@hidden logarithm for any
+value of @var{b}.
+
address@hidden
address@hidden
+2: 1000 1: 3 1: 1000. 2: 1000. 1: 6.9077
+1: 10 . . 1: 2.71828 .
+ . .
+
+ 1000 @key{RET} 10 B H E H P B
address@hidden group
address@hidden smallexample
+
address@hidden
+Here we first use @kbd{B} to compute the base-10 logarithm, then use
+the ``hyperbolic'' exponential as a cheap hack to recover the number
+1000, then use @kbd{B} again to compute the natural logarithm. Note
+that @kbd{P} with the hyperbolic prefix pushes the constant @expr{e}
+onto the stack.
+
+You may have noticed that both times we took the base-10 logarithm
+of 1000, we got an exact integer result. Calc always tries to give
+an exact rational result for calculations involving rational numbers
+where possible. But when we used @kbd{H E}, the result was a
+floating-point number for no apparent reason. In fact, if we had
+computed @kbd{10 @key{RET} 3 ^} we @emph{would} have gotten an
+exact integer 1000. But the @kbd{H E} command is rigged to generate
+a floating-point result all of the time so that @kbd{1000 H E} will
+not waste time computing a thousand-digit integer when all you
+probably wanted was @samp{1e1000}.
+
+(@bullet{}) @strong{Exercise 2.} Find a pair of integer inputs to
+the @kbd{B} command for which Calc could find an exact rational
+result but doesn't. @xref{Arithmetic Answer 2, 2}. (@bullet{})
+
+The Calculator also has a set of functions relating to combinatorics
+and statistics. You may be familiar with the @dfn{factorial} function,
+which computes the product of all the integers up to a given number.
+
address@hidden
address@hidden
+1: 100 1: 93326215443... 1: 100. 1: 9.3326e157
+ . . . .
+
+ 100 ! U c f !
address@hidden group
address@hidden smallexample
+
address@hidden
+Recall, the @kbd{c f} command converts the integer or fraction at the
+top of the stack to floating-point format. If you take the factorial
+of a floating-point number, you get a floating-point result
+accurate to the current precision. But if you give @kbd{!} an
+exact integer, you get an exact integer result (158 digits long
+in this case).
+
+If you take the factorial of a non-integer, Calc uses a generalized
+factorial function defined in terms of Euler's Gamma function
address@hidden @math{\Gamma(n)}
address@hidden @expr{gamma(n)}
+(which is itself available as the @kbd{f g} command).
+
address@hidden
address@hidden
+3: 4. 3: 24. 1: 5.5 1: 52.342777847
+2: 4.5 2: 52.3427777847 . .
+1: 5. 1: 120.
+ . .
+
+ M-3 ! M-0 @key{DEL} 5.5 f g
address@hidden group
address@hidden smallexample
+
address@hidden
+Here we verify the identity
address@hidden @math{n! = \Gamma(n+1)}.
address@hidden @address@hidden@: = gamma(@var{n}+1)}.
+
+The binomial coefficient @address@hidden
address@hidden or @math{\displaystyle {n \choose m}}
+is defined by
address@hidden @math{\displaystyle {n! \over m! \, (n-m)!}}
address@hidden @expr{n!@: / m!@: (n-m)!}
+for all reals @expr{n} and @expr{m}. The intermediate results in this
+formula can become quite large even if the final result is small; the
address@hidden c} command computes a binomial coefficient in a way that avoids
+large intermediate values.
+
+The @kbd{k} prefix key defines several common functions out of
+combinatorics and number theory. Here we compute the binomial
+coefficient 30-choose-20, then determine its prime factorization.
+
address@hidden
address@hidden
+2: 30 1: 30045015 1: [3, 3, 5, 7, 11, 13, 23, 29]
+1: 20 . .
+ .
+
+ 30 @key{RET} 20 k c k f
address@hidden group
address@hidden smallexample
+
address@hidden
+You can verify these prime factors by using @kbd{v u} to ``unpack''
+this vector into 8 separate stack entries, then @kbd{M-8 *} to
+multiply them back together. The result is the original number,
+30045015.
+
address@hidden Hash tables
+Suppose a program you are writing needs a hash table with at least
+10000 entries. It's best to use a prime number as the actual size
+of a hash table. Calc can compute the next prime number after 10000:
+
address@hidden
address@hidden
+1: 10000 1: 10007 1: 9973
+ . . .
+
+ 10000 k n I k n
address@hidden group
address@hidden smallexample
+
address@hidden
+Just for kicks we've also computed the next prime @emph{less} than
+10000.
+
address@hidden [fix-ref Financial Functions]
address@hidden Functions}, for a description of the Calculator
+commands that deal with business and financial calculations (functions
+like @code{pv}, @code{rate}, and @code{sln}).
+
address@hidden [fix-ref Binary Number Functions]
address@hidden Functions}, to read about the commands for operating
+on binary numbers (like @code{and}, @code{xor}, and @code{lsh}).
+
address@hidden Vector/Matrix Tutorial, Types Tutorial, Arithmetic Tutorial,
Tutorial
address@hidden Vector/Matrix Tutorial
+
address@hidden
+A @dfn{vector} is a list of numbers or other Calc data objects.
+Calc provides a large set of commands that operate on vectors. Some
+are familiar operations from vector analysis. Others simply treat
+a vector as a list of objects.
+
address@hidden
+* Vector Analysis Tutorial::
+* Matrix Tutorial::
+* List Tutorial::
address@hidden menu
+
address@hidden Vector Analysis Tutorial, Matrix Tutorial, Vector/Matrix
Tutorial, Vector/Matrix Tutorial
address@hidden Vector Analysis
+
address@hidden
+If you add two vectors, the result is a vector of the sums of the
+elements, taken pairwise.
+
address@hidden
address@hidden
+1: [1, 2, 3] 2: [1, 2, 3] 1: [8, 8, 3]
+ . 1: [7, 6, 0] .
+ .
+
+ [1,2,3] s 1 [7 6 0] s 2 +
address@hidden group
address@hidden smallexample
+
address@hidden
+Note that we can separate the vector elements with either commas or
+spaces. This is true whether we are using incomplete vectors or
+algebraic entry. The @kbd{s 1} and @kbd{s 2} commands save these
+vectors so we can easily reuse them later.
+
+If you multiply two vectors, the result is the sum of the products
+of the elements taken pairwise. This is called the @dfn{dot product}
+of the vectors.
+
address@hidden
address@hidden
+2: [1, 2, 3] 1: 19
+1: [7, 6, 0] .
+ .
+
+ r 1 r 2 *
address@hidden group
address@hidden smallexample
+
address@hidden Dot product
+The dot product of two vectors is equal to the product of their
+lengths times the cosine of the angle between them. (Here the vector
+is interpreted as a line from the origin @expr{(0,0,0)} to the
+specified point in three-dimensional space.) The @kbd{A}
+(absolute value) command can be used to compute the length of a
+vector.
+
address@hidden
address@hidden
+3: 19 3: 19 1: 0.550782 1: 56.579
+2: [1, 2, 3] 2: 3.741657 . .
+1: [7, 6, 0] 1: 9.219544
+ . .
+
+ address@hidden M-2 A * / I C
address@hidden group
address@hidden smallexample
+
address@hidden
+First we recall the arguments to the dot product command, then
+we compute the absolute values of the top two stack entries to
+obtain the lengths of the vectors, then we divide the dot product
+by the product of the lengths to get the cosine of the angle.
+The inverse cosine finds that the angle between the vectors
+is about 56 degrees.
+
address@hidden Cross product
address@hidden Perpendicular vectors
+The @dfn{cross product} of two vectors is a vector whose length
+is the product of the lengths of the inputs times the sine of the
+angle between them, and whose direction is perpendicular to both
+input vectors. Unlike the dot product, the cross product is
+defined only for three-dimensional vectors. Let's double-check
+our computation of the angle using the cross product.
+
address@hidden
address@hidden
+2: [1, 2, 3] 3: [-18, 21, -8] 1: [-0.52, 0.61, -0.23] 1: 56.579
+1: [7, 6, 0] 2: [1, 2, 3] . .
+ . 1: [7, 6, 0]
+ .
+
+ r 1 r 2 V C s 3 address@hidden M-2 A * / A I S
address@hidden group
address@hidden smallexample
+
address@hidden
+First we recall the original vectors and compute their cross product,
+which we also store for later reference. Now we divide the vector
+by the product of the lengths of the original vectors. The length of
+this vector should be the sine of the angle; sure enough, it is!
+
address@hidden [fix-ref General Mode Commands]
+Vector-related commands generally begin with the @kbd{v} prefix key.
+Some are uppercase letters and some are lowercase. To make it easier
+to type these commands, the address@hidden prefix key acts the same as
+the @kbd{v} key. (@xref{General Mode Commands}, for a way to make all
+prefix keys have this property.)
+
+If we take the dot product of two perpendicular vectors we expect
+to get zero, since the cosine of 90 degrees is zero. Let's check
+that the cross product is indeed perpendicular to both inputs:
+
address@hidden
address@hidden
+2: [1, 2, 3] 1: 0 2: [7, 6, 0] 1: 0
+1: [-18, 21, -8] . 1: [-18, 21, -8] .
+ . .
+
+ r 1 r 3 * @key{DEL} r 2 r 3 *
address@hidden group
address@hidden smallexample
+
address@hidden Normalizing a vector
address@hidden Unit vectors
+(@bullet{}) @strong{Exercise 1.} Given a vector on the top of the
+stack, what keystrokes would you use to @dfn{normalize} the
+vector, i.e., to reduce its length to one without changing its
+direction? @xref{Vector Answer 1, 1}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 2.} Suppose a certain particle can be
+at any of several positions along a ruler. You have a list of
+those positions in the form of a vector, and another list of the
+probabilities for the particle to be at the corresponding positions.
+Find the average position of the particle.
address@hidden Answer 2, 2}. (@bullet{})
+
address@hidden Matrix Tutorial, List Tutorial, Vector Analysis Tutorial,
Vector/Matrix Tutorial
address@hidden Matrices
+
address@hidden
+A @dfn{matrix} is just a vector of vectors, all the same length.
+This means you can enter a matrix using nested brackets. You can
+also use the semicolon character to enter a matrix. We'll show
+both methods here:
+
address@hidden
address@hidden
+1: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ]
+ [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
+ . .
+
+ [[1 2 3] [4 5 6]] ' [1 2 3; 4 5 6] @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+We'll be using this matrix again, so type @kbd{s 4} to save it now.
+
+Note that semicolons work with incomplete vectors, but they work
+better in algebraic entry. That's why we use the apostrophe in
+the second example.
+
+When two matrices are multiplied, the lefthand matrix must have
+the same number of columns as the righthand matrix has rows.
+Row @expr{i}, column @expr{j} of the result is effectively the
+dot product of row @expr{i} of the left matrix by column @expr{j}
+of the right matrix.
+
+If we try to duplicate this matrix and multiply it by itself,
+the dimensions are wrong and the multiplication cannot take place:
+
address@hidden
address@hidden
+1: [ [ 1, 2, 3 ] * [ [ 1, 2, 3 ]
+ [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
+ .
+
+ @key{RET} *
address@hidden group
address@hidden smallexample
+
address@hidden
+Though rather hard to read, this is a formula which shows the product
+of two matrices. The @samp{*} function, having invalid arguments, has
+been left in symbolic form.
+
+We can multiply the matrices if we @dfn{transpose} one of them first.
+
address@hidden
address@hidden
+2: [ [ 1, 2, 3 ] 1: [ [ 14, 32 ] 1: [ [ 17, 22, 27 ]
+ [ 4, 5, 6 ] ] [ 32, 77 ] ] [ 22, 29, 36 ]
+1: [ [ 1, 4 ] . [ 27, 36, 45 ] ]
+ [ 2, 5 ] .
+ [ 3, 6 ] ]
+ .
+
+ U v t * U @key{TAB} *
address@hidden group
address@hidden smallexample
+
+Matrix multiplication is not commutative; indeed, switching the
+order of the operands can even change the dimensions of the result
+matrix, as happened here!
+
+If you multiply a plain vector by a matrix, it is treated as a
+single row or column depending on which side of the matrix it is
+on. The result is a plain vector which should also be interpreted
+as a row or column as appropriate.
+
address@hidden
address@hidden
+2: [ [ 1, 2, 3 ] 1: [14, 32]
+ [ 4, 5, 6 ] ] .
+1: [1, 2, 3]
+ .
+
+ r 4 r 1 *
address@hidden group
address@hidden smallexample
+
+Multiplying in the other order wouldn't work because the number of
+rows in the matrix is different from the number of elements in the
+vector.
+
+(@bullet{}) @strong{Exercise 1.} Use @samp{*} to sum along the rows
+of the above
address@hidden @math{2\times3}
address@hidden 2x3
+matrix to get @expr{[6, 15]}. Now use @samp{*} to sum along the columns
+to get @expr{[5, 7, 9]}.
address@hidden Answer 1, 1}. (@bullet{})
+
address@hidden Identity matrix
+An @dfn{identity matrix} is a square matrix with ones along the
+diagonal and zeros elsewhere. It has the property that multiplication
+by an identity matrix, on the left or on the right, always produces
+the original matrix.
+
address@hidden
address@hidden
+1: [ [ 1, 2, 3 ] 2: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ]
+ [ 4, 5, 6 ] ] [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
+ . 1: [ [ 1, 0, 0 ] .
+ [ 0, 1, 0 ]
+ [ 0, 0, 1 ] ]
+ .
+
+ r 4 v i 3 @key{RET} *
address@hidden group
address@hidden smallexample
+
+If a matrix is square, it is often possible to find its @dfn{inverse},
+that is, a matrix which, when multiplied by the original matrix, yields
+an identity matrix. The @kbd{&} (reciprocal) key also computes the
+inverse of a matrix.
+
address@hidden
address@hidden
+1: [ [ 1, 2, 3 ] 1: [ [ -2.4, 1.2, -0.2 ]
+ [ 4, 5, 6 ] [ 2.8, -1.4, 0.4 ]
+ [ 7, 6, 0 ] ] [ -0.73333, 0.53333, -0.2 ] ]
+ . .
+
+ r 4 r 2 | s 5 &
address@hidden group
address@hidden smallexample
+
address@hidden
+The vertical bar @kbd{|} @dfn{concatenates} numbers, vectors, and
+matrices together. Here we have used it to add a new row onto
+our matrix to make it square.
+
+We can multiply these two matrices in either order to get an identity.
+
address@hidden
address@hidden
+1: [ [ 1., 0., 0. ] 1: [ [ 1., 0., 0. ]
+ [ 0., 1., 0. ] [ 0., 1., 0. ]
+ [ 0., 0., 1. ] ] [ 0., 0., 1. ] ]
+ . .
+
+ address@hidden * U @key{TAB} *
address@hidden group
address@hidden smallexample
+
address@hidden Systems of linear equations
address@hidden Linear equations, systems of
+Matrix inverses are related to systems of linear equations in algebra.
+Suppose we had the following set of equations:
+
address@hidden
address@hidden
address@hidden
+ a + 2b + 3c = 6
+ 4a + 5b + 6c = 2
+ 7a + 6b = 3
address@hidden example
address@hidden group
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplayh
+$$ \openup1\jot \tabskip=0pt plus1fil
+\halign to\displaywidth{\tabskip=0pt
+ $\hfil#$&$\hfil{}#{}$&
+ $\hfil#$&$\hfil{}#{}$&
+ $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
+ a&+&2b&+&3c&=6 \cr
+ 4a&+&5b&+&6c&=2 \cr
+ 7a&+&6b& & &=3 \cr}
+$$
+\afterdisplayh
address@hidden tex
+
address@hidden
+This can be cast into the matrix equation,
+
address@hidden
address@hidden
address@hidden
+ [ [ 1, 2, 3 ] [ [ a ] [ [ 6 ]
+ [ 4, 5, 6 ] * [ b ] = [ 2 ]
+ [ 7, 6, 0 ] ] [ c ] ] [ 3 ] ]
address@hidden example
address@hidden group
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ \pmatrix{ 1 & 2 & 3 \cr 4 & 5 & 6 \cr 7 & 6 & 0 }
+ \times
+ \pmatrix{ a \cr b \cr c } = \pmatrix{ 6 \cr 2 \cr 3 }
+$$
+\afterdisplay
address@hidden tex
+
+We can solve this system of equations by multiplying both sides by the
+inverse of the matrix. Calc can do this all in one step:
+
address@hidden
address@hidden
+2: [6, 2, 3] 1: [-12.6, 15.2, -3.93333]
+1: [ [ 1, 2, 3 ] .
+ [ 4, 5, 6 ]
+ [ 7, 6, 0 ] ]
+ .
+
+ [6,2,3] r 5 /
address@hidden group
address@hidden smallexample
+
address@hidden
+The result is the @expr{[a, b, c]} vector that solves the equations.
+(Dividing by a square matrix is equivalent to multiplying by its
+inverse.)
+
+Let's verify this solution:
+
address@hidden
address@hidden
+2: [ [ 1, 2, 3 ] 1: [6., 2., 3.]
+ [ 4, 5, 6 ] .
+ [ 7, 6, 0 ] ]
+1: [-12.6, 15.2, -3.93333]
+ .
+
+ r 5 @key{TAB} *
address@hidden group
address@hidden smallexample
+
address@hidden
+Note that we had to be careful about the order in which we multiplied
+the matrix and vector. If we multiplied in the other order, Calc would
+assume the vector was a row vector in order to make the dimensions
+come out right, and the answer would be incorrect. If you
+don't feel safe letting Calc take either interpretation of your
+vectors, use explicit
address@hidden @math{N\times1}
address@hidden Nx1
+or
address@hidden @math{1\times N}
address@hidden 1xN
+matrices instead. In this case, you would enter the original column
+vector as @samp{[[6], [2], [3]]} or @samp{[6; 2; 3]}.
+
+(@bullet{}) @strong{Exercise 2.} Algebraic entry allows you to make
+vectors and matrices that include variables. Solve the following
+system of equations to get expressions for @expr{x} and @expr{y}
+in terms of @expr{a} and @expr{b}.
+
address@hidden
address@hidden
address@hidden
+ x + a y = 6
+ x + b y = 10
address@hidden example
address@hidden group
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ \eqalign{ x &+ a y = 6 \cr
+ x &+ b y = 10}
+$$
+\afterdisplay
address@hidden tex
+
address@hidden
address@hidden Answer 2, 2}. (@bullet{})
+
address@hidden Least-squares for over-determined systems
address@hidden Over-determined systems of equations
+(@bullet{}) @strong{Exercise 3.} A system of equations is ``over-determined''
+if it has more equations than variables. It is often the case that
+there are no values for the variables that will satisfy all the
+equations at once, but it is still useful to find a set of values
+which ``nearly'' satisfy all the equations. In terms of matrix equations,
+you can't solve @expr{A X = B} directly because the matrix @expr{A}
+is not square for an over-determined system. Matrix inversion works
+only for square matrices. One common trick is to multiply both sides
+on the left by the transpose of @expr{A}:
address@hidden
address@hidden(A)*A*X = trn(A)*B}.
address@hidden ifnottex
address@hidden
+\turnoffactive
+$A^T A \, X = A^T B$, where $A^T$ is the transpose \samp{trn(A)}.
address@hidden tex
+Now
address@hidden @math{A^T A}
address@hidden @expr{trn(A)*A}
+is a square matrix so a solution is possible. It turns out that the
address@hidden vector you compute in this way will be a ``least-squares''
+solution, which can be regarded as the ``closest'' solution to the set
+of equations. Use Calc to solve the following over-determined
+system:
+
address@hidden
address@hidden
address@hidden
+ a + 2b + 3c = 6
+ 4a + 5b + 6c = 2
+ 7a + 6b = 3
+ 2a + 4b + 6c = 11
address@hidden example
address@hidden group
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplayh
+$$ \openup1\jot \tabskip=0pt plus1fil
+\halign to\displaywidth{\tabskip=0pt
+ $\hfil#$&$\hfil{}#{}$&
+ $\hfil#$&$\hfil{}#{}$&
+ $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
+ a&+&2b&+&3c&=6 \cr
+ 4a&+&5b&+&6c&=2 \cr
+ 7a&+&6b& & &=3 \cr
+ 2a&+&4b&+&6c&=11 \cr}
+$$
+\afterdisplayh
address@hidden tex
+
address@hidden
address@hidden Answer 3, 3}. (@bullet{})
+
address@hidden List Tutorial, , Matrix Tutorial, Vector/Matrix Tutorial
address@hidden Vectors as Lists
+
address@hidden
address@hidden Lists
+Although Calc has a number of features for manipulating vectors and
+matrices as mathematical objects, you can also treat vectors as
+simple lists of values. For example, we saw that the @kbd{k f}
+command returns a vector which is a list of the prime factors of a
+number.
+
+You can pack and unpack stack entries into vectors:
+
address@hidden
address@hidden
+3: 10 1: [10, 20, 30] 3: 10
+2: 20 . 2: 20
+1: 30 1: 30
+ . .
+
+ M-3 v p v u
address@hidden group
address@hidden smallexample
+
+You can also build vectors out of consecutive integers, or out
+of many copies of a given value:
+
address@hidden
address@hidden
+1: [1, 2, 3, 4] 2: [1, 2, 3, 4] 2: [1, 2, 3, 4]
+ . 1: 17 1: [17, 17, 17, 17]
+ . .
+
+ v x 4 @key{RET} 17 v b 4 @key{RET}
address@hidden group
address@hidden smallexample
+
+You can apply an operator to every element of a vector using the
address@hidden command.
+
address@hidden
address@hidden
+1: [17, 34, 51, 68] 1: [289, 1156, 2601, 4624] 1: [17, 34, 51, 68]
+ . . .
+
+ V M * 2 V M ^ V M Q
address@hidden group
address@hidden smallexample
+
address@hidden
+In the first step, we multiply the vector of integers by the vector
+of 17's elementwise. In the second step, we raise each element to
+the power two. (The general rule is that both operands must be
+vectors of the same length, or else one must be a vector and the
+other a plain number.) In the final step, we take the square root
+of each element.
+
+(@bullet{}) @strong{Exercise 1.} Compute a vector of powers of two
+from
address@hidden @math{2^{-4}}
address@hidden @expr{2^-4}
+to @expr{2^4}. @xref{List Answer 1, 1}. (@bullet{})
+
+You can also @dfn{reduce} a binary operator across a vector.
+For example, reducing @samp{*} computes the product of all the
+elements in the vector:
+
address@hidden
address@hidden
+1: 123123 1: [3, 7, 11, 13, 41] 1: 123123
+ . . .
+
+ 123123 k f V R *
address@hidden group
address@hidden smallexample
+
address@hidden
+In this example, we decompose 123123 into its prime factors, then
+multiply those factors together again to yield the original number.
+
+We could compute a dot product ``by hand'' using mapping and
+reduction:
+
address@hidden
address@hidden
+2: [1, 2, 3] 1: [7, 12, 0] 1: 19
+1: [7, 6, 0] . .
+ .
+
+ r 1 r 2 V M * V R +
address@hidden group
address@hidden smallexample
+
address@hidden
+Recalling two vectors from the previous section, we compute the
+sum of pairwise products of the elements to get the same answer
+for the dot product as before.
+
+A slight variant of vector reduction is the @dfn{accumulate} operation,
address@hidden U}. This produces a vector of the intermediate results from
+a corresponding reduction. Here we compute a table of factorials:
+
address@hidden
address@hidden
+1: [1, 2, 3, 4, 5, 6] 1: [1, 2, 6, 24, 120, 720]
+ . .
+
+ v x 6 @key{RET} V U *
address@hidden group
address@hidden smallexample
+
+Calc allows vectors to grow as large as you like, although it gets
+rather slow if vectors have more than about a hundred elements.
+Actually, most of the time is spent formatting these large vectors
+for display, not calculating on them. Try the following experiment
+(if your computer is very fast you may need to substitute a larger
+vector size).
+
address@hidden
address@hidden
+1: [1, 2, 3, 4, ... 1: [2, 3, 4, 5, ...
+ . .
+
+ v x 500 @key{RET} 1 V M +
address@hidden group
address@hidden smallexample
+
+Now press @kbd{v .} (the letter @kbd{v}, then a period) and try the
+experiment again. In @kbd{v .} mode, long vectors are displayed
+``abbreviated'' like this:
+
address@hidden
address@hidden
+1: [1, 2, 3, ..., 500] 1: [2, 3, 4, ..., 501]
+ . .
+
+ v x 500 @key{RET} 1 V M +
address@hidden group
address@hidden smallexample
+
address@hidden
+(where now the @samp{...} is actually part of the Calc display).
+You will find both operations are now much faster. But notice that
+even in @address@hidden .}} mode, the full vectors are still shown in the
Trail.
+Type @address@hidden .}} to cause the trail to abbreviate as well, and try the
+experiment one more time. Operations on long vectors are now quite
+fast! (But of course if you use @kbd{t .} you will lose the ability
+to get old vectors back using the @kbd{t y} command.)
+
+An easy way to view a full vector when @kbd{v .} mode is active is
+to press @kbd{`} (back-quote) to edit the vector; editing always works
+with the full, unabbreviated value.
+
address@hidden Least-squares for fitting a straight line
address@hidden Fitting data to a line
address@hidden Line, fitting data to
address@hidden Data, extracting from buffers
address@hidden Columns of data, extracting
+As a larger example, let's try to fit a straight line to some data,
+using the method of least squares. (Calc has a built-in command for
+least-squares curve fitting, but we'll do it by hand here just to
+practice working with vectors.) Suppose we have the following list
+of values in a file we have loaded into Emacs:
+
address@hidden
+ x y
+ --- ---
+ 1.34 0.234
+ 1.41 0.298
+ 1.49 0.402
+ 1.56 0.412
+ 1.64 0.466
+ 1.73 0.473
+ 1.82 0.601
+ 1.91 0.519
+ 2.01 0.603
+ 2.11 0.637
+ 2.22 0.645
+ 2.33 0.705
+ 2.45 0.917
+ 2.58 1.009
+ 2.71 0.971
+ 2.85 1.062
+ 3.00 1.148
+ 3.15 1.157
+ 3.32 1.354
address@hidden smallexample
+
address@hidden
+If you are reading this tutorial in printed form, you will find it
+easiest to press @kbd{C-x * i} to enter the on-line Info version of
+the manual and find this table there. (Press @kbd{g}, then type
address@hidden Tutorial}, to jump straight to this section.)
+
+Position the cursor at the upper-left corner of this table, just
+to the left of the @expr{1.34}. Press @kbd{C-@@} to set the mark.
+(On your system this may be @kbd{C-2}, @address@hidden, or @kbd{NUL}.)
+Now position the cursor to the lower-right, just after the @expr{1.354}.
+You have now defined this region as an Emacs ``rectangle.'' Still
+in the Info buffer, type @kbd{C-x * r}. This command
+(@code{calc-grab-rectangle}) will pop you back into the Calculator, with
+the contents of the rectangle you specified in the form of a matrix.
+
address@hidden
address@hidden
+1: [ [ 1.34, 0.234 ]
+ [ 1.41, 0.298 ]
+ @dots{}
address@hidden group
address@hidden smallexample
+
address@hidden
+(You may wish to use @kbd{v .} mode to abbreviate the display of this
+large matrix.)
+
+We want to treat this as a pair of lists. The first step is to
+transpose this matrix into a pair of rows. Remember, a matrix is
+just a vector of vectors. So we can unpack the matrix into a pair
+of row vectors on the stack.
+
address@hidden
address@hidden
+1: [ [ 1.34, 1.41, 1.49, ... ] 2: [1.34, 1.41, 1.49, ... ]
+ [ 0.234, 0.298, 0.402, ... ] ] 1: [0.234, 0.298, 0.402, ... ]
+ . .
+
+ v t v u
address@hidden group
address@hidden smallexample
+
address@hidden
+Let's store these in quick variables 1 and 2, respectively.
+
address@hidden
address@hidden
+1: [1.34, 1.41, 1.49, ... ] .
+ .
+
+ t 2 t 1
address@hidden group
address@hidden smallexample
+
address@hidden
+(Recall that @kbd{t 2} is a variant of @kbd{s 2} that removes the
+stored value from the stack.)
+
+In a least squares fit, the slope @expr{m} is given by the formula
+
address@hidden
address@hidden
+m = (N sum(x y) - sum(x) sum(y)) / (N sum(x^2) - sum(x)^2)
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ m = {N \sum x y - \sum x \sum y \over
+ N \sum x^2 - \left( \sum x \right)^2} $$
+\afterdisplay
address@hidden tex
+
address@hidden
+where
address@hidden @math{\sum x}
address@hidden @expr{sum(x)}
+represents the sum of all the values of @expr{x}. While there is an
+actual @code{sum} function in Calc, it's easier to sum a vector using a
+simple reduction. First, let's compute the four different sums that
+this formula uses.
+
address@hidden
address@hidden
+1: 41.63 1: 98.0003
+ . .
+
+ r 1 V R + t 3 r 1 2 V M ^ V R + t 4
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: 13.613 1: 33.36554
+ . .
+
+ r 2 V R + t 5 r 1 r 2 V M * V R + t 6
address@hidden group
address@hidden smallexample
+
address@hidden
address@hidden
+These are @samp{sum(x)}, @samp{sum(x^2)}, @samp{sum(y)}, and @samp{sum(x y)},
+respectively. (We could have used @kbd{*} to compute @samp{sum(x^2)} and
address@hidden(x y)}.)
address@hidden ifnottex
address@hidden
+\turnoffactive
+These are $\sum x$, $\sum x^2$, $\sum y$, and $\sum x y$,
+respectively. (We could have used \kbd{*} to compute $\sum x^2$ and
+$\sum x y$.)
address@hidden tex
+
+Finally, we also need @expr{N}, the number of data points. This is just
+the length of either of our lists.
+
address@hidden
address@hidden
+1: 19
+ .
+
+ r 1 v l t 7
address@hidden group
address@hidden smallexample
+
address@hidden
+(That's @kbd{v} followed by a lower-case @kbd{l}.)
+
+Now we grind through the formula:
+
address@hidden
address@hidden
+1: 633.94526 2: 633.94526 1: 67.23607
+ . 1: 566.70919 .
+ .
+
+ r 7 r 6 * r 3 r 5 * -
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+2: 67.23607 3: 67.23607 2: 67.23607 1: 0.52141679
+1: 1862.0057 2: 1862.0057 1: 128.9488 .
+ . 1: 1733.0569 .
+ .
+
+ r 7 r 4 * r 3 2 ^ - / t 8
address@hidden group
address@hidden smallexample
+
+That gives us the slope @expr{m}. The y-intercept @expr{b} can now
+be found with the simple formula,
+
address@hidden
address@hidden
+b = (sum(y) - m sum(x)) / N
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ b = {\sum y - m \sum x \over N} $$
+\afterdisplay
+\vskip10pt
address@hidden tex
+
address@hidden
address@hidden
+1: 13.613 2: 13.613 1: -8.09358 1: -0.425978
+ . 1: 21.70658 . .
+ .
+
+ r 5 r 8 r 3 * - r 7 / t 9
address@hidden group
address@hidden smallexample
+
+Let's ``plot'' this straight line approximation,
address@hidden @math{y \approx m x + b},
address@hidden @expr{m x + b},
+and compare it with the original data.
+
address@hidden
address@hidden
+1: [0.699, 0.735, ... ] 1: [0.273, 0.309, ... ]
+ . .
+
+ r 1 r 8 * r 9 + s 0
address@hidden group
address@hidden smallexample
+
address@hidden
+Notice that multiplying a vector by a constant, and adding a constant
+to a vector, can be done without mapping commands since these are
+common operations from vector algebra. As far as Calc is concerned,
+we've just been doing geometry in 19-dimensional space!
+
+We can subtract this vector from our original @expr{y} vector to get
+a feel for the error of our fit. Let's find the maximum error:
+
address@hidden
address@hidden
+1: [0.0387, 0.0112, ... ] 1: [0.0387, 0.0112, ... ] 1: 0.0897
+ . . .
+
+ r 2 - V M A V R X
address@hidden group
address@hidden smallexample
+
address@hidden
+First we compute a vector of differences, then we take the absolute
+values of these differences, then we reduce the @code{max} function
+across the vector. (The @code{max} function is on the two-key sequence
address@hidden x}; because it is so common to use @code{max} in a vector
+operation, the letters @kbd{X} and @kbd{N} are also accepted for
address@hidden and @code{min} in this context. In general, you answer
+the @kbd{V M} or @kbd{V R} prompt with the actual key sequence that
+invokes the function you want. You could have typed @kbd{V R f x} or
+even @kbd{V R x max @key{RET}} if you had preferred.)
+
+If your system has the GNUPLOT program, you can see graphs of your
+data and your straight line to see how well they match. (If you have
+GNUPLOT 3.0 or higher, the following instructions will work regardless
+of the kind of display you have. Some GNUPLOT 2.0, non-X-windows systems
+may require additional steps to view the graphs.)
+
+Let's start by plotting the original data. Recall the address@hidden'' and
address@hidden''
+vectors onto the stack and press @kbd{g f}. This ``fast'' graphing
+command does everything you need to do for simple, straightforward
+plotting of data.
+
address@hidden
address@hidden
+2: [1.34, 1.41, 1.49, ... ]
+1: [0.234, 0.298, 0.402, ... ]
+ .
+
+ r 1 r 2 g f
address@hidden group
address@hidden smallexample
+
+If all goes well, you will shortly get a new window containing a graph
+of the data. (If not, contact your GNUPLOT or Calc installer to find
+out what went wrong.) In the X window system, this will be a separate
+graphics window. For other kinds of displays, the default is to
+display the graph in Emacs itself using rough character graphics.
+Press @kbd{q} when you are done viewing the character graphics.
+
+Next, let's add the line we got from our least-squares fit.
address@hidden
+(If you are reading this tutorial on-line while running Calc, typing
address@hidden a} may cause the tutorial to disappear from its window and be
+replaced by a buffer named @samp{*Gnuplot Commands*}. The tutorial
+will reappear when you terminate GNUPLOT by typing @kbd{g q}.)
address@hidden ifinfo
+
address@hidden
address@hidden
+2: [1.34, 1.41, 1.49, ... ]
+1: [0.273, 0.309, 0.351, ... ]
+ .
+
+ @key{DEL} r 0 g a g p
address@hidden group
address@hidden smallexample
+
+It's not very useful to get symbols to mark the data points on this
+second curve; you can type @kbd{g S g p} to remove them. Type @kbd{g q}
+when you are done to remove the X graphics window and terminate GNUPLOT.
+
+(@bullet{}) @strong{Exercise 2.} An earlier exercise showed how to do
+least squares fitting to a general system of equations. Our 19 data
+points are really 19 equations of the form @expr{y_i = m x_i + b} for
+different pairs of @expr{(x_i,y_i)}. Use the matrix-transpose method
+to solve for @expr{m} and @expr{b}, duplicating the above result.
address@hidden Answer 2, 2}. (@bullet{})
+
address@hidden Geometric mean
+(@bullet{}) @strong{Exercise 3.} If the input data do not form a
+rectangle, you can use @address@hidden * g}} (@code{calc-grab-region})
+to grab the data the way Emacs normally works with regions---it reads
+left-to-right, top-to-bottom, treating line breaks the same as spaces.
+Use this command to find the geometric mean of the following numbers.
+(The geometric mean is the @var{n}th root of the product of @var{n} numbers.)
+
address@hidden
+2.3 6 22 15.1 7
+ 15 14 7.5
+ 2.5
address@hidden example
+
address@hidden
+The @kbd{C-x * g} command accepts numbers separated by spaces or commas,
+with or without surrounding vector brackets.
address@hidden Answer 3, 3}. (@bullet{})
+
address@hidden
+As another example, a theorem about binomial coefficients tells
+us that the alternating sum of binomial coefficients
address@hidden minus @var{n}-choose-1 plus @var{n}-choose-2, and so
+on up to @address@hidden,
+always comes out to zero. Let's verify this
+for @expr{n=6}.
address@hidden ifnottex
address@hidden
+As another example, a theorem about binomial coefficients tells
+us that the alternating sum of binomial coefficients
+${n \choose 0} - {n \choose 1} + {n \choose 2} - \cdots \pm {n \choose n}$
+always comes out to zero. Let's verify this
+for \cite{n=6}.
address@hidden tex
+
address@hidden
address@hidden
+1: [1, 2, 3, 4, 5, 6, 7] 1: [0, 1, 2, 3, 4, 5, 6]
+ . .
+
+ v x 7 @key{RET} 1 -
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: [1, -6, 15, -20, 15, -6, 1] 1: 0
+ . .
+
+ V M ' (-1)^$ choose(6,$) @key{RET} V R +
address@hidden group
address@hidden smallexample
+
+The @kbd{V M '} command prompts you to enter any algebraic expression
+to define the function to map over the vector. The symbol @samp{$}
+inside this expression represents the argument to the function.
+The Calculator applies this formula to each element of the vector,
+substituting each element's value for the @samp{$} sign(s) in turn.
+
+To define a two-argument function, use @samp{$$} for the first
+argument and @samp{$} for the second: @kbd{V M ' $$-$ @key{RET}} is
+equivalent to @kbd{V M -}. This is analogous to regular algebraic
+entry, where @samp{$$} would refer to the next-to-top stack entry
+and @samp{$} would refer to the top stack entry, and @kbd{' $$-$ @key{RET}}
+would act exactly like @kbd{-}.
+
+Notice that the @kbd{V M '} command has recorded two things in the
+trail: The result, as usual, and also a funny-looking thing marked
address@hidden that represents the operator function you typed in.
+The function is enclosed in @samp{< >} brackets, and the argument is
+denoted by a @samp{#} sign. If there were several arguments, they
+would be shown as @samp{#1}, @samp{#2}, and so on. (For example,
address@hidden M ' $$-$} will put the function @samp{<#1 - #2>} on the
+trail.) This object is a ``nameless function''; you can use nameless
address@hidden@samp{< >}} notation to answer the @kbd{V M '} prompt if you like.
+Nameless function notation has the interesting, occasionally useful
+property that a nameless function is not actually evaluated until
+it is used. For example, @kbd{V M ' $+random(2.0)} evaluates
address@hidden(2.0)} once and adds that random number to all elements
+of the vector, but @kbd{V M ' <#+random(2.0)>} evaluates the
address@hidden(2.0)} separately for each vector element.
+
+Another group of operators that are often useful with @kbd{V M} are
+the relational operators: @kbd{a =}, for example, compares two numbers
+and gives the result 1 if they are equal, or 0 if not. Similarly,
address@hidden@kbd{a <}} checks for one number being less than another.
+
+Other useful vector operations include @kbd{v v}, to reverse a
+vector end-for-end; @kbd{V S}, to sort the elements of a vector
+into increasing order; and @kbd{v r} and @address@hidden c}}, to extract
+one row or column of a matrix, or (in both cases) to extract one
+element of a plain vector. With a negative argument, @kbd{v r}
+and @kbd{v c} instead delete one row, column, or vector element.
+
address@hidden Divisor functions
+(@bullet{}) @strong{Exercise 4.} The @expr{k}th @dfn{divisor function}
address@hidden
+$\sigma_k(n)$
address@hidden tex
+is the sum of the @expr{k}th powers of all the divisors of an
+integer @expr{n}. Figure out a method for computing the divisor
+function for reasonably small values of @expr{n}. As a test,
+the 0th and 1st divisor functions of 30 are 8 and 72, respectively.
address@hidden Answer 4, 4}. (@bullet{})
+
address@hidden Square-free numbers
address@hidden Duplicate values in a list
+(@bullet{}) @strong{Exercise 5.} The @kbd{k f} command produces a
+list of prime factors for a number. Sometimes it is important to
+know that a number is @dfn{square-free}, i.e., that no prime occurs
+more than once in its list of prime factors. Find a sequence of
+keystrokes to tell if a number is square-free; your method should
+leave 1 on the stack if it is, or 0 if it isn't.
address@hidden Answer 5, 5}. (@bullet{})
+
address@hidden Triangular lists
+(@bullet{}) @strong{Exercise 6.} Build a list of lists that looks
+like the following diagram. (You may wish to use the @kbd{v /}
+command to enable multi-line display of vectors.)
+
address@hidden
address@hidden
+1: [ [1],
+ [1, 2],
+ [1, 2, 3],
+ [1, 2, 3, 4],
+ [1, 2, 3, 4, 5],
+ [1, 2, 3, 4, 5, 6] ]
address@hidden group
address@hidden smallexample
+
address@hidden
address@hidden Answer 6, 6}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 7.} Build the following list of lists.
+
address@hidden
address@hidden
+1: [ [0],
+ [1, 2],
+ [3, 4, 5],
+ [6, 7, 8, 9],
+ [10, 11, 12, 13, 14],
+ [15, 16, 17, 18, 19, 20] ]
address@hidden group
address@hidden smallexample
+
address@hidden
address@hidden Answer 7, 7}. (@bullet{})
+
address@hidden Maximizing a function over a list of values
address@hidden [fix-ref Numerical Solutions]
+(@bullet{}) @strong{Exercise 8.} Compute a list of values of Bessel's
address@hidden @math{J_1(x)}
address@hidden @expr{J1}
+function @samp{besJ(1,x)} for @expr{x} from 0 to 5 in steps of 0.25.
+Find the value of @expr{x} (from among the above set of values) for
+which @samp{besJ(1,x)} is a maximum. Use an ``automatic'' method,
+i.e., just reading along the list by hand to find the largest value
+is not allowed! (There is an @kbd{a X} command which does this kind
+of thing automatically; @pxref{Numerical Solutions}.)
address@hidden Answer 8, 8}. (@bullet{})
+
address@hidden Digits, vectors of
+(@bullet{}) @strong{Exercise 9.} You are given an integer in the range
address@hidden @math{0 \le N < 10^m}
address@hidden @expr{0 <= N < 10^m}
+for @expr{m=12} (i.e., an integer of less than
+twelve digits). Convert this integer into a vector of @expr{m}
+digits, each in the range from 0 to 9. In vector-of-digits notation,
+add one to this integer to produce a vector of @expr{m+1} digits
+(since there could be a carry out of the most significant digit).
+Convert this vector back into a regular integer. A good integer
+to try is 25129925999. @xref{List Answer 9, 9}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 10.} Your friend Joe tried to use
address@hidden R a =} to test if all numbers in a list were equal. What
+happened? How would you do this test? @xref{List Answer 10, 10}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 11.} The area of a circle of radius one
+is @cpi{}. The area of the
address@hidden @math{2\times2}
address@hidden 2x2
+square that encloses that circle is 4. So if we throw @var{n} darts at
+random points in the square, about @cpiover{4} of them will land inside
+the circle. This gives us an entertaining way to estimate the value of
address@hidden The @address@hidden r}}
+command picks a random number between zero and the value on the stack.
+We could get a random floating-point number between @mathit{-1} and 1 by typing
address@hidden@kbd{2.0 k r 1 -}}. Build a vector of 100 random @expr{(x,y)}
points in
+this square, then use vector mapping and reduction to count how many
+points lie inside the unit circle. Hint: Use the @kbd{v b} command.
address@hidden Answer 11, 11}. (@bullet{})
+
address@hidden Matchstick problem
+(@bullet{}) @strong{Exercise 12.} The @dfn{matchstick problem} provides
+another way to calculate @cpi{}. Say you have an infinite field
+of vertical lines with a spacing of one inch. Toss a one-inch matchstick
+onto the field. The probability that the matchstick will land crossing
+a line turns out to be
address@hidden @math{2/\pi}.
address@hidden @expr{2/pi}.
+Toss 100 matchsticks to estimate @cpi{}. (If you want still more fun,
+the probability that the GCD (@address@hidden g}}) of two large integers is
+one turns out to be
address@hidden @math{6/\pi^2}.
address@hidden @expr{6/pi^2}.
+That provides yet another way to estimate @cpi{}.)
address@hidden Answer 12, 12}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 13.} An algebraic entry of a string in
+double-quote marks, @samp{"hello"}, creates a vector of the numerical
+(ASCII) codes of the characters (here, @expr{[104, 101, 108, 108, 111]}).
+Sometimes it is convenient to compute a @dfn{hash code} of a string,
+which is just an integer that represents the value of that string.
+Two equal strings have the same hash code; two different strings
address@hidden have different hash codes. (For example, Calc has
+over 400 function names, but Emacs can quickly find the definition for
+any given name because it has sorted the functions into ``buckets'' by
+their hash codes. Sometimes a few names will hash into the same bucket,
+but it is easier to search among a few names than among all the names.)
+One popular hash function is computed as follows: First set @expr{h = 0}.
+Then, for each character from the string in turn, set @expr{h = 3h + c_i}
+where @expr{c_i} is the character's ASCII code. If we have 511 buckets,
+we then take the hash code modulo 511 to get the bucket number. Develop a
+simple command or commands for converting string vectors into hash codes.
+The hash code for @samp{"Testing, 1, 2, 3"} is 1960915098, which modulo
+511 is 121. @xref{List Answer 13, 13}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 14.} The @kbd{H V R} and @kbd{H V U}
+commands do nested function evaluations. @kbd{H V U} takes a starting
+value and a number of steps @var{n} from the stack; it then applies the
+function you give to the starting value 0, 1, 2, up to @var{n} times
+and returns a vector of the results. Use this command to create a
+``random walk'' of 50 steps. Start with the two-dimensional point
address@hidden(0,0)}; then take one step a random distance between @mathit{-1}
and 1
+in both @expr{x} and @expr{y}; then take another step, and so on. Use the
address@hidden f} command to display this random walk. Now modify your random
+walk to walk a unit distance, but in a random direction, at each step.
+(Hint: The @code{sincos} function returns a vector of the cosine and
+sine of an angle.) @xref{List Answer 14, 14}. (@bullet{})
+
address@hidden Types Tutorial, Algebra Tutorial, Vector/Matrix Tutorial,
Tutorial
address@hidden Types Tutorial
+
address@hidden
+Calc understands a variety of data types as well as simple numbers.
+In this section, we'll experiment with each of these types in turn.
+
+The numbers we've been using so far have mainly been either @dfn{integers}
+or @dfn{floats}. We saw that floats are usually a good approximation to
+the mathematical concept of real numbers, but they are only approximations
+and are susceptible to roundoff error. Calc also supports @dfn{fractions},
+which can exactly represent any rational number.
+
address@hidden
address@hidden
+1: 3628800 2: 3628800 1: 518400:7 1: 518414:7 1: 7:518414
+ . 1: 49 . . .
+ .
+
+ 10 ! 49 @key{RET} : 2 + &
address@hidden group
address@hidden smallexample
+
address@hidden
+The @kbd{:} command divides two integers to get a fraction; @kbd{/}
+would normally divide integers to get a floating-point result.
+Notice we had to type @key{RET} between the @kbd{49} and the @kbd{:}
+since the @kbd{:} would otherwise be interpreted as part of a
+fraction beginning with 49.
+
+You can convert between floating-point and fractional format using
address@hidden f} and @kbd{c F}:
+
address@hidden
address@hidden
+1: 1.35027217629e-5 1: 7:518414
+ . .
+
+ c f c F
address@hidden group
address@hidden smallexample
+
+The @kbd{c F} command replaces a floating-point number with the
+``simplest'' fraction whose floating-point representation is the
+same, to within the current precision.
+
address@hidden
address@hidden
+1: 3.14159265359 1: 1146408:364913 1: 3.1416 1: 355:113
+ . . . .
+
+ P c F @key{DEL} p 5 @key{RET} P c F
address@hidden group
address@hidden smallexample
+
+(@bullet{}) @strong{Exercise 1.} A calculation has produced the
+result 1.26508260337. You suspect it is the square root of the
+product of @cpi{} and some rational number. Is it? (Be sure
+to allow for roundoff error!) @xref{Types Answer 1, 1}. (@bullet{})
+
address@hidden numbers} can be stored in both rectangular and polar form.
+
address@hidden
address@hidden
+1: -9 1: (0, 3) 1: (3; 90.) 1: (6; 90.) 1: (2.4495; 45.)
+ . . . . .
+
+ 9 n Q c p 2 * Q
address@hidden group
address@hidden smallexample
+
address@hidden
+The square root of @mathit{-9} is by default rendered in rectangular form
+(@address@hidden + 3i}}), but we can convert it to polar form (3 with a
+phase angle of 90 degrees). All the usual arithmetic and scientific
+operations are defined on both types of complex numbers.
+
+Another generalized kind of number is @dfn{infinity}. Infinity
+isn't really a number, but it can sometimes be treated like one.
+Calc uses the symbol @code{inf} to represent positive infinity,
+i.e., a value greater than any real number. Naturally, you can
+also write @samp{-inf} for minus infinity, a value less than any
+real number. The word @code{inf} can only be input using
+algebraic entry.
+
address@hidden
address@hidden
+2: inf 2: -inf 2: -inf 2: -inf 1: nan
+1: -17 1: -inf 1: -inf 1: inf .
+ . . . .
+
+' inf @key{RET} 17 n * @key{RET} 72 + A +
address@hidden group
address@hidden smallexample
+
address@hidden
+Since infinity is infinitely large, multiplying it by any finite
+number (like @mathit{-17}) has no effect, except that since @mathit{-17}
+is negative, it changes a plus infinity to a minus infinity.
+(``A huge positive number, multiplied by @mathit{-17}, yields a huge
+negative number.'') Adding any finite number to infinity also
+leaves it unchanged. Taking an absolute value gives us plus
+infinity again. Finally, we add this plus infinity to the minus
+infinity we had earlier. If you work it out, you might expect
+the answer to be @mathit{-72} for this. But the 72 has been completely
+lost next to the infinities; by the time we compute @address@hidden - inf}}
+the finite difference between them, if any, is undetectable.
+So we say the result is @dfn{indeterminate}, which Calc writes
+with the symbol @code{nan} (for Not A Number).
+
+Dividing by zero is normally treated as an error, but you can get
+Calc to write an answer in terms of infinity by pressing @kbd{m i}
+to turn on Infinite mode.
+
address@hidden
address@hidden
+3: nan 2: nan 2: nan 2: nan 1: nan
+2: 1 1: 1 / 0 1: uinf 1: uinf .
+1: 0 . . .
+ .
+
+ 1 @key{RET} 0 / m i U / 17 n * +
address@hidden group
address@hidden smallexample
+
address@hidden
+Dividing by zero normally is left unevaluated, but after @kbd{m i}
+it instead gives an infinite result. The answer is actually
address@hidden, ``undirected infinity.'' If you look at a graph of
address@hidden / x} around @address@hidden = 0}}, you'll see that it goes toward
+plus infinity as you approach zero from above, but toward minus
+infinity as you approach from below. Since we said only @expr{1 / 0},
+Calc knows that the answer is infinite but not in which direction.
+That's what @code{uinf} means. Notice that multiplying @code{uinf}
+by a negative number still leaves plain @code{uinf}; there's no
+point in saying @samp{-uinf} because the sign of @code{uinf} is
+unknown anyway. Finally, we add @code{uinf} to our @code{nan},
+yielding @code{nan} again. It's easy to see that, because
address@hidden means ``totally unknown'' while @code{uinf} means
+``unknown sign but known to be infinite,'' the more mysterious
address@hidden wins out when it is combined with @code{uinf}, or, for
+that matter, with anything else.
+
+(@bullet{}) @strong{Exercise 2.} Predict what Calc will answer
+for each of these formulas: @samp{inf / inf}, @samp{exp(inf)},
address@hidden(-inf)}, @samp{sqrt(-inf)}, @samp{sqrt(uinf)},
address@hidden(uinf)}, @samp{ln(0)}.
address@hidden Answer 2, 2}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 3.} We saw that @samp{inf - inf = nan},
+which stands for an unknown value. Can @code{nan} stand for
+a complex number? Can it stand for infinity?
address@hidden Answer 3, 3}. (@bullet{})
+
address@hidden forms} represent a value in terms of hours, minutes, and
+seconds.
+
address@hidden
address@hidden
+1: 2@@ 30' 0" 1: 3@@ 30' 0" 2: 3@@ 30' 0" 1: 2.
+ . . 1: 1@@ 45' 0." .
+ .
+
+ 2@@ 30' @key{RET} 1 + @key{RET} 2 / /
address@hidden group
address@hidden smallexample
+
+HMS forms can also be used to hold angles in degrees, minutes, and
+seconds.
+
address@hidden
address@hidden
+1: 0.5 1: 26.56505 1: 26@@ 33' 54.18" 1: 0.44721
+ . . . .
+
+ 0.5 I T c h S
address@hidden group
address@hidden smallexample
+
address@hidden
+First we convert the inverse tangent of 0.5 to degrees-minutes-seconds
+form, then we take the sine of that angle. Note that the trigonometric
+functions will accept HMS forms directly as input.
+
address@hidden Beatles
+(@bullet{}) @strong{Exercise 4.} The Beatles' @emph{Abbey Road} is
+47 minutes and 26 seconds long, and contains 17 songs. What is the
+average length of a song on @emph{Abbey Road}? If the Extended Disco
+Version of @emph{Abbey Road} added 20 seconds to the length of each
+song, how long would the album be? @xref{Types Answer 4, 4}. (@bullet{})
+
+A @dfn{date form} represents a date, or a date and time. Dates must
+be entered using algebraic entry. Date forms are surrounded by
address@hidden< >} symbols; most standard formats for dates are recognized.
+
address@hidden
address@hidden
+2: <Sun Jan 13, 1991> 1: 2.25
+1: <6:00pm Thu Jan 10, 1991> .
+ .
+
+' <13 Jan 1991>, <1/10/91, 6pm> @key{RET} -
address@hidden group
address@hidden smallexample
+
address@hidden
+In this example, we enter two dates, then subtract to find the
+number of days between them. It is also possible to add an
+HMS form or a number (of days) to a date form to get another
+date form.
+
address@hidden
address@hidden
+1: <4:45:59pm Mon Jan 14, 1991> 1: <2:50:59am Thu Jan 17, 1991>
+ . .
+
+ t N 2 + 10@@ 5' +
address@hidden group
address@hidden smallexample
+
address@hidden [fix-ref Date Arithmetic]
address@hidden
+The @kbd{t N} (``now'') command pushes the current date and time on the
+stack; then we add two days, ten hours and five minutes to the date and
+time. Other date-and-time related commands include @kbd{t J}, which
+does Julian day conversions, @kbd{t W}, which finds the beginning of
+the week in which a date form lies, and @kbd{t I}, which increments a
+date by one or several months. @xref{Date Arithmetic}, for more.
+
+(@bullet{}) @strong{Exercise 5.} How many days until the next
+Friday the 13th? @xref{Types Answer 5, 5}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 6.} How many leap years will there be
+between now and the year 10001 A.D.? @xref{Types Answer 6, 6}. (@bullet{})
+
address@hidden Slope and angle of a line
address@hidden Angle and slope of a line
+An @dfn{error form} represents a mean value with an attached standard
+deviation, or error estimate. Suppose our measurements indicate that
+a certain telephone pole is about 30 meters away, with an estimated
+error of 1 meter, and 8 meters tall, with an estimated error of 0.2
+meters. What is the slope of a line from here to the top of the
+pole, and what is the equivalent angle in degrees?
+
address@hidden
address@hidden
+1: 8 +/- 0.2 2: 8 +/- 0.2 1: 0.266 +/- 0.011 1: 14.93 +/- 0.594
+ . 1: 30 +/- 1 . .
+ .
+
+ 8 p .2 @key{RET} 30 p 1 / I T
address@hidden group
address@hidden smallexample
+
address@hidden
+This means that the angle is about 15 degrees, and, assuming our
+original error estimates were valid standard deviations, there is about
+a 60% chance that the result is correct within 0.59 degrees.
+
address@hidden Torus, volume of
+(@bullet{}) @strong{Exercise 7.} The volume of a torus (a donut shape) is
address@hidden @math{2 \pi^2 R r^2}
address@hidden @address@hidden pi^2 R r^2}}
+where @expr{R} is the radius of the circle that
+defines the center of the tube and @expr{r} is the radius of the tube
+itself. Suppose @expr{R} is 20 cm and @expr{r} is 4 cm, each known to
+within 5 percent. What is the volume and the relative uncertainty of
+the volume? @xref{Types Answer 7, 7}. (@bullet{})
+
+An @dfn{interval form} represents a range of values. While an
+error form is best for making statistical estimates, intervals give
+you exact bounds on an answer. Suppose we additionally know that
+our telephone pole is definitely between 28 and 31 meters away,
+and that it is between 7.7 and 8.1 meters tall.
+
address@hidden
address@hidden
+1: [7.7 .. 8.1] 2: [7.7 .. 8.1] 1: [0.24 .. 0.28] 1: [13.9 .. 16.1]
+ . 1: [28 .. 31] . .
+ .
+
+ [ 7.7 .. 8.1 ] [ 28 .. 31 ] / I T
address@hidden group
address@hidden smallexample
+
address@hidden
+If our bounds were correct, then the angle to the top of the pole
+is sure to lie in the range shown.
+
+The square brackets around these intervals indicate that the endpoints
+themselves are allowable values. In other words, the distance to the
+telephone pole is between 28 and 31, @emph{inclusive}. You can also
+make an interval that is exclusive of its endpoints by writing
+parentheses instead of square brackets. You can even make an interval
+which is inclusive (``closed'') on one end and exclusive (``open'') on
+the other.
+
address@hidden
address@hidden
+1: [1 .. 10) 1: (0.1 .. 1] 2: (0.1 .. 1] 1: (0.2 .. 3)
+ . . 1: [2 .. 3) .
+ .
+
+ [ 1 .. 10 ) & [ 2 .. 3 ) *
address@hidden group
address@hidden smallexample
+
address@hidden
+The Calculator automatically keeps track of which end values should
+be open and which should be closed. You can also make infinite or
+semi-infinite intervals by using @samp{-inf} or @samp{inf} for one
+or both endpoints.
+
+(@bullet{}) @strong{Exercise 8.} What answer would you expect from
address@hidden@w{1 /} @w{(0 .. 10)}}? What about @address@hidden /} @w{(-10 ..
0)}}? What
+about @address@hidden /} @w{[0 .. 10]}} (where the interval actually includes
+zero)? What about @address@hidden /} @w{(-10 .. 10)}}?
address@hidden Answer 8, 8}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 9.} Two easy ways of squaring a number
+are @address@hidden *} and @address@hidden ^}}. Normally these produce the
same
+answer. Would you expect this still to hold true for interval forms?
+If not, which of these will result in a larger interval?
address@hidden Answer 9, 9}. (@bullet{})
+
+A @dfn{modulo form} is used for performing arithmetic modulo @var{m}.
+For example, arithmetic involving time is generally done modulo 12
+or 24 hours.
+
address@hidden
address@hidden
+1: 17 mod 24 1: 3 mod 24 1: 21 mod 24 1: 9 mod 24
+ . . . .
+
+ 17 M 24 @key{RET} 10 + n 5 /
address@hidden group
address@hidden smallexample
+
address@hidden
+In this last step, Calc has divided by 5 modulo 24; i.e., it has found a
+new number which, when multiplied by 5 modulo 24, produces the original
+number, 21. If @var{m} is prime and the divisor is not a multiple of
address@hidden, it is always possible to find such a number. For non-prime
address@hidden like 24, it is only sometimes possible.
+
address@hidden
address@hidden
+1: 10 mod 24 1: 16 mod 24 1: 1000000... 1: 16
+ . . . .
+
+ 10 M 24 @key{RET} 100 ^ 10 @key{RET} 100 ^ 24 %
address@hidden group
address@hidden smallexample
+
address@hidden
+These two calculations get the same answer, but the first one is
+much more efficient because it avoids the huge intermediate value
+that arises in the second one.
+
address@hidden Fermat, primality test of
+(@bullet{}) @strong{Exercise 10.} A theorem of Pierre de Fermat
+says that
address@hidden @address@hidden \bmod n = 1}}
address@hidden @expr{x^(n-1) mod n = 1}
+if @expr{n} is a prime number and @expr{x} is an integer less than
address@hidden If @expr{n} is @emph{not} a prime number, this will
address@hidden be true for most values of @expr{x}. Thus we can test
+informally if a number is prime by trying this formula for several
+values of @expr{x}. Use this test to tell whether the following numbers
+are prime: 811749613, 15485863. @xref{Types Answer 10, 10}. (@bullet{})
+
+It is possible to use HMS forms as parts of error forms, intervals,
+modulo forms, or as the phase part of a polar complex number.
+For example, the @code{calc-time} command pushes the current time
+of day on the stack as an HMS/modulo form.
+
address@hidden
address@hidden
+1: 17@@ 34' 45" mod 24@@ 0' 0" 1: 6@@ 22' 15" mod 24@@ 0' 0"
+ . .
+
+ x time @key{RET} n
address@hidden group
address@hidden smallexample
+
address@hidden
+This calculation tells me it is six hours and 22 minutes until midnight.
+
+(@bullet{}) @strong{Exercise 11.} A rule of thumb is that one year
+is about
address@hidden @math{\pi \times 10^7}
address@hidden @address@hidden * 10^7}}
+seconds. What time will it be that many seconds from right now?
address@hidden Answer 11, 11}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 12.} You are preparing to order packaging
+for the CD release of the Extended Disco Version of @emph{Abbey Road}.
+You are told that the songs will actually be anywhere from 20 to 60
+seconds longer than the originals. One CD can hold about 75 minutes
+of music. Should you order single or double packages?
address@hidden Answer 12, 12}. (@bullet{})
+
+Another kind of data the Calculator can manipulate is numbers with
address@hidden This isn't strictly a new data type; it's simply an
+application of algebraic expressions, where we use variables with
+suggestive names like @samp{cm} and @samp{in} to represent units
+like centimeters and inches.
+
address@hidden
address@hidden
+1: 2 in 1: 5.08 cm 1: 0.027778 fath 1: 0.0508 m
+ . . . .
+
+ ' 2in @key{RET} u c cm @key{RET} u c fath @key{RET} u b
address@hidden group
address@hidden smallexample
+
address@hidden
+We enter the quantity ``2 inches'' (actually an algebraic expression
+which means two times the variable @samp{in}), then we convert it
+first to centimeters, then to fathoms, then finally to ``base'' units,
+which in this case means meters.
+
address@hidden
address@hidden
+1: 9 acre 1: 3 sqrt(acre) 1: 190.84 m 1: 190.84 m + 30 cm
+ . . . .
+
+ ' 9 acre @key{RET} Q u s ' $+30 cm @key{RET}
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: 191.14 m 1: 36536.3046 m^2 1: 365363046 cm^2
+ . . .
+
+ u s 2 ^ u c cgs
address@hidden group
address@hidden smallexample
+
address@hidden
+Since units expressions are really just formulas, taking the square
+root of @samp{acre} is undefined. After all, @code{acre} might be an
+algebraic variable that you will someday assign a value. We use the
+``units-simplify'' command to simplify the expression with variables
+being interpreted as unit names.
+
+In the final step, we have converted not to a particular unit, but to a
+units system. The ``cgs'' system uses centimeters instead of meters
+as its standard unit of length.
+
+There is a wide variety of units defined in the Calculator.
+
address@hidden
address@hidden
+1: 55 mph 1: 88.5139 kph 1: 88.5139 km / hr 1: 8.201407e-8 c
+ . . . .
+
+ ' 55 mph @key{RET} u c kph @key{RET} u c km/hr @key{RET}
u c c @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+We express a speed first in miles per hour, then in kilometers per
+hour, then again using a slightly more explicit notation, then
+finally in terms of fractions of the speed of light.
+
+Temperature conversions are a bit more tricky. There are two ways to
+interpret ``20 degrees Fahrenheit''---it could mean an actual
+temperature, or it could mean a change in temperature. For normal
+units there is no difference, but temperature units have an offset
+as well as a scale factor and so there must be two explicit commands
+for them.
+
address@hidden
address@hidden
+1: 20 degF 1: 11.1111 degC 1: -20:3 degC 1: -6.666 degC
+ . . . .
+
+ ' 20 degF @key{RET} u c degC @key{RET} U u t degC @key{RET}
c f
address@hidden group
address@hidden smallexample
+
address@hidden
+First we convert a change of 20 degrees Fahrenheit into an equivalent
+change in degrees Celsius (or Centigrade). Then, we convert the
+absolute temperature 20 degrees Fahrenheit into Celsius. Since
+this comes out as an exact fraction, we then convert to floating-point
+for easier comparison with the other result.
+
+For simple unit conversions, you can put a plain number on the stack.
+Then @kbd{u c} and @kbd{u t} will prompt for both old and new units.
+When you use this method, you're responsible for remembering which
+numbers are in which units:
+
address@hidden
address@hidden
+1: 55 1: 88.5139 1: 8.201407e-8
+ . . .
+
+ 55 u c mph @key{RET} kph @key{RET} u c km/hr @key{RET} c
@key{RET}
address@hidden group
address@hidden smallexample
+
+To see a complete list of built-in units, type @kbd{u v}. Press
address@hidden@kbd{C-x * c}} again to re-enter the Calculator when you're done
looking
+at the units table.
+
+(@bullet{}) @strong{Exercise 13.} How many seconds are there really
+in a year? @xref{Types Answer 13, 13}. (@bullet{})
+
address@hidden Speed of light
+(@bullet{}) @strong{Exercise 14.} Supercomputer designs are limited by
+the speed of light (and of electricity, which is nearly as fast).
+Suppose a computer has a 4.1 ns (nanosecond) clock cycle, and its
+cabinet is one meter across. Is speed of light going to be a
+significant factor in its design? @xref{Types Answer 14, 14}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 15.} Sam the Slug normally travels about
+five yards in an hour. He has obtained a supply of Power Pills; each
+Power Pill he eats doubles his speed. How many Power Pills can he
+swallow and still travel legally on most US highways?
address@hidden Answer 15, 15}. (@bullet{})
+
address@hidden Algebra Tutorial, Programming Tutorial, Types Tutorial, Tutorial
address@hidden Algebra and Calculus Tutorial
+
address@hidden
+This section shows how to use Calc's algebra facilities to solve
+equations, do simple calculus problems, and manipulate algebraic
+formulas.
+
address@hidden
+* Basic Algebra Tutorial::
+* Rewrites Tutorial::
address@hidden menu
+
address@hidden Basic Algebra Tutorial, Rewrites Tutorial, Algebra Tutorial,
Algebra Tutorial
address@hidden Basic Algebra
+
address@hidden
+If you enter a formula in Algebraic mode that refers to variables,
+the formula itself is pushed onto the stack. You can manipulate
+formulas as regular data objects.
+
address@hidden
address@hidden
+1: 2 x^2 - 6 1: 6 - 2 x^2 1: (6 - 2 x^2) (3 x^2 + y)
+ . . .
+
+ ' 2x^2-6 @key{RET} n ' 3x^2+y @key{RET} *
address@hidden group
address@hidden smallexample
+
+(@bullet{}) @strong{Exercise 1.} Do @kbd{' x @key{RET} Q 2 ^} and
address@hidden' x @key{RET} 2 ^ Q} both wind up with the same result (@samp{x})?
+Why or why not? @xref{Algebra Answer 1, 1}. (@bullet{})
+
+There are also commands for doing common algebraic operations on
+formulas. Continuing with the formula from the last example,
+
address@hidden
address@hidden
+1: 18 x^2 + 6 y - 6 x^4 - 2 x^2 y 1: (18 - 2 y) x^2 - 6 x^4 + 6 y
+ . .
+
+ a x a c x @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+First we ``expand'' using the distributive law, then we ``collect''
+terms involving like powers of @expr{x}.
+
+Let's find the value of this expression when @expr{x} is 2 and @expr{y}
+is one-half.
+
address@hidden
address@hidden
+1: 17 x^2 - 6 x^4 + 3 1: -25
+ . .
+
+ 1:2 s l y @key{RET} 2 s l x @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+The @kbd{s l} command means ``let''; it takes a number from the top of
+the stack and temporarily assigns it as the value of the variable
+you specify. It then evaluates (as if by the @kbd{=} key) the
+next expression on the stack. After this command, the variable goes
+back to its original value, if any.
+
+(An earlier exercise in this tutorial involved storing a value in the
+variable @code{x}; if this value is still there, you will have to
+unstore it with @kbd{s u x @key{RET}} before the above example will work
+properly.)
+
address@hidden Maximum of a function using Calculus
+Let's find the maximum value of our original expression when @expr{y}
+is one-half and @expr{x} ranges over all possible values. We can
+do this by taking the derivative with respect to @expr{x} and examining
+values of @expr{x} for which the derivative is zero. If the second
+derivative of the function at that value of @expr{x} is negative,
+the function has a local maximum there.
+
address@hidden
address@hidden
+1: 17 x^2 - 6 x^4 + 3 1: 34 x - 24 x^3
+ . .
+
+ U @key{DEL} s 1 a d x @key{RET} s 2
address@hidden group
address@hidden smallexample
+
address@hidden
+Well, the derivative is clearly zero when @expr{x} is zero. To find
+the other root(s), let's divide through by @expr{x} and then solve:
+
address@hidden
address@hidden
+1: (34 x - 24 x^3) / x 1: 34 x / x - 24 x^3 / x 1: 34 - 24 x^2
+ . . .
+
+ ' x @key{RET} / a x a s
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: 34 - 24 x^2 = 0 1: x = 1.19023
+ . .
+
+ 0 a = s 3 a S x @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+Notice the use of @kbd{a s} to ``simplify'' the formula. When the
+default algebraic simplifications don't do enough, you can use
address@hidden s} to tell Calc to spend more time on the job.
+
+Now we compute the second derivative and plug in our values of @expr{x}:
+
address@hidden
address@hidden
+1: 1.19023 2: 1.19023 2: 1.19023
+ . 1: 34 x - 24 x^3 1: 34 - 72 x^2
+ . .
+
+ a . r 2 a d x @key{RET} s 4
address@hidden group
address@hidden smallexample
+
address@hidden
+(The @kbd{a .} command extracts just the righthand side of an equation.
+Another method would have been to use @kbd{v u} to unpack the equation
address@hidden@samp{x = 1.19}} to @samp{x} and @samp{1.19}, then use @kbd{M--
M-2 @key{DEL}}
+to delete the @samp{x}.)
+
address@hidden
address@hidden
+2: 34 - 72 x^2 1: -68. 2: 34 - 72 x^2 1: 34
+1: 1.19023 . 1: 0 .
+ . .
+
+ @key{TAB} s l x @key{RET} U @key{DEL} 0 s
l x @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+The first of these second derivatives is negative, so we know the function
+has a maximum value at @expr{x = 1.19023}. (The function also has a
+local @emph{minimum} at @expr{x = 0}.)
+
+When we solved for @expr{x}, we got only one value even though
address@hidden - 24 x^2 = 0} is a quadratic equation that ought to have
+two solutions. The reason is that @address@hidden S}} normally returns a
+single ``principal'' solution. If it needs to come up with an
+arbitrary sign (as occurs in the quadratic formula) it picks @expr{+}.
+If it needs an arbitrary integer, it picks zero. We can get a full
+solution by pressing @kbd{H} (the Hyperbolic flag) before @kbd{a S}.
+
address@hidden
address@hidden
+1: 34 - 24 x^2 = 0 1: x = 1.19023 s1 1: x = -1.19023
+ . . .
+
+ r 3 H a S x @key{RET} s 5 1 n s l s1 @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+Calc has invented the variable @samp{s1} to represent an unknown sign;
+it is supposed to be either @mathit{+1} or @mathit{-1}. Here we have used
+the ``let'' command to evaluate the expression when the sign is negative.
+If we plugged this into our second derivative we would get the same,
+negative, answer, so @expr{x = -1.19023} is also a maximum.
+
+To find the actual maximum value, we must plug our two values of @expr{x}
+into the original formula.
+
address@hidden
address@hidden
+2: 17 x^2 - 6 x^4 + 3 1: 24.08333 s1^2 - 12.04166 s1^4 + 3
+1: x = 1.19023 s1 .
+ .
+
+ r 1 r 5 s l @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+(Here we see another way to use @kbd{s l}; if its input is an equation
+with a variable on the lefthand side, then @kbd{s l} treats the equation
+like an assignment to that variable if you don't give a variable name.)
+
+It's clear that this will have the same value for either sign of
address@hidden, but let's work it out anyway, just for the exercise:
+
address@hidden
address@hidden
+2: [-1, 1] 1: [15.04166, 15.04166]
+1: 24.08333 s1^2 ... .
+ .
+
+ [ 1 n , 1 ] @key{TAB} V M $ @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+Here we have used a vector mapping operation to evaluate the function
+at several values of @samp{s1} at once. @kbd{V M $} is like @kbd{V M '}
+except that it takes the formula from the top of the stack. The
+formula is interpreted as a function to apply across the vector at the
+next-to-top stack level. Since a formula on the stack can't contain
address@hidden signs, Calc assumes the variables in the formula stand for
+different arguments. It prompts you for an @dfn{argument list}, giving
+the list of all variables in the formula in alphabetical order as the
+default list. In this case the default is @samp{(s1)}, which is just
+what we want so we simply press @key{RET} at the prompt.
+
+If there had been several different values, we could have used
address@hidden@kbd{V R X}} to find the global maximum.
+
+Calc has a built-in @kbd{a P} command that solves an equation using
address@hidden@kbd{H a S}} and returns a vector of all the solutions. It simply
+automates the job we just did by hand. Applied to our original
+cubic polynomial, it would produce the vector of solutions
address@hidden, -1.19023, 0]}. (There is also an @kbd{a X} command
+which finds a local maximum of a function. It uses a numerical search
+method rather than examining the derivatives, and thus requires you
+to provide some kind of initial guess to show it where to look.)
+
+(@bullet{}) @strong{Exercise 2.} Given a vector of the roots of a
+polynomial (such as the output of an @kbd{a P} command), what
+sequence of commands would you use to reconstruct the original
+polynomial? (The answer will be unique to within a constant
+multiple; choose the solution where the leading coefficient is one.)
address@hidden Answer 2, 2}. (@bullet{})
+
+The @kbd{m s} command enables Symbolic mode, in which formulas
+like @samp{sqrt(5)} that can't be evaluated exactly are left in
+symbolic form rather than giving a floating-point approximate answer.
+Fraction mode (@kbd{m f}) is also useful when doing algebra.
+
address@hidden
address@hidden
+2: 34 x - 24 x^3 2: 34 x - 24 x^3
+1: 34 x - 24 x^3 1: [sqrt(51) / 6, sqrt(51) / -6, 0]
+ . .
+
+ r 2 @key{RET} m s m f a P x @key{RET}
address@hidden group
address@hidden smallexample
+
+One more mode that makes reading formulas easier is Big mode.
+
address@hidden
address@hidden
+ 3
+2: 34 x - 24 x
+
+ ____ ____
+ V 51 V 51
+1: [-----, -----, 0]
+ 6 -6
+
+ .
+
+ d B
address@hidden group
address@hidden smallexample
+
+Here things like powers, square roots, and quotients and fractions
+are displayed in a two-dimensional pictorial form. Calc has other
+language modes as well, such as C mode, FORTRAN mode, @TeX{} mode
+and address@hidden mode.
+
address@hidden
address@hidden
+2: 34*x - 24*pow(x, 3) 2: 34*x - 24*x**3
+1: @{sqrt(51) / 6, sqrt(51) / -6, address@hidden 1: /sqrt(51) / 6, sqrt(51)
/ -6, 0/
+ . .
+
+ d C d F
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+3: 34 x - 24 x^3
+2: address@hidden@address@hidden \over address@hidden, @address@hidden@}
\over address@hidden, 0]
+1: @{2 \over address@hidden address@hidden@}
+ .
+
+ d T ' 2 address@hidden@} \over 3 @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+As you can see, language modes affect both entry and display of
+formulas. They affect such things as the names used for built-in
+functions, the set of arithmetic operators and their precedences,
+and notations for vectors and matrices.
+
+Notice that @samp{sqrt(51)} may cause problems with older
+implementations of C and FORTRAN, which would require something more
+like @samp{sqrt(51.0)}. It is always wise to check over the formulas
+produced by the various language modes to make sure they are fully
+correct.
+
+Type @kbd{m s}, @kbd{m f}, and @kbd{d N} to reset these modes. (You
+may prefer to remain in Big mode, but all the examples in the tutorial
+are shown in normal mode.)
+
address@hidden Area under a curve
+What is the area under the portion of this curve from @expr{x = 1} to @expr{2}?
+This is simply the integral of the function:
+
address@hidden
address@hidden
+1: 17 x^2 - 6 x^4 + 3 1: 5.6666 x^3 - 1.2 x^5 + 3 x
+ . .
+
+ r 1 a i x
address@hidden group
address@hidden smallexample
+
address@hidden
+We want to evaluate this at our two values for @expr{x} and subtract.
+One way to do it is again with vector mapping and reduction:
+
address@hidden
address@hidden
+2: [2, 1] 1: [12.93333, 7.46666] 1: 5.46666
+1: 5.6666 x^3 ... . .
+
+ [ 2 , 1 ] @key{TAB} V M $ @key{RET} V R -
address@hidden group
address@hidden smallexample
+
+(@bullet{}) @strong{Exercise 3.} Find the integral from 1 to @expr{y}
+of
address@hidden @math{x \sin \pi x}
address@hidden @address@hidden sin(pi x)}}
+(where the sine is calculated in radians). Find the values of the
+integral for integers @expr{y} from 1 to 5. @xref{Algebra Answer 3,
+3}. (@bullet{})
+
+Calc's integrator can do many simple integrals symbolically, but many
+others are beyond its capabilities. Suppose we wish to find the area
+under the curve
address@hidden @math{\sin x \ln x}
address@hidden @expr{sin(x) ln(x)}
+over the same range of @expr{x}. If you entered this formula and typed
address@hidden i x @key{RET}} (don't bother to try this), Calc would work for a
+long time but would be unable to find a solution. In fact, there is no
+closed-form solution to this integral. Now what do we do?
+
address@hidden Integration, numerical
address@hidden Numerical integration
+One approach would be to do the integral numerically. It is not hard
+to do this by hand using vector mapping and reduction. It is rather
+slow, though, since the sine and logarithm functions take a long time.
+We can save some time by reducing the working precision.
+
address@hidden
address@hidden
+3: 10 1: [1, 1.1, 1.2, ... , 1.8, 1.9]
+2: 1 .
+1: 0.1
+ .
+
+ 10 @key{RET} 1 @key{RET} .1 @key{RET} C-u v x
address@hidden group
address@hidden smallexample
+
address@hidden
+(Note that we have used the extended version of @kbd{v x}; we could
+also have used plain @kbd{v x} as follows: @kbd{v x 10 @key{RET} 9 + .1 *}.)
+
address@hidden
address@hidden
+2: [1, 1.1, ... ] 1: [0., 0.084941, 0.16993, ... ]
+1: sin(x) ln(x) .
+ .
+
+ ' sin(x) ln(x) @key{RET} s 1 m r p 5 @key{RET} V M $ @key{RET}
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: 3.4195 0.34195
+ . .
+
+ V R + 0.1 *
address@hidden group
address@hidden smallexample
+
address@hidden
+(If you got wildly different results, did you remember to switch
+to Radians mode?)
+
+Here we have divided the curve into ten segments of equal width;
+approximating these segments as rectangular boxes (i.e., assuming
+the curve is nearly flat at that resolution), we compute the areas
+of the boxes (height times width), then sum the areas. (It is
+faster to sum first, then multiply by the width, since the width
+is the same for every box.)
+
+The true value of this integral turns out to be about 0.374, so
+we're not doing too well. Let's try another approach.
+
address@hidden
address@hidden
+1: sin(x) ln(x) 1: 0.84147 x - 0.84147 + 0.11957 (x - 1)^2 - ...
+ . .
+
+ r 1 a t x=1 @key{RET} 4 @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+Here we have computed the Taylor series expansion of the function
+about the point @expr{x=1}. We can now integrate this polynomial
+approximation, since polynomials are easy to integrate.
+
address@hidden
address@hidden
+1: 0.42074 x^2 + ... 1: [-0.0446, -0.42073] 1: 0.3761
+ . . .
+
+ a i x @key{RET} [ 2 , 1 ] @key{TAB} V M $ @key{RET} V
R -
address@hidden group
address@hidden smallexample
+
address@hidden
+Better! By increasing the precision and/or asking for more terms
+in the Taylor series, we can get a result as accurate as we like.
+(Taylor series converge better away from singularities in the
+function such as the one at @code{ln(0)}, so it would also help to
+expand the series about the points @expr{x=2} or @expr{x=1.5} instead
+of @expr{x=1}.)
+
address@hidden Simpson's rule
address@hidden Integration by Simpson's rule
+(@bullet{}) @strong{Exercise 4.} Our first method approximated the
+curve by stairsteps of width 0.1; the total area was then the sum
+of the areas of the rectangles under these stairsteps. Our second
+method approximated the function by a polynomial, which turned out
+to be a better approximation than stairsteps. A third method is
address@hidden's rule}, which is like the stairstep method except
+that the steps are not required to be flat. Simpson's rule boils
+down to the formula,
+
address@hidden
address@hidden
+(h/3) * (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + ...
+ + 2 f(a+(n-2)*h) + 4 f(a+(n-1)*h) + f(a+n*h))
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ \displaylines{
+ \qquad {h \over 3} (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + \cdots
+ \hfill \cr \hfill {} + 2 f(a+(n-2)h) + 4 f(a+(n-1)h) + f(a+n h)) \qquad
+} $$
+\afterdisplay
address@hidden tex
+
address@hidden
+where @expr{n} (which must be even) is the number of slices and @expr{h}
+is the width of each slice. These are 10 and 0.1 in our example.
+For reference, here is the corresponding formula for the stairstep
+method:
+
address@hidden
address@hidden
+h * (f(a) + f(a+h) + f(a+2h) + f(a+3h) + ...
+ + f(a+(n-2)*h) + f(a+(n-1)*h))
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ h (f(a) + f(a+h) + f(a+2h) + f(a+3h) + \cdots
+ + f(a+(n-2)h) + f(a+(n-1)h)) $$
+\afterdisplay
address@hidden tex
+
+Compute the integral from 1 to 2 of
address@hidden @math{\sin x \ln x}
address@hidden @expr{sin(x) ln(x)}
+using Simpson's rule with 10 slices.
address@hidden Answer 4, 4}. (@bullet{})
+
+Calc has a built-in @kbd{a I} command for doing numerical integration.
+It uses @dfn{Romberg's method}, which is a more sophisticated cousin
+of Simpson's rule. In particular, it knows how to keep refining the
+result until the current precision is satisfied.
+
address@hidden [fix-ref Selecting Sub-Formulas]
+Aside from the commands we've seen so far, Calc also provides a
+large set of commands for operating on parts of formulas. You
+indicate the desired sub-formula by placing the cursor on any part
+of the formula before giving a @dfn{selection} command. Selections won't
+be covered in the tutorial; @pxref{Selecting Subformulas}, for
+details and examples.
+
address@hidden hard exercise: simplify (2^(n r) - 2^(r*(n - 1))) / (2^r - 1)
2^(n - 1)
address@hidden to 2^((n-1)*(r-1)).
+
address@hidden Rewrites Tutorial, , Basic Algebra Tutorial, Algebra Tutorial
address@hidden Rewrite Rules
+
address@hidden
+No matter how many built-in commands Calc provided for doing algebra,
+there would always be something you wanted to do that Calc didn't have
+in its repertoire. So Calc also provides a @dfn{rewrite rule} system
+that you can use to define your own algebraic manipulations.
+
+Suppose we want to simplify this trigonometric formula:
+
address@hidden
address@hidden
+1: 1 / cos(x) - sin(x) tan(x)
+ .
+
+ ' 1/cos(x) - sin(x) tan(x) @key{RET} s 1
address@hidden group
address@hidden smallexample
+
address@hidden
+If we were simplifying this by hand, we'd probably replace the
address@hidden with a @samp{sin/cos} first, then combine over a common
+denominator. There is no Calc command to do the former; the @kbd{a n}
+algebra command will do the latter but we'll do both with rewrite
+rules just for practice.
+
+Rewrite rules are written with the @samp{:=} symbol.
+
address@hidden
address@hidden
+1: 1 / cos(x) - sin(x)^2 / cos(x)
+ .
+
+ a r tan(a) := sin(a)/cos(a) @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+(The ``assignment operator'' @samp{:=} has several uses in Calc. All
+by itself the formula @samp{tan(a) := sin(a)/cos(a)} doesn't do anything,
+but when it is given to the @kbd{a r} command, that command interprets
+it as a rewrite rule.)
+
+The lefthand side, @samp{tan(a)}, is called the @dfn{pattern} of the
+rewrite rule. Calc searches the formula on the stack for parts that
+match the pattern. Variables in a rewrite pattern are called
address@hidden, and when matching the pattern each meta-variable
+can match any sub-formula. Here, the meta-variable @samp{a} matched
+the actual variable @samp{x}.
+
+When the pattern part of a rewrite rule matches a part of the formula,
+that part is replaced by the righthand side with all the meta-variables
+substituted with the things they matched. So the result is
address@hidden(x) / cos(x)}. Calc's normal algebraic simplifications then
+mix this in with the rest of the original formula.
+
+To merge over a common denominator, we can use another simple rule:
+
address@hidden
address@hidden
+1: (1 - sin(x)^2) / cos(x)
+ .
+
+ a r a/x + b/x := (a+b)/x @key{RET}
address@hidden group
address@hidden smallexample
+
+This rule points out several interesting features of rewrite patterns.
+First, if a meta-variable appears several times in a pattern, it must
+match the same thing everywhere. This rule detects common denominators
+because the same meta-variable @samp{x} is used in both of the
+denominators.
+
+Second, meta-variable names are independent from variables in the
+target formula. Notice that the meta-variable @samp{x} here matches
+the subformula @samp{cos(x)}; Calc never confuses the two meanings of
address@hidden
+
+And third, rewrite patterns know a little bit about the algebraic
+properties of formulas. The pattern called for a sum of two quotients;
+Calc was able to match a difference of two quotients by matching
address@hidden = 1}, @samp{b = -sin(x)^2}, and @samp{x = cos(x)}.
+
address@hidden [fix-ref Algebraic Properties of Rewrite Rules]
+We could just as easily have written @samp{a/x - b/x := (a-b)/x} for
+the rule. It would have worked just the same in all cases. (If we
+really wanted the rule to apply only to @samp{+} or only to @samp{-},
+we could have used the @code{plain} symbol. @xref{Algebraic Properties
+of Rewrite Rules}, for some examples of this.)
+
+One more rewrite will complete the job. We want to use the identity
address@hidden(x)^2 + cos(x)^2 = 1}, but of course we must first rearrange
+the identity in a way that matches our formula. The obvious rule
+would be @address@hidden - sin(x)^2} := cos(x)^2}, but a little thought shows
+that the rule @samp{sin(x)^2 := 1 - cos(x)^2} will also work. The
+latter rule has a more general pattern so it will work in many other
+situations, too.
+
address@hidden
address@hidden
+1: (1 + cos(x)^2 - 1) / cos(x) 1: cos(x)
+ . .
+
+ a r sin(x)^2 := 1 - cos(x)^2 @key{RET} a s
address@hidden group
address@hidden smallexample
+
+You may ask, what's the point of using the most general rule if you
+have to type it in every time anyway? The answer is that Calc allows
+you to store a rewrite rule in a variable, then give the variable
+name in the @kbd{a r} command. In fact, this is the preferred way to
+use rewrites. For one, if you need a rule once you'll most likely
+need it again later. Also, if the rule doesn't work quite right you
+can simply Undo, edit the variable, and run the rule again without
+having to retype it.
+
address@hidden
address@hidden
+' tan(x) := sin(x)/cos(x) @key{RET} s t tsc @key{RET}
+' a/x + b/x := (a+b)/x @key{RET} s t merge @key{RET}
+' sin(x)^2 := 1 - cos(x)^2 @key{RET} s t sinsqr @key{RET}
+
+1: 1 / cos(x) - sin(x) tan(x) 1: cos(x)
+ . .
+
+ r 1 a r tsc @key{RET} a r merge @key{RET} a r sinsqr
@key{RET} a s
address@hidden group
address@hidden smallexample
+
+To edit a variable, type @kbd{s e} and the variable name, use regular
+Emacs editing commands as necessary, then type @kbd{C-c C-c} to store
+the edited value back into the variable.
+You can also use @address@hidden e}} to create a new variable if you wish.
+
+Notice that the first time you use each rule, Calc puts up a ``compiling''
+message briefly. The pattern matcher converts rules into a special
+optimized pattern-matching language rather than using them directly.
+This allows @kbd{a r} to apply even rather complicated rules very
+efficiently. If the rule is stored in a variable, Calc compiles it
+only once and stores the compiled form along with the variable. That's
+another good reason to store your rules in variables rather than
+entering them on the fly.
+
+(@bullet{}) @strong{Exercise 1.} Type @kbd{m s} to get Symbolic
+mode, then enter the formula @address@hidden(2 + sqrt(2))} / @w{(1 +
sqrt(2))}}.
+Using a rewrite rule, simplify this formula by multiplying the top and
+bottom by the conjugate @address@hidden - sqrt(2)}}. The result will have
+to be expanded by the distributive law; do this with another
+rewrite. @xref{Rewrites Answer 1, 1}. (@bullet{})
+
+The @kbd{a r} command can also accept a vector of rewrite rules, or
+a variable containing a vector of rules.
+
address@hidden
address@hidden
+1: [tsc, merge, sinsqr] 1: [tan(x) := sin(x) / cos(x), ... ]
+ . .
+
+ ' [tsc,merge,sinsqr] @key{RET} =
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: 1 / cos(x) - sin(x) tan(x) 1: cos(x)
+ . .
+
+ s t trig @key{RET} r 1 a r trig @key{RET} a s
address@hidden group
address@hidden smallexample
+
address@hidden [fix-ref Nested Formulas with Rewrite Rules]
+Calc tries all the rules you give against all parts of the formula,
+repeating until no further change is possible. (The exact order in
+which things are tried is rather complex, but for simple rules like
+the ones we've used here the order doesn't really matter.
address@hidden Formulas with Rewrite Rules}.)
+
+Calc actually repeats only up to 100 times, just in case your rule set
+has gotten into an infinite loop. You can give a numeric prefix argument
+to @kbd{a r} to specify any limit. In particular, @kbd{M-1 a r} does
+only one rewrite at a time.
+
address@hidden
address@hidden
+1: 1 / cos(x) - sin(x)^2 / cos(x) 1: (1 - sin(x)^2) / cos(x)
+ . .
+
+ r 1 M-1 a r trig @key{RET} M-1 a r trig @key{RET}
address@hidden group
address@hidden smallexample
+
+You can type @kbd{M-0 a r} if you want no limit at all on the number
+of rewrites that occur.
+
+Rewrite rules can also be @dfn{conditional}. Simply follow the rule
+with a @samp{::} symbol and the desired condition. For example,
+
address@hidden
address@hidden
+1: exp(2 pi i) + exp(3 pi i) + exp(4 pi i)
+ .
+
+ ' exp(2 pi i) + exp(3 pi i) + exp(4 pi i) @key{RET}
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: 1 + exp(3 pi i) + 1
+ .
+
+ a r exp(k pi i) := 1 :: k % 2 = 0 @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+(Recall, @samp{k % 2} is the remainder from dividing @samp{k} by 2,
+which will be zero only when @samp{k} is an even integer.)
+
+An interesting point is that the variables @samp{pi} and @samp{i}
+were matched literally rather than acting as meta-variables.
+This is because they are special-constant variables. The special
+constants @samp{e}, @samp{phi}, and so on also match literally.
+A common error with rewrite
+rules is to write, say, @samp{f(a,b,c,d,e) := g(a+b+c+d+e)}, expecting
+to match any @samp{f} with five arguments but in fact matching
+only when the fifth argument is literally @samp{e}!
+
address@hidden Fibonacci numbers
address@hidden
address@hidden
address@hidden ignore
address@hidden fib
+Rewrite rules provide an interesting way to define your own functions.
+Suppose we want to define @samp{fib(n)} to produce the @var{n}th
+Fibonacci number. The first two Fibonacci numbers are each 1;
+later numbers are formed by summing the two preceding numbers in
+the sequence. This is easy to express in a set of three rules:
+
address@hidden
address@hidden
+' [fib(1) := 1, fib(2) := 1, fib(n) := fib(n-1) + fib(n-2)] @key{RET} s t fib
+
+1: fib(7) 1: 13
+ . .
+
+ ' fib(7) @key{RET} a r fib @key{RET}
address@hidden group
address@hidden smallexample
+
+One thing that is guaranteed about the order that rewrites are tried
+is that, for any given subformula, earlier rules in the rule set will
+be tried for that subformula before later ones. So even though the
+first and third rules both match @samp{fib(1)}, we know the first will
+be used preferentially.
+
+This rule set has one dangerous bug: Suppose we apply it to the
+formula @samp{fib(x)}? (Don't actually try this.) The third rule
+will match @samp{fib(x)} and replace it with @address@hidden(x-1) + fib(x-2)}}.
+Each of these will then be replaced to get @samp{fib(x-2) + 2 fib(x-3) +
+fib(x-4)}, and so on, expanding forever. What we really want is to apply
+the third rule only when @samp{n} is an integer greater than two. Type
address@hidden@kbd{s e fib @key{RET}}}, then edit the third rule to:
+
address@hidden
+fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2
address@hidden smallexample
+
address@hidden
+Now:
+
address@hidden
address@hidden
+1: fib(6) + fib(x) + fib(0) 1: 8 + fib(x) + fib(0)
+ . .
+
+ ' fib(6)+fib(x)+fib(0) @key{RET} a r fib @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+We've created a new function, @code{fib}, and a new command,
address@hidden@kbd{a r fib @key{RET}}}, which means ``evaluate all @code{fib}
calls in
+this formula.'' To make things easier still, we can tell Calc to
+apply these rules automatically by storing them in the special
+variable @code{EvalRules}.
+
address@hidden
address@hidden
+1: [fib(1) := ...] . 1: [8, 13]
+ . .
+
+ s r fib @key{RET} s t EvalRules @key{RET} ' [fib(6), fib(7)]
@key{RET}
address@hidden group
address@hidden smallexample
+
+It turns out that this rule set has the problem that it does far
+more work than it needs to when @samp{n} is large. Consider the
+first few steps of the computation of @samp{fib(6)}:
+
address@hidden
address@hidden
+fib(6) =
+fib(5) + fib(4) =
+fib(4) + fib(3) + fib(3) + fib(2) =
+fib(3) + fib(2) + fib(2) + fib(1) + fib(2) + fib(1) + 1 = ...
address@hidden group
address@hidden smallexample
+
address@hidden
+Note that @samp{fib(3)} appears three times here. Unless Calc's
+algebraic simplifier notices the multiple @samp{fib(3)}s and combines
+them (and, as it happens, it doesn't), this rule set does lots of
+needless recomputation. To cure the problem, type @code{s e EvalRules}
+to edit the rules (or just @kbd{s E}, a shorthand command for editing
address@hidden) and add another condition:
+
address@hidden
+fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2 :: remember
address@hidden smallexample
+
address@hidden
+If a @samp{:: remember} condition appears anywhere in a rule, then if
+that rule succeeds Calc will add another rule that describes that match
+to the front of the rule set. (Remembering works in any rule set, but
+for technical reasons it is most effective in @code{EvalRules}.) For
+example, if the rule rewrites @samp{fib(7)} to something that evaluates
+to 13, then the rule @samp{fib(7) := 13} will be added to the rule set.
+
+Type @kbd{' fib(8) @key{RET}} to compute the eighth Fibonacci number, then
+type @kbd{s E} again to see what has happened to the rule set.
+
+With the @code{remember} feature, our rule set can now compute
address@hidden(@var{n})} in just @var{n} steps. In the process it builds
+up a table of all Fibonacci numbers up to @var{n}. After we have
+computed the result for a particular @var{n}, we can get it back
+(and the results for all smaller @var{n}) later in just one step.
+
+All Calc operations will run somewhat slower whenever @code{EvalRules}
+contains any rules. You should type @kbd{s u EvalRules @key{RET}} now to
+un-store the variable.
+
+(@bullet{}) @strong{Exercise 2.} Sometimes it is possible to reformulate
+a problem to reduce the amount of recursion necessary to solve it.
+Create a rule that, in about @var{n} simple steps and without recourse
+to the @code{remember} option, replaces @samp{fib(@var{n}, 1, 1)} with
address@hidden(1, @var{x}, @var{y})} where @var{x} and @var{y} are the
address@hidden and @var{n+1}st Fibonacci numbers, respectively. This rule is
+rather clunky to use, so add a couple more rules to make the ``user
+interface'' the same as for our first version: enter @samp{fib(@var{n})},
+get back a plain number. @xref{Rewrites Answer 2, 2}. (@bullet{})
+
+There are many more things that rewrites can do. For example, there
+are @samp{&&&} and @samp{|||} pattern operators that create ``and''
+and ``or'' combinations of rules. As one really simple example, we
+could combine our first two Fibonacci rules thusly:
+
address@hidden
+[fib(1 ||| 2) := 1, fib(n) := ... ]
address@hidden example
+
address@hidden
+That means address@hidden of something matching either 1 or 2 rewrites
+to 1.''
+
+You can also make meta-variables optional by enclosing them in @code{opt}.
+For example, the pattern @samp{a + b x} matches @samp{2 + 3 x} but not
address@hidden + x} or @samp{3 x} or @samp{x}. The pattern @samp{opt(a) +
opt(b) x}
+matches all of these forms, filling in a default of zero for @samp{a}
+and one for @samp{b}.
+
+(@bullet{}) @strong{Exercise 3.} Your friend Joe had @samp{2 + 3 x}
+on the stack and tried to use the rule
address@hidden(a) + opt(b) x := f(a, b, x)}. What happened?
address@hidden Answer 3, 3}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 4.} Starting with a positive integer @expr{a},
+divide @expr{a} by two if it is even, otherwise compute @expr{3 a + 1}.
+Now repeat this step over and over. A famous unproved conjecture
+is that for any starting @expr{a}, the sequence always eventually
+reaches 1. Given the formula @samp{seq(@var{a}, 0)}, write a set of
+rules that convert this into @samp{seq(1, @var{n})} where @var{n}
+is the number of steps it took the sequence to reach the value 1.
+Now enhance the rules to accept @samp{seq(@var{a})} as a starting
+configuration, and to stop with just the number @var{n} by itself.
+Now make the result be a vector of values in the sequence, from @var{a}
+to 1. (The formula @address@hidden|@var{y}} appends the vectors @var{x}
+and @var{y}.) For example, rewriting @samp{seq(6)} should yield the
+vector @expr{[6, 3, 10, 5, 16, 8, 4, 2, 1]}.
address@hidden Answer 4, 4}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 5.} Define, using rewrite rules, a function
address@hidden(@var{x})} that returns the number of terms in the sum
address@hidden, or 1 if @var{x} is not a sum. (A @dfn{sum} for our purposes
+is one or more non-sum terms separated by @samp{+} or @samp{-} signs,
+so that @expr{2 - 3 (x + y) + x y} is a sum of three terms.)
address@hidden Answer 5, 5}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 6.} A Taylor series for a function is an
+infinite series that exactly equals the value of that function at
+values of @expr{x} near zero.
+
address@hidden
address@hidden
+cos(x) = 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ...
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ \cos x = 1 - {x^2 \over 2!} + {x^4 \over 4!} - {x^6 \over 6!} + \cdots $$
+\afterdisplay
address@hidden tex
+
+The @kbd{a t} command produces a @dfn{truncated Taylor series} which
+is obtained by dropping all the terms higher than, say, @expr{x^2}.
+Calc represents the truncated Taylor series as a polynomial in @expr{x}.
+Mathematicians often write a truncated series using a ``big-O'' notation
+that records what was the lowest term that was truncated.
+
address@hidden
address@hidden
+cos(x) = 1 - x^2 / 2! + O(x^3)
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ \cos x = 1 - {x^2 \over 2!} + O(x^3) $$
+\afterdisplay
address@hidden tex
+
address@hidden
+The meaning of @expr{O(x^3)} is ``a quantity which is negligibly small
+if @expr{x^3} is considered negligibly small as @expr{x} goes to zero.''
+
+The exercise is to create rewrite rules that simplify sums and products of
+power series represented as @address@hidden + O(@address@hidden)}.
+For example, given @samp{1 - x^2 / 2 + O(x^3)} and @samp{x - x^3 / 6 + O(x^4)}
+on the stack, we want to be able to type @kbd{*} and get the result
address@hidden - 2:3 x^3 + O(x^4)}. Don't worry if the terms of the sum are
+rearranged or if @kbd{a s} needs to be typed after rewriting. (This one
+is rather tricky; the solution at the end of this chapter uses 6 rewrite
+rules. Hint: The @samp{constant(x)} condition tests whether @samp{x} is
+a number.) @xref{Rewrites Answer 6, 6}. (@bullet{})
+
+Just for kicks, try adding the rule @code{2+3 := 6} to @code{EvalRules}.
+What happens? (Be sure to remove this rule afterward, or you might get
+a nasty surprise when you use Calc to balance your checkbook!)
+
address@hidden Rules}, for the whole story on rewrite rules.
+
address@hidden Programming Tutorial, Answers to Exercises, Algebra Tutorial,
Tutorial
address@hidden Programming Tutorial
+
address@hidden
+The Calculator is written entirely in Emacs Lisp, a highly extensible
+language. If you know Lisp, you can program the Calculator to do
+anything you like. Rewrite rules also work as a powerful programming
+system. But Lisp and rewrite rules take a while to master, and often
+all you want to do is define a new function or repeat a command a few
+times. Calc has features that allow you to do these things easily.
+
+One very limited form of programming is defining your own functions.
+Calc's @kbd{Z F} command allows you to define a function name and
+key sequence to correspond to any formula. Programming commands use
+the address@hidden prefix; the user commands they create use the lower
+case @kbd{z} prefix.
+
address@hidden
address@hidden
+1: 1 + x + x^2 / 2 + x^3 / 6 1: 1 + x + x^2 / 2 + x^3 / 6
+ . .
+
+ ' 1 + x + x^2/2! + x^3/3! @key{RET} Z F e myexp @key{RET}
@key{RET} @key{RET} y
address@hidden group
address@hidden smallexample
+
+This polynomial is a Taylor series approximation to @samp{exp(x)}.
+The @kbd{Z F} command asks a number of questions. The above answers
+say that the key sequence for our function should be @kbd{z e}; the
address@hidden equivalent should be @code{calc-myexp}; the name of the
+function in algebraic formulas should also be @code{myexp}; the
+default argument list @samp{(x)} is acceptable; and finally @kbd{y}
+answers the question ``leave it in symbolic form for non-constant
+arguments?''
+
address@hidden
address@hidden
+1: 1.3495 2: 1.3495 3: 1.3495
+ . 1: 1.34986 2: 1.34986
+ . 1: myexp(a + 1)
+ .
+
+ .3 z e .3 E ' a+1 @key{RET} z e
address@hidden group
address@hidden smallexample
+
address@hidden
+First we call our new @code{exp} approximation with 0.3 as an
+argument, and compare it with the true @code{exp} function. Then
+we note that, as requested, if we try to give @kbd{z e} an
+argument that isn't a plain number, it leaves the @code{myexp}
+function call in symbolic form. If we had answered @kbd{n} to the
+final question, @samp{myexp(a + 1)} would have evaluated by plugging
+in @samp{a + 1} for @samp{x} in the defining formula.
+
address@hidden Sine integral Si(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden Si
+(@bullet{}) @strong{Exercise 1.} The ``sine integral'' function
address@hidden @math{{\rm Si}(x)}
address@hidden @expr{Si(x)}
+is defined as the integral of @samp{sin(t)/t} for
address@hidden = 0} to @expr{x} in radians. (It was invented because this
+integral has no solution in terms of basic functions; if you give it
+to Calc's @kbd{a i} command, it will ponder it for a long time and then
+give up.) We can use the numerical integration command, however,
+which in algebraic notation is written like @samp{ninteg(f(t), t, 0, x)}
+with any integrand @samp{f(t)}. Define a @kbd{z s} command and
address@hidden function that implement this. You will need to edit the
+default argument list a bit. As a test, @samp{Si(1)} should return
+0.946083. (If you don't get this answer, you might want to check that
+Calc is in Radians mode. Also, @code{ninteg} will run a lot faster if
+you reduce the precision to, say, six digits beforehand.)
address@hidden Answer 1, 1}. (@bullet{})
+
+The simplest way to do real ``programming'' of Emacs is to define a
address@hidden macro}. A keyboard macro is simply a sequence of
+keystrokes which Emacs has stored away and can play back on demand.
+For example, if you find yourself typing @kbd{H a S x @key{RET}} often,
+you may wish to program a keyboard macro to type this for you.
+
address@hidden
address@hidden
+1: y = sqrt(x) 1: x = y^2
+ . .
+
+ ' y=sqrt(x) @key{RET} C-x ( H a S x @key{RET} C-x )
+
+1: y = cos(x) 1: x = s1 arccos(y) + 2 pi n1
+ . .
+
+ ' y=cos(x) @key{RET} X
address@hidden group
address@hidden smallexample
+
address@hidden
+When you type @kbd{C-x (}, Emacs begins recording. But it is also
+still ready to execute your keystrokes, so you're really ``training''
+Emacs by walking it through the procedure once. When you type
address@hidden@kbd{C-x )}}, the macro is recorded. You can now type @kbd{X} to
+re-execute the same keystrokes.
+
+You can give a name to your macro by typing @kbd{Z K}.
+
address@hidden
address@hidden
+1: . 1: y = x^4 1: x = s2 sqrt(s1 sqrt(y))
+ . .
+
+ Z K x @key{RET} ' y=x^4 @key{RET} z x
address@hidden group
address@hidden smallexample
+
address@hidden
+Notice that we use address@hidden to define the command, and lower-case
address@hidden to call it up.
+
+Keyboard macros can call other macros.
+
address@hidden
address@hidden
+1: abs(x) 1: x = s1 y 1: 2 / x 1: x = 2 / y
+ . . . .
+
+ ' abs(x) @key{RET} C-x ( ' y @key{RET} a = z x C-x ) ' 2/x @key{RET}
X
address@hidden group
address@hidden smallexample
+
+(@bullet{}) @strong{Exercise 2.} Define a keyboard macro to negate
+the item in level 3 of the stack, without disturbing the rest of
+the stack. @xref{Programming Answer 2, 2}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 3.} Define keyboard macros to compute
+the following functions:
+
address@hidden
address@hidden
+Compute
address@hidden @math{\displaystyle{\sin x \over x}},
address@hidden @expr{sin(x) / x},
+where @expr{x} is the number on the top of the stack.
+
address@hidden
+Compute the address@hidden logarithm, just like the @kbd{B} key except
+the arguments are taken in the opposite order.
+
address@hidden
+Produce a vector of integers from 1 to the integer on the top of
+the stack.
address@hidden enumerate
address@hidden
address@hidden Answer 3, 3}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 4.} Define a keyboard macro to compute
+the average (mean) value of a list of numbers.
address@hidden Answer 4, 4}. (@bullet{})
+
+In many programs, some of the steps must execute several times.
+Calc has @dfn{looping} commands that allow this. Loops are useful
+inside keyboard macros, but actually work at any time.
+
address@hidden
address@hidden
+1: x^6 2: x^6 1: 360 x^2
+ . 1: 4 .
+ .
+
+ ' x^6 @key{RET} 4 Z < a d x @key{RET} Z >
address@hidden group
address@hidden smallexample
+
address@hidden
+Here we have computed the fourth derivative of @expr{x^6} by
+enclosing a derivative command in a ``repeat loop'' structure.
+This structure pops a repeat count from the stack, then
+executes the body of the loop that many times.
+
+If you make a mistake while entering the body of the loop,
+type @address@hidden C-g}} to cancel the loop command.
+
address@hidden Fibonacci numbers
+Here's another example:
+
address@hidden
address@hidden
+3: 1 2: 10946
+2: 1 1: 17711
+1: 20 .
+ .
+
+1 @key{RET} @key{RET} 20 Z < @key{TAB} C-j + Z >
address@hidden group
address@hidden smallexample
+
address@hidden
+The numbers in levels 2 and 1 should be the 21st and 22nd Fibonacci
+numbers, respectively. (To see what's going on, try a few repetitions
+of the loop body by hand; @kbd{C-j}, also on the Line-Feed or @key{LFD}
+key if you have one, makes a copy of the number in level 2.)
+
address@hidden Golden ratio
address@hidden Phi, golden ratio
+A fascinating property of the Fibonacci numbers is that the @expr{n}th
+Fibonacci number can be found directly by computing
address@hidden @math{\phi^n / \sqrt{5}}
address@hidden @expr{phi^n / sqrt(5)}
+and then rounding to the nearest integer, where
address@hidden @math{\phi} (``phi''),
address@hidden @expr{phi},
+the ``golden ratio,'' is
address@hidden @math{(1 + \sqrt{5}) / 2}.
address@hidden @expr{(1 + sqrt(5)) / 2}.
+(For convenience, this constant is available from the @code{phi}
+variable, or the @kbd{I H P} command.)
+
address@hidden
address@hidden
+1: 1.61803 1: 24476.0000409 1: 10945.9999817 1: 10946
+ . . . .
+
+ I H P 21 ^ 5 Q / R
address@hidden group
address@hidden smallexample
+
address@hidden Continued fractions
+(@bullet{}) @strong{Exercise 5.} The @dfn{continued fraction}
+representation of
address@hidden @math{\phi}
address@hidden @expr{phi}
+is
address@hidden @math{1 + 1/(1 + 1/(1 + 1/( \ldots )))}.
address@hidden @expr{1 + 1/(1 + 1/(1 + 1/( ...@: )))}.
+We can compute an approximate value by carrying this however far
+and then replacing the innermost
address@hidden @math{1/( \ldots )}
address@hidden @expr{1/( ...@: )}
+by 1. Approximate
address@hidden @math{\phi}
address@hidden @expr{phi}
+using a twenty-term continued fraction.
address@hidden Answer 5, 5}. (@bullet{})
+
+(@bullet{}) @strong{Exercise 6.} Linear recurrences like the one for
+Fibonacci numbers can be expressed in terms of matrices. Given a
+vector @address@hidden, b]}} determine a matrix which, when multiplied by this
+vector, produces the vector @expr{[b, c]}, where @expr{a}, @expr{b} and
address@hidden are three successive Fibonacci numbers. Now write a program
+that, given an integer @expr{n}, computes the @expr{n}th Fibonacci number
+using matrix arithmetic. @xref{Programming Answer 6, 6}. (@bullet{})
+
address@hidden Harmonic numbers
+A more sophisticated kind of loop is the @dfn{for} loop. Suppose
+we wish to compute the 20th ``harmonic'' number, which is equal to
+the sum of the reciprocals of the integers from 1 to 20.
+
address@hidden
address@hidden
+3: 0 1: 3.597739
+2: 1 .
+1: 20
+ .
+
+0 @key{RET} 1 @key{RET} 20 Z ( & + 1 Z )
address@hidden group
address@hidden smallexample
+
address@hidden
+The ``for'' loop pops two numbers, the lower and upper limits, then
+repeats the body of the loop as an internal counter increases from
+the lower limit to the upper one. Just before executing the loop
+body, it pushes the current loop counter. When the loop body
+finishes, it pops the ``step,'' i.e., the amount by which to
+increment the loop counter. As you can see, our loop always
+uses a step of one.
+
+This harmonic number function uses the stack to hold the running
+total as well as for the various loop housekeeping functions. If
+you find this disorienting, you can sum in a variable instead:
+
address@hidden
address@hidden
+1: 0 2: 1 . 1: 3.597739
+ . 1: 20 .
+ .
+
+ 0 t 7 1 @key{RET} 20 Z ( & s + 7 1 Z ) r 7
address@hidden group
address@hidden smallexample
+
address@hidden
+The @kbd{s +} command adds the top-of-stack into the value in a
+variable (and removes that value from the stack).
+
+It's worth noting that many jobs that call for a ``for'' loop can
+also be done more easily by Calc's high-level operations. Two
+other ways to compute harmonic numbers are to use vector mapping
+and reduction (@kbd{v x 20}, then @address@hidden M &}}, then @kbd{V R +}),
+or to use the summation command @kbd{a +}. Both of these are
+probably easier than using loops. However, there are some
+situations where loops really are the way to go:
+
+(@bullet{}) @strong{Exercise 7.} Use a ``for'' loop to find the first
+harmonic number which is greater than 4.0.
address@hidden Answer 7, 7}. (@bullet{})
+
+Of course, if we're going to be using variables in our programs,
+we have to worry about the programs clobbering values that the
+caller was keeping in those same variables. This is easy to
+fix, though:
+
address@hidden
address@hidden
+ . 1: 0.6667 1: 0.6667 3: 0.6667
+ . . 2: 3.597739
+ 1: 0.6667
+ .
+
+ Z ` p 4 @key{RET} 2 @key{RET} 3 / s 7 s s a @key{RET} Z ' r 7 s r
a @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+When we type @kbd{Z `} (that's a back-quote character), Calc saves
+its mode settings and the contents of the ten ``quick variables''
+for later reference. When we type @kbd{Z '} (that's an apostrophe
+now), Calc restores those saved values. Thus the @kbd{p 4} and
address@hidden 7} commands have no effect outside this sequence. Wrapping
+this around the body of a keyboard macro ensures that it doesn't
+interfere with what the user of the macro was doing. Notice that
+the contents of the stack, and the values of named variables,
+survive past the @kbd{Z '} command.
+
address@hidden Bernoulli numbers, approximate
+The @dfn{Bernoulli numbers} are a sequence with the interesting
+property that all of the odd Bernoulli numbers are zero, and the
+even ones, while difficult to compute, can be roughly approximated
+by the formula
address@hidden @math{\displaystyle{2 n! \over (2 \pi)^n}}.
address@hidden @expr{2 n!@: / (2 pi)^n}.
+Let's write a keyboard macro to compute (approximate) Bernoulli numbers.
+(Calc has a command, @kbd{k b}, to compute exact Bernoulli numbers, but
+this command is very slow for large @expr{n} since the higher Bernoulli
+numbers are very large fractions.)
+
address@hidden
address@hidden
+1: 10 1: 0.0756823
+ . .
+
+ 10 C-x ( @key{RET} 2 % Z [ @key{DEL} 0 Z : ' 2 $! / (2 pi)^$ @key{RET}
= Z ] C-x )
address@hidden group
address@hidden smallexample
+
address@hidden
+You can read @kbd{Z [} as ``then,'' @kbd{Z :} as ``else,'' and
address@hidden ]} as ``end-if.'' There is no need for an explicit ``if''
+command. For the purposes of @address@hidden [}}, the condition is ``true''
+if the value it pops from the stack is a nonzero number, or ``false''
+if it pops zero or something that is not a number (like a formula).
+Here we take our integer argument modulo 2; this will be nonzero
+if we're asking for an odd Bernoulli number.
+
+The actual tenth Bernoulli number is @expr{5/66}.
+
address@hidden
address@hidden
+3: 0.0756823 1: 0 1: 0.25305 1: 0 1: 1.16659
+2: 5:66 . . . .
+1: 0.0757575
+ .
+
+10 k b @key{RET} c f M-0 @key{DEL} 11 X @key{DEL} 12 X @key{DEL} 13
X @key{DEL} 14 X
address@hidden group
address@hidden smallexample
+
+Just to exercise loops a bit more, let's compute a table of even
+Bernoulli numbers.
+
address@hidden
address@hidden
+3: [] 1: [0.10132, 0.03079, 0.02340, 0.033197, ...]
+2: 2 .
+1: 30
+ .
+
+ [ ] 2 @key{RET} 30 Z ( X | 2 Z )
address@hidden group
address@hidden smallexample
+
address@hidden
+The vertical-bar @kbd{|} is the vector-concatenation command. When
+we execute it, the list we are building will be in stack level 2
+(initially this is an empty list), and the next Bernoulli number
+will be in level 1. The effect is to append the Bernoulli number
+onto the end of the list. (To create a table of exact fractional
+Bernoulli numbers, just replace @kbd{X} with @kbd{k b} in the above
+sequence of keystrokes.)
+
+With loops and conditionals, you can program essentially anything
+in Calc. One other command that makes looping easier is @kbd{Z /},
+which takes a condition from the stack and breaks out of the enclosing
+loop if the condition is true (non-zero). You can use this to make
+``while'' and ``until'' style loops.
+
+If you make a mistake when entering a keyboard macro, you can edit
+it using @kbd{Z E}. First, you must attach it to a key with @kbd{Z K}.
+One technique is to enter a throwaway dummy definition for the macro,
+then enter the real one in the edit command.
+
address@hidden
address@hidden
+1: 3 1: 3 Calc Macro Edit Mode.
+ . . Original keys: 1 <return> 2 +
+
+ 1 ;; calc
digits
+ RET ;;
calc-enter
+ 2 ;; calc
digits
+ + ;; calc-plus
+
+C-x ( 1 @key{RET} 2 + C-x ) Z K h @key{RET} Z E h
address@hidden group
address@hidden smallexample
+
address@hidden
+A keyboard macro is stored as a pure keystroke sequence. The
address@hidden package (invoked by @kbd{Z E}) scans along the
+macro and tries to decode it back into human-readable steps.
+Descriptions of the keystrokes are given as comments, which begin with
address@hidden;;}, and which are ignored when the edited macro is saved.
+Spaces and line breaks are also ignored when the edited macro is saved.
+To enter a space into the macro, type @code{SPC}. All the special
+characters @code{RET}, @code{LFD}, @code{TAB}, @code{SPC}, @code{DEL},
+and @code{NUL} must be written in all uppercase, as must the prefixes
address@hidden and @code{M-}.
+
+Let's edit in a new definition, for computing harmonic numbers.
+First, erase the four lines of the old definition. Then, type
+in the new definition (or use Emacs @kbd{M-w} and @kbd{C-y} commands
+to copy it from this page of the Info file; you can of course skip
+typing the comments, which begin with @samp{;;}).
+
address@hidden
+Z` ;; calc-kbd-push (Save local values)
+0 ;; calc digits (Push a zero onto the stack)
+st ;; calc-store-into (Store it in the following
variable)
+1 ;; calc quick variable (Quick variable q1)
+1 ;; calc digits (Initial value for the loop)
+TAB ;; calc-roll-down (Swap initial and final)
+Z( ;; calc-kbd-for (Begin the "for" loop)
+& ;; calc-inv (Take the reciprocal)
+s+ ;; calc-store-plus (Add to the following variable)
+1 ;; calc quick variable (Quick variable q1)
+1 ;; calc digits (The loop step is 1)
+Z) ;; calc-kbd-end-for (End the "for" loop)
+sr ;; calc-recall (Recall the final accumulated
value)
+1 ;; calc quick variable (Quick variable q1)
+Z' ;; calc-kbd-pop (Restore values)
address@hidden smallexample
+
address@hidden
+Press @kbd{C-c C-c} to finish editing and return to the Calculator.
+
address@hidden
address@hidden
+1: 20 1: 3.597739
+ . .
+
+ 20 z h
address@hidden group
address@hidden smallexample
+
+The @file{edmacro} package defines a handy @code{read-kbd-macro} command
+which reads the current region of the current buffer as a sequence of
+keystroke names, and defines that sequence on the @kbd{X}
+(and @kbd{C-x e}) key. Because this is so useful, Calc puts this
+command on the @kbd{C-x * m} key. Try reading in this macro in the
+following form: Press @kbd{C-@@} (or @address@hidden) at
+one end of the text below, then type @kbd{C-x * m} at the other.
+
address@hidden
address@hidden
+Z ` 0 t 1
+ 1 TAB
+ Z ( & s + 1 1 Z )
+ r 1
+Z '
address@hidden group
address@hidden example
+
+(@bullet{}) @strong{Exercise 8.} A general algorithm for solving
+equations numerically is @dfn{Newton's Method}. Given the equation
address@hidden(x) = 0} for any function @expr{f}, and an initial guess
address@hidden which is reasonably close to the desired solution, apply
+this formula over and over:
+
address@hidden
address@hidden
+new_x = x - f(x)/f'(x)
address@hidden example
address@hidden ifnottex
address@hidden
+\beforedisplay
+$$ x_{\rm new} = x - {f(x) \over f'(x)} $$
+\afterdisplay
address@hidden tex
+
address@hidden
+where @expr{f'(x)} is the derivative of @expr{f}. The @expr{x}
+values will quickly converge to a solution, i.e., eventually
address@hidden @math{x_{\rm new}}
address@hidden @expr{new_x}
+and @expr{x} will be equal to within the limits
+of the current precision. Write a program which takes a formula
+involving the variable @expr{x}, and an initial guess @expr{x_0},
+on the stack, and produces a value of @expr{x} for which the formula
+is zero. Use it to find a solution of
address@hidden @math{\sin(\cos x) = 0.5}
address@hidden @expr{sin(cos(x)) = 0.5}
+near @expr{x = 4.5}. (Use angles measured in radians.) Note that
+the built-in @address@hidden R}} (@code{calc-find-root}) command uses Newton's
+method when it is able. @xref{Programming Answer 8, 8}. (@bullet{})
+
address@hidden Digamma function
address@hidden Gamma constant, Euler's
address@hidden Euler's gamma constant
+(@bullet{}) @strong{Exercise 9.} The @dfn{digamma} function
address@hidden @math{\psi(z) (``psi'')}
address@hidden @expr{psi(z)}
+is defined as the derivative of
address@hidden @math{\ln \Gamma(z)}.
address@hidden @expr{ln(gamma(z))}.
+For large values of @expr{z}, it can be approximated by the infinite sum
+
address@hidden
address@hidden
+psi(z) ~= ln(z) - 1/2z - sum(bern(2 n) / 2 n z^(2 n), n, 1, inf)
address@hidden example
address@hidden ifnottex
address@hidden
+\beforedisplay
+$$ \psi(z) \approx \ln z - {1\over2z} -
+ \sum_{n=1}^\infty {\code{bern}(2 n) \over 2 n z^{2n}}
+$$
+\afterdisplay
address@hidden tex
+
address@hidden
+where
address@hidden @math{\sum}
address@hidden @expr{sum}
+represents the sum over @expr{n} from 1 to infinity
+(or to some limit high enough to give the desired accuracy), and
+the @code{bern} function produces (exact) Bernoulli numbers.
+While this sum is not guaranteed to converge, in practice it is safe.
+An interesting mathematical constant is Euler's gamma, which is equal
+to about 0.5772. One way to compute it is by the formula,
address@hidden @math{\gamma = -\psi(1)}.
address@hidden @expr{gamma = -psi(1)}.
+Unfortunately, 1 isn't a large enough argument
+for the above formula to work (5 is a much safer value for @expr{z}).
+Fortunately, we can compute
address@hidden @math{\psi(1)}
address@hidden @expr{psi(1)}
+from
address@hidden @math{\psi(5)}
address@hidden @expr{psi(5)}
+using the recurrence
address@hidden @math{\psi(z+1) = \psi(z) + {1 \over z}}.
address@hidden @expr{psi(z+1) = psi(z) + 1/z}.
+Your task: Develop a program to compute
address@hidden @math{\psi(z)};
address@hidden @expr{psi(z)};
+it should ``pump up'' @expr{z}
+if necessary to be greater than 5, then use the above summation
+formula. Use looping commands to compute the sum. Use your function
+to compute
address@hidden @math{\gamma}
address@hidden @expr{gamma}
+to twelve decimal places. (Calc has a built-in command
+for Euler's constant, @kbd{I P}, which you can use to check your answer.)
address@hidden Answer 9, 9}. (@bullet{})
+
address@hidden Polynomial, list of coefficients
+(@bullet{}) @strong{Exercise 10.} Given a polynomial in @expr{x} and
+a number @expr{m} on the stack, where the polynomial is of degree
address@hidden or less (i.e., does not have any terms higher than @expr{x^m}),
+write a program to convert the polynomial into a list-of-coefficients
+notation. For example, @expr{5 x^4 + (x + 1)^2} with @expr{m = 6}
+should produce the list @expr{[1, 2, 1, 0, 5, 0, 0]}. Also develop
+a way to convert from this form back to the standard algebraic form.
address@hidden Answer 10, 10}. (@bullet{})
+
address@hidden Recursion
+(@bullet{}) @strong{Exercise 11.} The @dfn{Stirling numbers of the
+first kind} are defined by the recurrences,
+
address@hidden
address@hidden
+s(n,n) = 1 for n >= 0,
+s(n,0) = 0 for n > 0,
+s(n+1,m) = s(n,m-1) - n s(n,m) for n >= m >= 1.
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ \eqalign{ s(n,n) &= 1 \qquad \hbox{for } n \ge 0, \cr
+ s(n,0) &= 0 \qquad \hbox{for } n > 0, \cr
+ s(n+1,m) &= s(n,m-1) - n \, s(n,m) \qquad
+ \hbox{for } n \ge m \ge 1.}
+$$
+\afterdisplay
+\vskip5pt
+(These numbers are also sometimes written $\displaystyle{n \brack m}$.)
address@hidden tex
+
+This can be implemented using a @dfn{recursive} program in Calc; the
+program must invoke itself in order to calculate the two righthand
+terms in the general formula. Since it always invokes itself with
+``simpler'' arguments, it's easy to see that it must eventually finish
+the computation. Recursion is a little difficult with Emacs keyboard
+macros since the macro is executed before its definition is complete.
+So here's the recommended strategy: Create a ``dummy macro'' and assign
+it to a key with, e.g., @kbd{Z K s}. Now enter the true definition,
+using the @kbd{z s} command to call itself recursively, then assign it
+to the same key with @kbd{Z K s}. Now the @kbd{z s} command will run
+the complete recursive program. (Another way is to use @address@hidden E}}
+or @kbd{C-x * m} (@code{read-kbd-macro}) to read the whole macro at once,
+thus avoiding the ``training'' phase.) The task: Write a program
+that computes Stirling numbers of the first kind, given @expr{n} and
address@hidden on the stack. Test it with @emph{small} inputs like
address@hidden(4,2)}. (There is a built-in command for Stirling numbers,
address@hidden s}, which you can use to check your answers.)
address@hidden Answer 11, 11}. (@bullet{})
+
+The programming commands we've seen in this part of the tutorial
+are low-level, general-purpose operations. Often you will find
+that a higher-level function, such as vector mapping or rewrite
+rules, will do the job much more easily than a detailed, step-by-step
+program can:
+
+(@bullet{}) @strong{Exercise 12.} Write another program for
+computing Stirling numbers of the first kind, this time using
+rewrite rules. Once again, @expr{n} and @expr{m} should be taken
+from the stack. @xref{Programming Answer 12, 12}. (@bullet{})
+
address@hidden
+
address@hidden example
+This ends the tutorial section of the Calc manual. Now you know enough
+about Calc to use it effectively for many kinds of calculations. But
+Calc has many features that were not even touched upon in this tutorial.
address@hidden [not-split]
+The rest of this manual tells the whole story.
address@hidden [when-split]
address@hidden Volume II of this manual, the @dfn{Calc Reference}, tells the
whole story.
+
address@hidden
address@hidden Answers to Exercises, , Programming Tutorial, Tutorial
address@hidden Answers to Exercises
+
address@hidden
+This section includes answers to all the exercises in the Calc tutorial.
+
address@hidden
+* RPN Answer 1:: 1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -
+* RPN Answer 2:: 2*4 + 7*9.5 + 5/4
+* RPN Answer 3:: Operating on levels 2 and 3
+* RPN Answer 4:: Joe's complex problems
+* Algebraic Answer 1:: Simulating Q command
+* Algebraic Answer 2:: Joe's algebraic woes
+* Algebraic Answer 3:: 1 / 0
+* Modes Answer 1:: 3#0.1 = 3#0.0222222?
+* Modes Answer 2:: 16#f.e8fe15
+* Modes Answer 3:: Joe's rounding bug
+* Modes Answer 4:: Why floating point?
+* Arithmetic Answer 1:: Why the \ command?
+* Arithmetic Answer 2:: Tripping up the B command
+* Vector Answer 1:: Normalizing a vector
+* Vector Answer 2:: Average position
+* Matrix Answer 1:: Row and column sums
+* Matrix Answer 2:: Symbolic system of equations
+* Matrix Answer 3:: Over-determined system
+* List Answer 1:: Powers of two
+* List Answer 2:: Least-squares fit with matrices
+* List Answer 3:: Geometric mean
+* List Answer 4:: Divisor function
+* List Answer 5:: Duplicate factors
+* List Answer 6:: Triangular list
+* List Answer 7:: Another triangular list
+* List Answer 8:: Maximum of Bessel function
+* List Answer 9:: Integers the hard way
+* List Answer 10:: All elements equal
+* List Answer 11:: Estimating pi with darts
+* List Answer 12:: Estimating pi with matchsticks
+* List Answer 13:: Hash codes
+* List Answer 14:: Random walk
+* Types Answer 1:: Square root of pi times rational
+* Types Answer 2:: Infinities
+* Types Answer 3:: What can "nan" be?
+* Types Answer 4:: Abbey Road
+* Types Answer 5:: Friday the 13th
+* Types Answer 6:: Leap years
+* Types Answer 7:: Erroneous donut
+* Types Answer 8:: Dividing intervals
+* Types Answer 9:: Squaring intervals
+* Types Answer 10:: Fermat's primality test
+* Types Answer 11:: pi * 10^7 seconds
+* Types Answer 12:: Abbey Road on CD
+* Types Answer 13:: Not quite pi * 10^7 seconds
+* Types Answer 14:: Supercomputers and c
+* Types Answer 15:: Sam the Slug
+* Algebra Answer 1:: Squares and square roots
+* Algebra Answer 2:: Building polynomial from roots
+* Algebra Answer 3:: Integral of x sin(pi x)
+* Algebra Answer 4:: Simpson's rule
+* Rewrites Answer 1:: Multiplying by conjugate
+* Rewrites Answer 2:: Alternative fib rule
+* Rewrites Answer 3:: Rewriting opt(a) + opt(b) x
+* Rewrites Answer 4:: Sequence of integers
+* Rewrites Answer 5:: Number of terms in sum
+* Rewrites Answer 6:: Truncated Taylor series
+* Programming Answer 1:: Fresnel's C(x)
+* Programming Answer 2:: Negate third stack element
+* Programming Answer 3:: Compute sin(x) / x, etc.
+* Programming Answer 4:: Average value of a list
+* Programming Answer 5:: Continued fraction phi
+* Programming Answer 6:: Matrix Fibonacci numbers
+* Programming Answer 7:: Harmonic number greater than 4
+* Programming Answer 8:: Newton's method
+* Programming Answer 9:: Digamma function
+* Programming Answer 10:: Unpacking a polynomial
+* Programming Answer 11:: Recursive Stirling numbers
+* Programming Answer 12:: Stirling numbers with rewrites
address@hidden menu
+
address@hidden The following kludgery prevents the individual answers from
address@hidden being entered on the table of contents.
address@hidden
+\global\let\oldwrite=\write
+\gdef\skipwrite#1#2{\let\write=\oldwrite}
+\global\let\oldchapternofonts=\chapternofonts
+\gdef\chapternofonts{\let\write=\skipwrite\oldchapternofonts}
address@hidden tex
+
address@hidden RPN Answer 1, RPN Answer 2, Answers to Exercises, Answers to
Exercises
address@hidden RPN Tutorial Exercise 1
+
address@hidden
address@hidden @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -}
+
+The result is
address@hidden @math{1 - (2 \times (3 + 4)) = -13}.
address@hidden @expr{1 - (2 * (3 + 4)) = -13}.
+
address@hidden RPN Answer 2, RPN Answer 3, RPN Answer 1, Answers to Exercises
address@hidden RPN Tutorial Exercise 2
+
address@hidden
address@hidden @math{2\times4 + 7\times9.5 + {5\over4} = 75.75}
address@hidden @expr{2*4 + 7*9.5 + 5/4 = 75.75}
+
+After computing the intermediate term
address@hidden @math{2\times4 = 8},
address@hidden @expr{2*4 = 8},
+you can leave that result on the stack while you compute the second
+term. With both of these results waiting on the stack you can then
+compute the final term, then press @kbd{+ +} to add everything up.
+
address@hidden
address@hidden
+2: 2 1: 8 3: 8 2: 8
+1: 4 . 2: 7 1: 66.5
+ . 1: 9.5 .
+ .
+
+ 2 @key{RET} 4 * 7 @key{RET} 9.5 *
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+4: 8 3: 8 2: 8 1: 75.75
+3: 66.5 2: 66.5 1: 67.75 .
+2: 5 1: 1.25 .
+1: 4 .
+ .
+
+ 5 @key{RET} 4 / + +
address@hidden group
address@hidden smallexample
+
+Alternatively, you could add the first two terms before going on
+with the third term.
+
address@hidden
address@hidden
+2: 8 1: 74.5 3: 74.5 2: 74.5 1: 75.75
+1: 66.5 . 2: 5 1: 1.25 .
+ . 1: 4 .
+ .
+
+ ... + 5 @key{RET} 4 / +
address@hidden group
address@hidden smallexample
+
+On an old-style RPN calculator this second method would have the
+advantage of using only three stack levels. But since Calc's stack
+can grow arbitrarily large this isn't really an issue. Which method
+you choose is purely a matter of taste.
+
address@hidden RPN Answer 3, RPN Answer 4, RPN Answer 2, Answers to Exercises
address@hidden RPN Tutorial Exercise 3
+
address@hidden
+The @key{TAB} key provides a way to operate on the number in level 2.
+
address@hidden
address@hidden
+3: 10 3: 10 4: 10 3: 10 3: 10
+2: 20 2: 30 3: 30 2: 30 2: 21
+1: 30 1: 20 2: 20 1: 21 1: 30
+ . . 1: 1 . .
+ .
+
+ @key{TAB} 1 + @key{TAB}
address@hidden group
address@hidden smallexample
+
+Similarly, @address@hidden gives you access to the number in level 3.
+
address@hidden
address@hidden
+3: 10 3: 21 3: 21 3: 30 3: 11
+2: 21 2: 30 2: 30 2: 11 2: 21
+1: 30 1: 10 1: 11 1: 21 1: 30
+ . . . . .
+
+ address@hidden 1 + address@hidden
address@hidden
address@hidden group
address@hidden smallexample
+
address@hidden RPN Answer 4, Algebraic Answer 1, RPN Answer 3, Answers to
Exercises
address@hidden RPN Tutorial Exercise 4
+
address@hidden
+Either @kbd{( 2 , 3 )} or @kbd{( 2 @key{SPC} 3 )} would have worked,
+but using both the comma and the space at once yields:
+
address@hidden
address@hidden
+1: ( ... 2: ( ... 1: (2, ... 2: (2, ... 2: (2, ...
+ . 1: 2 . 1: (2, ... 1: (2, 3)
+ . . .
+
+ ( 2 , @key{SPC} 3 )
address@hidden group
address@hidden smallexample
+
+Joe probably tried to type @address@hidden @key{DEL}} to swap the
+extra incomplete object to the top of the stack and delete it.
+But a feature of Calc is that @key{DEL} on an incomplete object
+deletes just one component out of that object, so he had to press
address@hidden twice to finish the job.
+
address@hidden
address@hidden
+2: (2, ... 2: (2, 3) 2: (2, 3) 1: (2, 3)
+1: (2, 3) 1: (2, ... 1: ( ... .
+ . . .
+
+ @key{TAB} @key{DEL} @key{DEL}
address@hidden group
address@hidden smallexample
+
+(As it turns out, deleting the second-to-top stack entry happens often
+enough that Calc provides a special key, @address@hidden, to do just that.
address@hidden@key{DEL}} is just like @address@hidden @key{DEL}}, except that
it doesn't exhibit
+the ``feature'' that tripped poor Joe.)
+
address@hidden Algebraic Answer 1, Algebraic Answer 2, RPN Answer 4, Answers to
Exercises
address@hidden Algebraic Entry Tutorial Exercise 1
+
address@hidden
+Type @kbd{' sqrt($) @key{RET}}.
+
+If the @kbd{Q} key is broken, you could use @kbd{' $^0.5 @key{RET}}.
+Or, RPN style, @kbd{0.5 ^}.
+
+(Actually, @samp{$^1:2}, using the fraction one-half as the power, is
+a closer equivalent, since @samp{9^0.5} yields @expr{3.0} whereas
address@hidden(9)} and @samp{9^1:2} yield the exact integer @expr{3}.)
+
address@hidden Algebraic Answer 2, Algebraic Answer 3, Algebraic Answer 1,
Answers to Exercises
address@hidden Algebraic Entry Tutorial Exercise 2
+
address@hidden
+In the formula @samp{2 x (1+y)}, @samp{x} was interpreted as a function
+name with @samp{1+y} as its argument. Assigning a value to a variable
+has no relation to a function by the same name. Joe needed to use an
+explicit @samp{*} symbol here: @samp{2 x*(1+y)}.
+
address@hidden Algebraic Answer 3, Modes Answer 1, Algebraic Answer 2, Answers
to Exercises
address@hidden Algebraic Entry Tutorial Exercise 3
+
address@hidden
+The result from @kbd{1 @key{RET} 0 /} will be the formula @expr{1 / 0}.
+The ``function'' @samp{/} cannot be evaluated when its second argument
+is zero, so it is left in symbolic form. When you now type @kbd{0 *},
+the result will be zero because Calc uses the general rule that ``zero
+times anything is zero.''
+
address@hidden [fix-ref Infinities]
+The @kbd{m i} command enables an @dfn{Infinite mode} in which @expr{1 / 0}
+results in a special symbol that represents ``infinity.'' If you
+multiply infinity by zero, Calc uses another special new symbol to
+show that the answer is ``indeterminate.'' @xref{Infinities}, for
+further discussion of infinite and indeterminate values.
+
address@hidden Modes Answer 1, Modes Answer 2, Algebraic Answer 3, Answers to
Exercises
address@hidden Modes Tutorial Exercise 1
+
address@hidden
+Calc always stores its numbers in decimal, so even though one-third has
+an exact base-3 representation (@samp{3#0.1}), it is still stored as
+0.3333333 (chopped off after 12 or however many decimal digits) inside
+the calculator's memory. When this inexact number is converted back
+to base 3 for display, it may still be slightly inexact. When we
+multiply this number by 3, we get 0.999999, also an inexact value.
+
+When Calc displays a number in base 3, it has to decide how many digits
+to show. If the current precision is 12 (decimal) digits, that corresponds
+to @samp{12 / log10(3) = 25.15} base-3 digits. Because 25.15 is not an
+exact integer, Calc shows only 25 digits, with the result that stored
+numbers carry a little bit of extra information that may not show up on
+the screen. When Joe entered @samp{3#0.2}, the stored number 0.666666
+happened to round to a pleasing value when it lost that last 0.15 of a
+digit, but it was still inexact in Calc's memory. When he divided by 2,
+he still got the dreaded inexact value 0.333333. (Actually, he divided
+0.666667 by 2 to get 0.333334, which is why he got something a little
+higher than @code{3#0.1} instead of a little lower.)
+
+If Joe didn't want to be bothered with all this, he could have typed
address@hidden d n} to display with one less digit than the default. (If
+you give @kbd{d n} a negative argument, it uses default-minus-that,
+so @kbd{M-- d n} would be an easier way to get the same effect.) Those
+inexact results would still be lurking there, but they would now be
+rounded to nice, natural-looking values for display purposes. (Remember,
address@hidden in base 3 is like @samp{0.099999} in base 10; rounding
+off one digit will round the number up to @samp{0.1}.) Depending on the
+nature of your work, this hiding of the inexactness may be a benefit or
+a danger. With the @kbd{d n} command, Calc gives you the choice.
+
+Incidentally, another consequence of all this is that if you type
address@hidden d n} to display more digits than are ``really there,''
+you'll see garbage digits at the end of the number. (In decimal
+display mode, with decimally-stored numbers, these garbage digits are
+always zero so they vanish and you don't notice them.) Because Calc
+rounds off that 0.15 digit, there is the danger that two numbers could
+be slightly different internally but still look the same. If you feel
+uneasy about this, set the @kbd{d n} precision to be a little higher
+than normal; you'll get ugly garbage digits, but you'll always be able
+to tell two distinct numbers apart.
+
+An interesting side note is that most computers store their
+floating-point numbers in binary, and convert to decimal for display.
+Thus everyday programs have the same problem: Decimal 0.1 cannot be
+represented exactly in binary (try it: @kbd{0.1 d 2}), so @samp{0.1 * 10}
+comes out as an inexact approximation to 1 on some machines (though
+they generally arrange to hide it from you by rounding off one digit as
+we did above). Because Calc works in decimal instead of binary, you can
+be sure that numbers that look exact @emph{are} exact as long as you stay
+in decimal display mode.
+
+It's not hard to show that any number that can be represented exactly
+in binary, octal, or hexadecimal is also exact in decimal, so the kinds
+of problems we saw in this exercise are likely to be severe only when
+you use a relatively unusual radix like 3.
+
address@hidden Modes Answer 2, Modes Answer 3, Modes Answer 1, Answers to
Exercises
address@hidden Modes Tutorial Exercise 2
+
+If the radix is 15 or higher, we can't use the letter @samp{e} to mark
+the exponent because @samp{e} is interpreted as a digit. When Calc
+needs to display scientific notation in a high radix, it writes
address@hidden You can enter a number like this as an
+algebraic entry. Also, pressing @kbd{e} without any digits before it
+normally types @kbd{1e}, but in a high radix it types @kbd{16.^} and
+puts you in algebraic entry: @kbd{16#f.e8f @key{RET} e 15 @key{RET} *} is
another
+way to enter this number.
+
+The reason Calc puts a decimal point in the @samp{16.^} is to prevent
+huge integers from being generated if the exponent is large (consider
address@hidden, where we compute @samp{16^1000} as a giant
+exact integer and then throw away most of the digits when we multiply
+it by the floating-point @samp{16#1.23}). While this wouldn't normally
+matter for display purposes, it could give you a nasty surprise if you
+copied that number into a file and later moved it back into Calc.
+
address@hidden Modes Answer 3, Modes Answer 4, Modes Answer 2, Answers to
Exercises
address@hidden Modes Tutorial Exercise 3
+
address@hidden
+The answer he got was @expr{0.5000000000006399}.
+
+The problem is not that the square operation is inexact, but that the
+sine of 45 that was already on the stack was accurate to only 12 places.
+Arbitrary-precision calculations still only give answers as good as
+their inputs.
+
+The real problem is that there is no 12-digit number which, when
+squared, comes out to 0.5 exactly. The @kbd{f [} and @kbd{f ]}
+commands decrease or increase a number by one unit in the last
+place (according to the current precision). They are useful for
+determining facts like this.
+
address@hidden
address@hidden
+1: 0.707106781187 1: 0.500000000001
+ . .
+
+ 45 S 2 ^
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: 0.707106781187 1: 0.707106781186 1: 0.499999999999
+ . . .
+
+ U @key{DEL} f [ 2 ^
address@hidden group
address@hidden smallexample
+
+A high-precision calculation must be carried out in high precision
+all the way. The only number in the original problem which was known
+exactly was the quantity 45 degrees, so the precision must be raised
+before anything is done after the number 45 has been entered in order
+for the higher precision to be meaningful.
+
address@hidden Modes Answer 4, Arithmetic Answer 1, Modes Answer 3, Answers to
Exercises
address@hidden Modes Tutorial Exercise 4
+
address@hidden
+Many calculations involve real-world quantities, like the width and
+height of a piece of wood or the volume of a jar. Such quantities
+can't be measured exactly anyway, and if the data that is input to
+a calculation is inexact, doing exact arithmetic on it is a waste
+of time.
+
+Fractions become unwieldy after too many calculations have been
+done with them. For example, the sum of the reciprocals of the
+integers from 1 to 10 is 7381:2520. The sum from 1 to 30 is
+9304682830147:2329089562800. After a point it will take a long
+time to add even one more term to this sum, but a floating-point
+calculation of the sum will not have this problem.
+
+Also, rational numbers cannot express the results of all calculations.
+There is no fractional form for the square root of two, so if you type
address@hidden@kbd{2 Q}}, Calc has no choice but to give you a floating-point
answer.
+
address@hidden Arithmetic Answer 1, Arithmetic Answer 2, Modes Answer 4,
Answers to Exercises
address@hidden Arithmetic Tutorial Exercise 1
+
address@hidden
+Dividing two integers that are larger than the current precision may
+give a floating-point result that is inaccurate even when rounded
+down to an integer. Consider @expr{123456789 / 2} when the current
+precision is 6 digits. The true answer is @expr{61728394.5}, but
+with a precision of 6 this will be rounded to
address@hidden @math{12345700.0/2.0 = 61728500.0}.
address@hidden @expr{12345700.@: / 2.@: = 61728500.}.
+The result, when converted to an integer, will be off by 106.
+
+Here are two solutions: Raise the precision enough that the
+floating-point round-off error is strictly to the right of the
+decimal point. Or, convert to Fraction mode so that @expr{123456789 / 2}
+produces the exact fraction @expr{123456789:2}, which can be rounded
+down by the @kbd{F} command without ever switching to floating-point
+format.
+
address@hidden Arithmetic Answer 2, Vector Answer 1, Arithmetic Answer 1,
Answers to Exercises
address@hidden Arithmetic Tutorial Exercise 2
+
address@hidden
address@hidden @key{RET} 9 B} could give the exact result @expr{3:2}, but it
+does a floating-point calculation instead and produces @expr{1.5}.
+
+Calc will find an exact result for a logarithm if the result is an integer
+or (when in Fraction mode) the reciprocal of an integer. But there is
+no efficient way to search the space of all possible rational numbers
+for an exact answer, so Calc doesn't try.
+
address@hidden Vector Answer 1, Vector Answer 2, Arithmetic Answer 2, Answers
to Exercises
address@hidden Vector Tutorial Exercise 1
+
address@hidden
+Duplicate the vector, compute its length, then divide the vector
+by its length: @address@hidden A /}.
+
address@hidden
address@hidden
+1: [1, 2, 3] 2: [1, 2, 3] 1: [0.27, 0.53, 0.80] 1: 1.
+ . 1: 3.74165738677 . .
+ .
+
+ r 1 @key{RET} A / A
address@hidden group
address@hidden smallexample
+
+The final @kbd{A} command shows that the normalized vector does
+indeed have unit length.
+
address@hidden Vector Answer 2, Matrix Answer 1, Vector Answer 1, Answers to
Exercises
address@hidden Vector Tutorial Exercise 2
+
address@hidden
+The average position is equal to the sum of the products of the
+positions times their corresponding probabilities. This is the
+definition of the dot product operation. So all you need to do
+is to put the two vectors on the stack and press @kbd{*}.
+
address@hidden Matrix Answer 1, Matrix Answer 2, Vector Answer 2, Answers to
Exercises
address@hidden Matrix Tutorial Exercise 1
+
address@hidden
+The trick is to multiply by a vector of ones. Use @kbd{r 4 [1 1 1] *} to
+get the row sum. Similarly, use @kbd{[1 1] r 4 *} to get the column sum.
+
address@hidden Matrix Answer 2, Matrix Answer 3, Matrix Answer 1, Answers to
Exercises
address@hidden Matrix Tutorial Exercise 2
+
address@hidden
address@hidden
address@hidden
+ x + a y = 6
+ x + b y = 10
address@hidden group
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ \eqalign{ x &+ a y = 6 \cr
+ x &+ b y = 10}
+$$
+\afterdisplay
address@hidden tex
+
+Just enter the righthand side vector, then divide by the lefthand side
+matrix as usual.
+
address@hidden
address@hidden
+1: [6, 10] 2: [6, 10] 1: [6 - 4 a / (b - a), 4 / (b - a) ]
+ . 1: [ [ 1, a ] .
+ [ 1, b ] ]
+ .
+
+' [6 10] @key{RET} ' [1 a; 1 b] @key{RET} /
address@hidden group
address@hidden smallexample
+
+This can be made more readable using @kbd{d B} to enable Big display
+mode:
+
address@hidden
address@hidden
+ 4 a 4
+1: [6 - -----, -----]
+ b - a b - a
address@hidden group
address@hidden smallexample
+
+Type @kbd{d N} to return to Normal display mode afterwards.
+
address@hidden Matrix Answer 3, List Answer 1, Matrix Answer 2, Answers to
Exercises
address@hidden Matrix Tutorial Exercise 3
+
address@hidden
+To solve
address@hidden @math{A^T A \, X = A^T B},
address@hidden @expr{trn(A) * A * X = trn(A) * B},
+first we compute
address@hidden @math{A' = A^T A}
address@hidden @expr{A2 = trn(A) * A}
+and
address@hidden @math{B' = A^T B};
address@hidden @expr{B2 = trn(A) * B};
+now, we have a system
address@hidden @math{A' X = B'}
address@hidden @expr{A2 * X = B2}
+which we can solve using Calc's @samp{/} command.
+
address@hidden
address@hidden
address@hidden
+ a + 2b + 3c = 6
+ 4a + 5b + 6c = 2
+ 7a + 6b = 3
+ 2a + 4b + 6c = 11
address@hidden group
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplayh
+$$ \openup1\jot \tabskip=0pt plus1fil
+\halign to\displaywidth{\tabskip=0pt
+ $\hfil#$&$\hfil{}#{}$&
+ $\hfil#$&$\hfil{}#{}$&
+ $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
+ a&+&2b&+&3c&=6 \cr
+ 4a&+&5b&+&6c&=2 \cr
+ 7a&+&6b& & &=3 \cr
+ 2a&+&4b&+&6c&=11 \cr}
+$$
+\afterdisplayh
address@hidden tex
+
+The first step is to enter the coefficient matrix. We'll store it in
+quick variable number 7 for later reference. Next, we compute the
address@hidden @math{B'}
address@hidden @expr{B2}
+vector.
+
address@hidden
address@hidden
+1: [ [ 1, 2, 3 ] 2: [ [ 1, 4, 7, 2 ] 1: [57, 84, 96]
+ [ 4, 5, 6 ] [ 2, 5, 6, 4 ] .
+ [ 7, 6, 0 ] [ 3, 6, 0, 6 ] ]
+ [ 2, 4, 6 ] ] 1: [6, 2, 3, 11]
+ . .
+
+' [1 2 3; 4 5 6; 7 6 0; 2 4 6] @key{RET} s 7 v t [6 2 3 11] *
address@hidden group
address@hidden smallexample
+
address@hidden
+Now we compute the matrix
address@hidden @math{A'}
address@hidden @expr{A2}
+and divide.
+
address@hidden
address@hidden
+2: [57, 84, 96] 1: [-11.64, 14.08, -3.64]
+1: [ [ 70, 72, 39 ] .
+ [ 72, 81, 60 ]
+ [ 39, 60, 81 ] ]
+ .
+
+ r 7 v t r 7 * /
address@hidden group
address@hidden smallexample
+
address@hidden
+(The actual computed answer will be slightly inexact due to
+round-off error.)
+
+Notice that the answers are similar to those for the
address@hidden @math{3\times3}
address@hidden 3x3
+system solved in the text. That's because the fourth equation that was
+added to the system is almost identical to the first one multiplied
+by two. (If it were identical, we would have gotten the exact same
+answer since the
address@hidden @math{4\times3}
address@hidden 4x3
+system would be equivalent to the original
address@hidden @math{3\times3}
address@hidden 3x3
+system.)
+
+Since the first and fourth equations aren't quite equivalent, they
+can't both be satisfied at once. Let's plug our answers back into
+the original system of equations to see how well they match.
+
address@hidden
address@hidden
+2: [-11.64, 14.08, -3.64] 1: [5.6, 2., 3., 11.2]
+1: [ [ 1, 2, 3 ] .
+ [ 4, 5, 6 ]
+ [ 7, 6, 0 ]
+ [ 2, 4, 6 ] ]
+ .
+
+ r 7 @key{TAB} *
address@hidden group
address@hidden smallexample
+
address@hidden
+This is reasonably close to our original @expr{B} vector,
address@hidden, 2, 3, 11]}.
+
address@hidden List Answer 1, List Answer 2, Matrix Answer 3, Answers to
Exercises
address@hidden List Tutorial Exercise 1
+
address@hidden
+We can use @kbd{v x} to build a vector of integers. This needs to be
+adjusted to get the range of integers we desire. Mapping @samp{-}
+across the vector will accomplish this, although it turns out the
+plain @samp{-} key will work just as well.
+
address@hidden
address@hidden
+2: 2 2: 2
+1: [1, 2, 3, 4, 5, 6, 7, 8, 9] 1: [-4, -3, -2, -1, 0, 1, 2, 3, 4]
+ . .
+
+ 2 v x 9 @key{RET} 5 V M - or 5 -
address@hidden group
address@hidden smallexample
+
address@hidden
+Now we use @kbd{V M ^} to map the exponentiation operator across the
+vector.
+
address@hidden
address@hidden
+1: [0.0625, 0.125, 0.25, 0.5, 1, 2, 4, 8, 16]
+ .
+
+ V M ^
address@hidden group
address@hidden smallexample
+
address@hidden List Answer 2, List Answer 3, List Answer 1, Answers to Exercises
address@hidden List Tutorial Exercise 2
+
address@hidden
+Given @expr{x} and @expr{y} vectors in quick variables 1 and 2 as before,
+the first job is to form the matrix that describes the problem.
+
address@hidden
address@hidden
+ m*x + b*1 = y
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ m \times x + b \times 1 = y $$
+\afterdisplay
address@hidden tex
+
+Thus we want a
address@hidden @math{19\times2}
address@hidden 19x2
+matrix with our @expr{x} vector as one column and
+ones as the other column. So, first we build the column of ones, then
+we combine the two columns to form our @expr{A} matrix.
+
address@hidden
address@hidden
+2: [1.34, 1.41, 1.49, ... ] 1: [ [ 1.34, 1 ]
+1: [1, 1, 1, ...] [ 1.41, 1 ]
+ . [ 1.49, 1 ]
+ @dots{}
+
+ r 1 1 v b 19 @key{RET} M-2 v p v t s 3
address@hidden group
address@hidden smallexample
+
address@hidden
+Now we compute
address@hidden @math{A^T y}
address@hidden @expr{trn(A) * y}
+and
address@hidden @math{A^T A}
address@hidden @expr{trn(A) * A}
+and divide.
+
address@hidden
address@hidden
+1: [33.36554, 13.613] 2: [33.36554, 13.613]
+ . 1: [ [ 98.0003, 41.63 ]
+ [ 41.63, 19 ] ]
+ .
+
+ v t r 2 * r 3 v t r 3 *
address@hidden group
address@hidden smallexample
+
address@hidden
+(Hey, those numbers look familiar!)
+
address@hidden
address@hidden
+1: [0.52141679, -0.425978]
+ .
+
+ /
address@hidden group
address@hidden smallexample
+
+Since we were solving equations of the form
address@hidden @math{m \times x + b \times 1 = y},
address@hidden @expr{m*x + b*1 = y},
+these numbers should be @expr{m} and @expr{b}, respectively. Sure
+enough, they agree exactly with the result computed using @kbd{V M} and
address@hidden R}!
+
+The moral of this story: @kbd{V M} and @kbd{V R} will probably solve
+your problem, but there is often an easier way using the higher-level
+arithmetic functions!
+
address@hidden [fix-ref Curve Fitting]
+In fact, there is a built-in @kbd{a F} command that does least-squares
+fits. @xref{Curve Fitting}.
+
address@hidden List Answer 3, List Answer 4, List Answer 2, Answers to Exercises
address@hidden List Tutorial Exercise 3
+
address@hidden
+Move to one end of the list and press @kbd{C-@@} (or @address@hidden or
+whatever) to set the mark, then move to the other end of the list
+and type @address@hidden * g}}.
+
address@hidden
address@hidden
+1: [2.3, 6, 22, 15.1, 7, 15, 14, 7.5, 2.5]
+ .
address@hidden group
address@hidden smallexample
+
+To make things interesting, let's assume we don't know at a glance
+how many numbers are in this list. Then we could type:
+
address@hidden
address@hidden
+2: [2.3, 6, 22, ... ] 2: [2.3, 6, 22, ... ]
+1: [2.3, 6, 22, ... ] 1: 126356422.5
+ . .
+
+ @key{RET} V R *
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+2: 126356422.5 2: 126356422.5 1: 7.94652913734
+1: [2.3, 6, 22, ... ] 1: 9 .
+ . .
+
+ @key{TAB} v l I ^
address@hidden group
address@hidden smallexample
+
address@hidden
+(The @kbd{I ^} command computes the @var{n}th root of a number.
+You could also type @kbd{& ^} to take the reciprocal of 9 and
+then raise the number to that power.)
+
address@hidden List Answer 4, List Answer 5, List Answer 3, Answers to Exercises
address@hidden List Tutorial Exercise 4
+
address@hidden
+A number @expr{j} is a divisor of @expr{n} if
address@hidden @math{n \mathbin{\hbox{\code{\%}}} j = 0}.
address@hidden @samp{n % j = 0}.
+The first step is to get a vector that identifies the divisors.
+
address@hidden
address@hidden
+2: 30 2: [0, 0, 0, 2, ...] 1: [1, 1, 1, 0, ...]
+1: [1, 2, 3, 4, ...] 1: 0 .
+ . .
+
+ 30 @key{RET} v x 30 @key{RET} s 1 V M % 0 V M a = s 2
address@hidden group
address@hidden smallexample
+
address@hidden
+This vector has 1's marking divisors of 30 and 0's marking non-divisors.
+
+The zeroth divisor function is just the total number of divisors.
+The first divisor function is the sum of the divisors.
+
address@hidden
address@hidden
+1: 8 3: 8 2: 8 2: 8
+ 2: [1, 2, 3, 4, ...] 1: [1, 2, 3, 0, ...] 1: 72
+ 1: [1, 1, 1, 0, ...] . .
+ .
+
+ V R + r 1 r 2 V M * V R +
address@hidden group
address@hidden smallexample
+
address@hidden
+Once again, the last two steps just compute a dot product for which
+a simple @kbd{*} would have worked equally well.
+
address@hidden List Answer 5, List Answer 6, List Answer 4, Answers to Exercises
address@hidden List Tutorial Exercise 5
+
address@hidden
+The obvious first step is to obtain the list of factors with @kbd{k f}.
+This list will always be in sorted order, so if there are duplicates
+they will be right next to each other. A suitable method is to compare
+the list with a copy of itself shifted over by one.
+
address@hidden
address@hidden
+1: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19, 0]
+ . 1: [3, 7, 7, 7, 19, 0] 1: [0, 3, 7, 7, 7, 19]
+ . .
+
+ 19551 k f @key{RET} 0 | @key{TAB} 0
@key{TAB} |
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: [0, 0, 1, 1, 0, 0] 1: 2 1: 0
+ . . .
+
+ V M a = V R + 0 a =
address@hidden group
address@hidden smallexample
+
address@hidden
+Note that we have to arrange for both vectors to have the same length
+so that the mapping operation works; no prime factor will ever be
+zero, so adding zeros on the left and right is safe. From then on
+the job is pretty straightforward.
+
+Incidentally, Calc provides the
address@hidden @dfn{M@"obius} @math{\mu}
address@hidden @dfn{Moebius mu}
+function which is zero if and only if its argument is square-free. It
+would be a much more convenient way to do the above test in practice.
+
address@hidden List Answer 6, List Answer 7, List Answer 5, Answers to Exercises
address@hidden List Tutorial Exercise 6
+
address@hidden
+First use @kbd{v x 6 @key{RET}} to get a list of integers, then @kbd{V M v x}
+to get a list of lists of integers!
+
address@hidden List Answer 7, List Answer 8, List Answer 6, Answers to Exercises
address@hidden List Tutorial Exercise 7
+
address@hidden
+Here's one solution. First, compute the triangular list from the previous
+exercise and type @kbd{1 -} to subtract one from all the elements.
+
address@hidden
address@hidden
+1: [ [0],
+ [0, 1],
+ [0, 1, 2],
+ @dots{}
+
+ 1 -
address@hidden group
address@hidden smallexample
+
+The numbers down the lefthand edge of the list we desire are called
+the ``triangular numbers'' (now you know why!). The @expr{n}th
+triangular number is the sum of the integers from 1 to @expr{n}, and
+can be computed directly by the formula
address@hidden @math{n (n+1) \over 2}.
address@hidden @expr{n * (n+1) / 2}.
+
address@hidden
address@hidden
+2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ]
+1: [0, 1, 2, 3, 4, 5] 1: [0, 1, 3, 6, 10, 15]
+ . .
+
+ v x 6 @key{RET} 1 - V M ' $ ($+1)/2 @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+Adding this list to the above list of lists produces the desired
+result:
+
address@hidden
address@hidden
+1: [ [0],
+ [1, 2],
+ [3, 4, 5],
+ [6, 7, 8, 9],
+ [10, 11, 12, 13, 14],
+ [15, 16, 17, 18, 19, 20] ]
+ .
+
+ V M +
address@hidden group
address@hidden smallexample
+
+If we did not know the formula for triangular numbers, we could have
+computed them using a @kbd{V U +} command. We could also have
+gotten them the hard way by mapping a reduction across the original
+triangular list.
+
address@hidden
address@hidden
+2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ]
+1: [ [0], [0, 1], ... ] 1: [0, 1, 3, 6, 10, 15]
+ . .
+
+ @key{RET} V M V R +
address@hidden group
address@hidden smallexample
+
address@hidden
+(This means ``map a @kbd{V R +} command across the vector,'' and
+since each element of the main vector is itself a small vector,
address@hidden R +} computes the sum of its elements.)
+
address@hidden List Answer 8, List Answer 9, List Answer 7, Answers to Exercises
address@hidden List Tutorial Exercise 8
+
address@hidden
+The first step is to build a list of values of @expr{x}.
+
address@hidden
address@hidden
+1: [1, 2, 3, ..., 21] 1: [0, 1, 2, ..., 20] 1: [0, 0.25, 0.5, ..., 5]
+ . . .
+
+ v x 21 @key{RET} 1 - 4 / s 1
address@hidden group
address@hidden smallexample
+
+Next, we compute the Bessel function values.
+
address@hidden
address@hidden
+1: [0., 0.124, 0.242, ..., -0.328]
+ .
+
+ V M ' besJ(1,$) @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+(Another way to do this would be @kbd{1 @key{TAB} V M f j}.)
+
+A way to isolate the maximum value is to compute the maximum using
address@hidden R X}, then compare all the Bessel values with that maximum.
+
address@hidden
address@hidden
+2: [0., 0.124, 0.242, ... ] 1: [0, 0, 0, ... ] 2: [0, 0, 0, ... ]
+1: 0.5801562 . 1: 1
+ . .
+
+ @key{RET} V R X V M a = @key{RET} V R
+ @key{DEL}
address@hidden group
address@hidden smallexample
+
address@hidden
+It's a good idea to verify, as in the last step above, that only
+one value is equal to the maximum. (After all, a plot of
address@hidden @math{\sin x}
address@hidden @expr{sin(x)}
+might have many points all equal to the maximum value, 1.)
+
+The vector we have now has a single 1 in the position that indicates
+the maximum value of @expr{x}. Now it is a simple matter to convert
+this back into the corresponding value itself.
+
address@hidden
address@hidden
+2: [0, 0, 0, ... ] 1: [0, 0., 0., ... ] 1: 1.75
+1: [0, 0.25, 0.5, ... ] . .
+ .
+
+ r 1 V M * V R +
address@hidden group
address@hidden smallexample
+
+If @kbd{a =} had produced more than one @expr{1} value, this method
+would have given the sum of all maximum @expr{x} values; not very
+useful! In this case we could have used @kbd{v m} (@code{calc-mask-vector})
+instead. This command deletes all elements of a ``data'' vector that
+correspond to zeros in a ``mask'' vector, leaving us with, in this
+example, a vector of maximum @expr{x} values.
+
+The built-in @kbd{a X} command maximizes a function using more
+efficient methods. Just for illustration, let's use @kbd{a X}
+to maximize @samp{besJ(1,x)} over this same interval.
+
address@hidden
address@hidden
+2: besJ(1, x) 1: [1.84115, 0.581865]
+1: [0 .. 5] .
+ .
+
+' besJ(1,x), [0..5] @key{RET} a X x @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+The output from @kbd{a X} is a vector containing the value of @expr{x}
+that maximizes the function, and the function's value at that maximum.
+As you can see, our simple search got quite close to the right answer.
+
address@hidden List Answer 9, List Answer 10, List Answer 8, Answers to
Exercises
address@hidden List Tutorial Exercise 9
+
address@hidden
+Step one is to convert our integer into vector notation.
+
address@hidden
address@hidden
+1: 25129925999 3: 25129925999
+ . 2: 10
+ 1: [11, 10, 9, ..., 1, 0]
+ .
+
+ 25129925999 @key{RET} 10 @key{RET} 12 @key{RET} v x 12 @key{RET}
-
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: 25129925999 1: [0, 2, 25, 251, 2512, ... ]
+2: [100000000000, ... ] .
+ .
+
+ V M ^ s 1 V M \
address@hidden group
address@hidden smallexample
+
address@hidden
+(Recall, the @kbd{\} command computes an integer quotient.)
+
address@hidden
address@hidden
+1: [0, 2, 5, 1, 2, 9, 9, 2, 5, 9, 9, 9]
+ .
+
+ 10 V M % s 2
address@hidden group
address@hidden smallexample
+
+Next we must increment this number. This involves adding one to
+the last digit, plus handling carries. There is a carry to the
+left out of a digit if that digit is a nine and all the digits to
+the right of it are nines.
+
address@hidden
address@hidden
+1: [0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1] 1: [1, 1, 1, 0, 0, 1, ... ]
+ . .
+
+ 9 V M a = v v
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: [1, 1, 1, 0, 0, 0, ... ] 1: [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]
+ . .
+
+ V U * v v 1 |
address@hidden group
address@hidden smallexample
+
address@hidden
+Accumulating @kbd{*} across a vector of ones and zeros will preserve
+only the initial run of ones. These are the carries into all digits
+except the rightmost digit. Concatenating a one on the right takes
+care of aligning the carries properly, and also adding one to the
+rightmost digit.
+
address@hidden
address@hidden
+2: [0, 0, 0, 0, ... ] 1: [0, 0, 2, 5, 1, 2, 9, 9, 2, 6, 0, 0, 0]
+1: [0, 0, 2, 5, ... ] .
+ .
+
+ 0 r 2 | V M + 10 V M %
address@hidden group
address@hidden smallexample
+
address@hidden
+Here we have concatenated 0 to the @emph{left} of the original number;
+this takes care of shifting the carries by one with respect to the
+digits that generated them.
+
+Finally, we must convert this list back into an integer.
+
address@hidden
address@hidden
+3: [0, 0, 2, 5, ... ] 2: [0, 0, 2, 5, ... ]
+2: 1000000000000 1: [1000000000000, 100000000000, ... ]
+1: [100000000000, ... ] .
+ .
+
+ 10 @key{RET} 12 ^ r 1 |
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: [0, 0, 20000000000, 5000000000, ... ] 1: 25129926000
+ . .
+
+ V M * V R +
address@hidden group
address@hidden smallexample
+
address@hidden
+Another way to do this final step would be to reduce the formula
address@hidden@samp{10 $$ + $}} across the vector of digits.
+
address@hidden
address@hidden
+1: [0, 0, 2, 5, ... ] 1: 25129926000
+ . .
+
+ V R ' 10 $$ + $ @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden List Answer 10, List Answer 11, List Answer 9, Answers to
Exercises
address@hidden List Tutorial Exercise 10
+
address@hidden
+For the list @expr{[a, b, c, d]}, the result is @expr{((a = b) = c) = d},
+which will compare @expr{a} and @expr{b} to produce a 1 or 0, which is
+then compared with @expr{c} to produce another 1 or 0, which is then
+compared with @expr{d}. This is not at all what Joe wanted.
+
+Here's a more correct method:
+
address@hidden
address@hidden
+1: [7, 7, 7, 8, 7] 2: [7, 7, 7, 8, 7]
+ . 1: 7
+ .
+
+ ' [7,7,7,8,7] @key{RET} @key{RET} v r 1 @key{RET}
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: [1, 1, 1, 0, 1] 1: 0
+ . .
+
+ V M a = V R *
address@hidden group
address@hidden smallexample
+
address@hidden List Answer 11, List Answer 12, List Answer 10, Answers to
Exercises
address@hidden List Tutorial Exercise 11
+
address@hidden
+The circle of unit radius consists of those points @expr{(x,y)} for which
address@hidden + y^2 < 1}. We start by generating a vector of @expr{x^2}
+and a vector of @expr{y^2}.
+
+We can make this go a bit faster by using the @kbd{v .} and @kbd{t .}
+commands.
+
address@hidden
address@hidden
+2: [2., 2., ..., 2.] 2: [2., 2., ..., 2.]
+1: [2., 2., ..., 2.] 1: [1.16, 1.98, ..., 0.81]
+ . .
+
+ v . t . 2. v b 100 @key{RET} @key{RET} V M k r
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+2: [2., 2., ..., 2.] 1: [0.026, 0.96, ..., 0.036]
+1: [0.026, 0.96, ..., 0.036] 2: [0.53, 0.81, ..., 0.094]
+ . .
+
+ 1 - 2 V M ^ @key{TAB} V M k r 1 - 2 V M ^
address@hidden group
address@hidden smallexample
+
+Now we sum the @expr{x^2} and @expr{y^2} values, compare with 1 to
+get a vector of 1/0 truth values, then sum the truth values.
+
address@hidden
address@hidden
+1: [0.56, 1.78, ..., 0.13] 1: [1, 0, ..., 1] 1: 84
+ . . .
+
+ + 1 V M a < V R +
address@hidden group
address@hidden smallexample
+
address@hidden
+The ratio @expr{84/100} should approximate the ratio @cpiover{4}.
+
address@hidden
address@hidden
+1: 0.84 1: 3.36 2: 3.36 1: 1.0695
+ . . 1: 3.14159 .
+
+ 100 / 4 * P /
address@hidden group
address@hidden smallexample
+
address@hidden
+Our estimate, 3.36, is off by about 7%. We could get a better estimate
+by taking more points (say, 1000), but it's clear that this method is
+not very efficient!
+
+(Naturally, since this example uses random numbers your own answer
+will be slightly different from the one shown here!)
+
+If you typed @kbd{v .} and @kbd{t .} before, type them again to
+return to full-sized display of vectors.
+
address@hidden List Answer 12, List Answer 13, List Answer 11, Answers to
Exercises
address@hidden List Tutorial Exercise 12
+
address@hidden
+This problem can be made a lot easier by taking advantage of some
+symmetries. First of all, after some thought it's clear that the
address@hidden axis can be ignored altogether. Just pick a random @expr{x}
+component for one end of the match, pick a random direction
address@hidden @math{\theta},
address@hidden @expr{theta},
+and see if @expr{x} and
address@hidden @math{x + \cos \theta}
address@hidden @expr{x + cos(theta)}
+(which is the @expr{x} coordinate of the other endpoint) cross a line.
+The lines are at integer coordinates, so this happens when the two
+numbers surround an integer.
+
+Since the two endpoints are equivalent, we may as well choose the leftmost
+of the two endpoints as @expr{x}. Then @expr{theta} is an angle pointing
+to the right, in the range -90 to 90 degrees. (We could use radians, but
+it would feel like cheating to refer to @cpiover{2} radians while trying
+to estimate @cpi{}!)
+
+In fact, since the field of lines is infinite we can choose the
+coordinates 0 and 1 for the lines on either side of the leftmost
+endpoint. The rightmost endpoint will be between 0 and 1 if the
+match does not cross a line, or between 1 and 2 if it does. So:
+Pick random @expr{x} and
address@hidden @math{\theta},
address@hidden @expr{theta},
+compute
address@hidden @math{x + \cos \theta},
address@hidden @expr{x + cos(theta)},
+and count how many of the results are greater than one. Simple!
+
+We can make this go a bit faster by using the @kbd{v .} and @kbd{t .}
+commands.
+
address@hidden
address@hidden
+1: [0.52, 0.71, ..., 0.72] 2: [0.52, 0.71, ..., 0.72]
+ . 1: [78.4, 64.5, ..., -42.9]
+ .
+
+v . t . 1. v b 100 @key{RET} V M k r 180. v b 100 @key{RET} V M k r 90 -
address@hidden group
address@hidden smallexample
+
address@hidden
+(The next step may be slow, depending on the speed of your computer.)
+
address@hidden
address@hidden
+2: [0.52, 0.71, ..., 0.72] 1: [0.72, 1.14, ..., 1.45]
+1: [0.20, 0.43, ..., 0.73] .
+ .
+
+ m d V M C +
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: [0, 1, ..., 1] 1: 0.64 1: 3.125
+ . . .
+
+ 1 V M a > V R + 100 / 2 @key{TAB} /
address@hidden group
address@hidden smallexample
+
+Let's try the third method, too. We'll use random integers up to
+one million. The @kbd{k r} command with an integer argument picks
+a random integer.
+
address@hidden
address@hidden
+2: [1000000, 1000000, ..., 1000000] 2: [78489, 527587, ..., 814975]
+1: [1000000, 1000000, ..., 1000000] 1: [324014, 358783, ..., 955450]
+ . .
+
+ 1000000 v b 100 @key{RET} @key{RET} V M k r @key{TAB} V M
k r
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: [1, 1, ..., 25] 1: [1, 1, ..., 0] 1: 0.56
+ . . .
+
+ V M k g 1 V M a = V R + 100 /
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: 10.714 1: 3.273
+ . .
+
+ 6 @key{TAB} / Q
address@hidden group
address@hidden smallexample
+
+For a proof of this property of the GCD function, see section 4.5.2,
+exercise 10, of Knuth's @emph{Art of Computer Programming}, volume II.
+
+If you typed @kbd{v .} and @kbd{t .} before, type them again to
+return to full-sized display of vectors.
+
address@hidden List Answer 13, List Answer 14, List Answer 12, Answers to
Exercises
address@hidden List Tutorial Exercise 13
+
address@hidden
+First, we put the string on the stack as a vector of ASCII codes.
+
address@hidden
address@hidden
+1: [84, 101, 115, ..., 51]
+ .
+
+ "Testing, 1, 2, 3 @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+Note that the @kbd{"} key, like @kbd{$}, initiates algebraic entry so
+there was no need to type an apostrophe. Also, Calc didn't mind that
+we omitted the closing @kbd{"}. (The same goes for all closing delimiters
+like @kbd{)} and @kbd{]} at the end of a formula.
+
+We'll show two different approaches here. In the first, we note that
+if the input vector is @expr{[a, b, c, d]}, then the hash code is
address@hidden (3 (3a + b) + c) + d = 27a + 9b + 3c + d}. In other words,
+it's a sum of descending powers of three times the ASCII codes.
+
address@hidden
address@hidden
+2: [84, 101, 115, ..., 51] 2: [84, 101, 115, ..., 51]
+1: 16 1: [15, 14, 13, ..., 0]
+ . .
+
+ @key{RET} v l v x 16 @key{RET} -
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+2: [84, 101, 115, ..., 51] 1: 1960915098 1: 121
+1: [14348907, ..., 1] . .
+ .
+
+ 3 @key{TAB} V M ^ * 511 %
address@hidden group
address@hidden smallexample
+
address@hidden
+Once again, @kbd{*} elegantly summarizes most of the computation.
+But there's an even more elegant approach: Reduce the formula
address@hidden $$ + $} across the vector. Recall that this represents a
+function of two arguments that computes its first argument times three
+plus its second argument.
+
address@hidden
address@hidden
+1: [84, 101, 115, ..., 51] 1: 1960915098
+ . .
+
+ "Testing, 1, 2, 3 @key{RET} V R ' 3$$+$ @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+If you did the decimal arithmetic exercise, this will be familiar.
+Basically, we're turning a base-3 vector of digits into an integer,
+except that our ``digits'' are much larger than real digits.
+
+Instead of typing @kbd{511 %} again to reduce the result, we can be
+cleverer still and notice that rather than computing a huge integer
+and taking the modulo at the end, we can take the modulo at each step
+without affecting the result. While this means there are more
+arithmetic operations, the numbers we operate on remain small so
+the operations are faster.
+
address@hidden
address@hidden
+1: [84, 101, 115, ..., 51] 1: 121
+ . .
+
+ "Testing, 1, 2, 3 @key{RET} V R ' (3$$+$)%511 @key{RET}
address@hidden group
address@hidden smallexample
+
+Why does this work? Think about a two-step computation:
address@hidden@expr{3 (3a + b) + c}}. Taking a result modulo 511 basically
means
+subtracting off enough 511's to put the result in the desired range.
+So the result when we take the modulo after every step is,
+
address@hidden
address@hidden
+3 (3 a + b - 511 m) + c - 511 n
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ 3 (3 a + b - 511 m) + c - 511 n $$
+\afterdisplay
address@hidden tex
+
address@hidden
+for some suitable integers @expr{m} and @expr{n}. Expanding out by
+the distributive law yields
+
address@hidden
address@hidden
+9 a + 3 b + c - 511*3 m - 511 n
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ 9 a + 3 b + c - 511\times3 m - 511 n $$
+\afterdisplay
address@hidden tex
+
address@hidden
+The @expr{m} term in the latter formula is redundant because any
+contribution it makes could just as easily be made by the @expr{n}
+term. So we can take it out to get an equivalent formula with
address@hidden' = 3m + n},
+
address@hidden
address@hidden
+9 a + 3 b + c - 511 n'
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ 9 a + 3 b + c - 511 n' $$
+\afterdisplay
address@hidden tex
+
address@hidden
+which is just the formula for taking the modulo only at the end of
+the calculation. Therefore the two methods are essentially the same.
+
+Later in the tutorial we will encounter @dfn{modulo forms}, which
+basically automate the idea of reducing every intermediate result
+modulo some value @var{m}.
+
address@hidden List Answer 14, Types Answer 1, List Answer 13, Answers to
Exercises
address@hidden List Tutorial Exercise 14
+
+We want to use @kbd{H V U} to nest a function which adds a random
+step to an @expr{(x,y)} coordinate. The function is a bit long, but
+otherwise the problem is quite straightforward.
+
address@hidden
address@hidden
+2: [0, 0] 1: [ [ 0, 0 ]
+1: 50 [ 0.4288, -0.1695 ]
+ . [ -0.4787, -0.9027 ]
+ ...
+
+ [0,0] 50 H V U ' <# + [random(2.0)-1, random(2.0)-1]> @key{RET}
address@hidden group
address@hidden smallexample
+
+Just as the text recommended, we used @samp{< >} nameless function
+notation to keep the two @code{random} calls from being evaluated
+before nesting even begins.
+
+We now have a vector of @expr{[x, y]} sub-vectors, which by Calc's
+rules acts like a matrix. We can transpose this matrix and unpack
+to get a pair of vectors, @expr{x} and @expr{y}, suitable for graphing.
+
address@hidden
address@hidden
+2: [ 0, 0.4288, -0.4787, ... ]
+1: [ 0, -0.1696, -0.9027, ... ]
+ .
+
+ v t v u g f
address@hidden group
address@hidden smallexample
+
+Incidentally, because the @expr{x} and @expr{y} are completely
+independent in this case, we could have done two separate commands
+to create our @expr{x} and @expr{y} vectors of numbers directly.
+
+To make a random walk of unit steps, we note that @code{sincos} of
+a random direction exactly gives us an @expr{[x, y]} step of unit
+length; in fact, the new nesting function is even briefer, though
+we might want to lower the precision a bit for it.
+
address@hidden
address@hidden
+2: [0, 0] 1: [ [ 0, 0 ]
+1: 50 [ 0.1318, 0.9912 ]
+ . [ -0.5965, 0.3061 ]
+ ...
+
+ [0,0] 50 m d p 6 @key{RET} H V U ' <# + sincos(random(360.0))>
@key{RET}
address@hidden group
address@hidden smallexample
+
+Another @kbd{v t v u g f} sequence will graph this new random walk.
+
+An interesting twist on these random walk functions would be to use
+complex numbers instead of 2-vectors to represent points on the plane.
+In the first example, we'd use something like @samp{random + random*(0,1)},
+and in the second we could use polar complex numbers with random phase
+angles. (This exercise was first suggested in this form by Randal
+Schwartz.)
+
address@hidden Types Answer 1, Types Answer 2, List Answer 14, Answers to
Exercises
address@hidden Types Tutorial Exercise 1
+
address@hidden
+If the number is the square root of @cpi{} times a rational number,
+then its square, divided by @cpi{}, should be a rational number.
+
address@hidden
address@hidden
+1: 1.26508260337 1: 0.509433962268 1: 2486645810:4881193627
+ . . .
+
+ 2 ^ P / c F
address@hidden group
address@hidden smallexample
+
address@hidden
+Technically speaking this is a rational number, but not one that is
+likely to have arisen in the original problem. More likely, it just
+happens to be the fraction which most closely represents some
+irrational number to within 12 digits.
+
+But perhaps our result was not quite exact. Let's reduce the
+precision slightly and try again:
+
address@hidden
address@hidden
+1: 0.509433962268 1: 27:53
+ . .
+
+ U p 10 @key{RET} c F
address@hidden group
address@hidden smallexample
+
address@hidden
+Aha! It's unlikely that an irrational number would equal a fraction
+this simple to within ten digits, so our original number was probably
address@hidden @math{\sqrt{27 \pi / 53}}.
address@hidden @expr{sqrt(27 pi / 53)}.
+
+Notice that we didn't need to re-round the number when we reduced the
+precision. Remember, arithmetic operations always round their inputs
+to the current precision before they begin.
+
address@hidden Types Answer 2, Types Answer 3, Types Answer 1, Answers to
Exercises
address@hidden Types Tutorial Exercise 2
+
address@hidden
address@hidden / inf = nan}. Perhaps @samp{1} is the ``obvious'' answer.
+But if @address@hidden inf = inf}}, then @samp{17 inf / inf = inf / inf = 17},
too.
+
address@hidden(inf) = inf}. It's tempting to say that the exponential
+of infinity must be ``bigger'' than ``regular'' infinity, but as
+far as Calc is concerned all infinities are as just as big.
+In other words, as @expr{x} goes to infinity, @expr{e^x} also goes
+to infinity, but the fact the @expr{e^x} grows much faster than
address@hidden is not relevant here.
+
address@hidden(-inf) = 0}. Here we have a finite answer even though
+the input is infinite.
+
address@hidden(-inf) = (0, 1) inf}. Remember that @expr{(0, 1)}
+represents the imaginary number @expr{i}. Here's a derivation:
address@hidden(-inf) = @w{sqrt((-1) * inf)} = sqrt(-1) * sqrt(inf)}.
+The first part is, by definition, @expr{i}; the second is @code{inf}
+because, once again, all infinities are the same size.
+
address@hidden(uinf) = uinf}. In fact, we do know something about the
+direction because @code{sqrt} is defined to return a value in the
+right half of the complex plane. But Calc has no notation for this,
+so it settles for the conservative answer @code{uinf}.
+
address@hidden(uinf) = inf}. No matter which direction @expr{x} points,
address@hidden(x)} always points along the positive real axis.
+
address@hidden(0) = -inf}. Here we have an infinite answer to a finite
+input. As in the @expr{1 / 0} case, Calc will only use infinities
+here if you have turned on Infinite mode. Otherwise, it will
+treat @samp{ln(0)} as an error.
+
address@hidden Types Answer 3, Types Answer 4, Types Answer 2, Answers to
Exercises
address@hidden Types Tutorial Exercise 3
+
address@hidden
+We can make @samp{inf - inf} be any real number we like, say,
address@hidden, just by claiming that we added @expr{a} to the first
+infinity but not to the second. This is just as true for complex
+values of @expr{a}, so @code{nan} can stand for a complex number.
+(And, similarly, @code{uinf} can stand for an infinity that points
+in any direction in the complex plane, such as @samp{(0, 1) inf}).
+
+In fact, we can multiply the first @code{inf} by two. Surely
address@hidden@samp{2 inf - inf = inf}}, but also @samp{2 inf - inf = inf - inf
= nan}.
+So @code{nan} can even stand for infinity. Obviously it's just
+as easy to make it stand for minus infinity as for plus infinity.
+
+The moral of this story is that ``infinity'' is a slippery fish
+indeed, and Calc tries to handle it by having a very simple model
+for infinities (only the direction counts, not the ``size''); but
+Calc is careful to write @code{nan} any time this simple model is
+unable to tell what the true answer is.
+
address@hidden Types Answer 4, Types Answer 5, Types Answer 3, Answers to
Exercises
address@hidden Types Tutorial Exercise 4
+
address@hidden
address@hidden
+2: 0@@ 47' 26" 1: 0@@ 2' 47.411765"
+1: 17 .
+ .
+
+ 0@@ 47' 26" @key{RET} 17 /
address@hidden group
address@hidden smallexample
+
address@hidden
+The average song length is two minutes and 47.4 seconds.
+
address@hidden
address@hidden
+2: 0@@ 2' 47.411765" 1: 0@@ 3' 7.411765" 1: 0@@ 53' 6.000005"
+1: 0@@ 0' 20" . .
+ .
+
+ 20" + 17 *
address@hidden group
address@hidden smallexample
+
address@hidden
+The album would be 53 minutes and 6 seconds long.
+
address@hidden Types Answer 5, Types Answer 6, Types Answer 4, Answers to
Exercises
address@hidden Types Tutorial Exercise 5
+
address@hidden
+Let's suppose it's January 14, 1991. The easiest thing to do is
+to keep trying 13ths of months until Calc reports a Friday.
+We can do this by manually entering dates, or by using @kbd{t I}:
+
address@hidden
address@hidden
+1: <Wed Feb 13, 1991> 1: <Wed Mar 13, 1991> 1: <Sat Apr 13, 1991>
+ . . .
+
+ ' <2/13> @key{RET} @key{DEL} ' <3/13> @key{RET} t I
address@hidden group
address@hidden smallexample
+
address@hidden
+(Calc assumes the current year if you don't say otherwise.)
+
+This is getting tedious---we can keep advancing the date by typing
address@hidden I} over and over again, but let's automate the job by using
+vector mapping. The @kbd{t I} command actually takes a second
+``how-many-months'' argument, which defaults to one. This
+argument is exactly what we want to map over:
+
address@hidden
address@hidden
+2: <Sat Apr 13, 1991> 1: [<Mon May 13, 1991>, <Thu Jun 13, 1991>,
+1: [1, 2, 3, 4, 5, 6] <Sat Jul 13, 1991>, <Tue Aug 13, 1991>,
+ . <Fri Sep 13, 1991>, <Sun Oct 13, 1991>]
+ .
+
+ v x 6 @key{RET} V M t I
address@hidden group
address@hidden smallexample
+
address@hidden
+Et address@hidden, September 13, 1991 is a Friday.
+
address@hidden
address@hidden
+1: 242
+ .
+
+' <sep 13> - <jan 14> @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+And the answer to our original question: 242 days to go.
+
address@hidden Types Answer 6, Types Answer 7, Types Answer 5, Answers to
Exercises
address@hidden Types Tutorial Exercise 6
+
address@hidden
+The full rule for leap years is that they occur in every year divisible
+by four, except that they don't occur in years divisible by 100, except
+that they @emph{do} in years divisible by 400. We could work out the
+answer by carefully counting the years divisible by four and the
+exceptions, but there is a much simpler way that works even if we
+don't know the leap year rule.
+
+Let's assume the present year is 1991. Years have 365 days, except
+that leap years (whenever they occur) have 366 days. So let's count
+the number of days between now and then, and compare that to the
+number of years times 365. The number of extra days we find must be
+equal to the number of leap years there were.
+
address@hidden
address@hidden
+1: <Mon Jan 1, 10001> 2: <Mon Jan 1, 10001> 1: 2925593
+ . 1: <Tue Jan 1, 1991> .
+ .
+
+ ' <jan 1 10001> @key{RET} ' <jan 1 1991> @key{RET} -
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+3: 2925593 2: 2925593 2: 2925593 1: 1943
+2: 10001 1: 8010 1: 2923650 .
+1: 1991 . .
+ .
+
+ 10001 @key{RET} 1991 - 365 * -
address@hidden group
address@hidden smallexample
+
address@hidden [fix-ref Date Forms]
address@hidden
+There will be 1943 leap years before the year 10001. (Assuming,
+of course, that the algorithm for computing leap years remains
+unchanged for that long. @xref{Date Forms}, for some interesting
+background information in that regard.)
+
address@hidden Types Answer 7, Types Answer 8, Types Answer 6, Answers to
Exercises
address@hidden Types Tutorial Exercise 7
+
address@hidden
+The relative errors must be converted to absolute errors so that
address@hidden/-} notation may be used.
+
address@hidden
address@hidden
+1: 1. 2: 1.
+ . 1: 0.2
+ .
+
+ 20 @key{RET} .05 * 4 @key{RET} .05 *
address@hidden group
address@hidden smallexample
+
+Now we simply chug through the formula.
+
address@hidden
address@hidden
+1: 19.7392088022 1: 394.78 +/- 19.739 1: 6316.5 +/- 706.21
+ . . .
+
+ 2 P 2 ^ * 20 p 1 * 4 p .2 @key{RET} 2 ^ *
address@hidden group
address@hidden smallexample
+
+It turns out the @kbd{v u} command will unpack an error form as
+well as a vector. This saves us some retyping of numbers.
+
address@hidden
address@hidden
+3: 6316.5 +/- 706.21 2: 6316.5 +/- 706.21
+2: 6316.5 1: 0.1118
+1: 706.21 .
+ .
+
+ @key{RET} v u @key{TAB} /
address@hidden group
address@hidden smallexample
+
address@hidden
+Thus the volume is 6316 cubic centimeters, within about 11 percent.
+
address@hidden Types Answer 8, Types Answer 9, Types Answer 7, Answers to
Exercises
address@hidden Types Tutorial Exercise 8
+
address@hidden
+The first answer is pretty simple: @samp{1 / (0 .. 10) = (0.1 .. inf)}.
+Since a number in the interval @samp{(0 .. 10)} can get arbitrarily
+close to zero, its reciprocal can get arbitrarily large, so the answer
+is an interval that effectively means, ``any number greater than 0.1''
+but with no upper bound.
+
+The second answer, similarly, is @samp{1 / (-10 .. 0) = (-inf .. -0.1)}.
+
+Calc normally treats division by zero as an error, so that the formula
address@hidden@samp{1 / 0}} is left unsimplified. Our third problem,
address@hidden@samp{1 / [0 .. 10]}}, also (potentially) divides by zero because
zero
+is now a member of the interval. So Calc leaves this one unevaluated, too.
+
+If you turn on Infinite mode by pressing @kbd{m i}, you will
+instead get the answer @samp{[0.1 .. inf]}, which includes infinity
+as a possible value.
+
+The fourth calculation, @samp{1 / (-10 .. 10)}, has the same problem.
+Zero is buried inside the interval, but it's still a possible value.
+It's not hard to see that the actual result of @samp{1 / (-10 .. 10)}
+will be either greater than @mathit{0.1}, or less than @mathit{-0.1}. Thus
+the interval goes from minus infinity to plus infinity, with a ``hole''
+in it from @mathit{-0.1} to @mathit{0.1}. Calc doesn't have any way to
+represent this, so it just reports @samp{[-inf .. inf]} as the answer.
+It may be disappointing to hear ``the answer lies somewhere between
+minus infinity and plus infinity, inclusive,'' but that's the best
+that interval arithmetic can do in this case.
+
address@hidden Types Answer 9, Types Answer 10, Types Answer 8, Answers to
Exercises
address@hidden Types Tutorial Exercise 9
+
address@hidden
address@hidden
+1: [-3 .. 3] 2: [-3 .. 3] 2: [0 .. 9]
+ . 1: [0 .. 9] 1: [-9 .. 9]
+ . .
+
+ [ 3 n .. 3 ] @key{RET} 2 ^ @key{TAB} @key{RET} *
address@hidden group
address@hidden smallexample
+
address@hidden
+In the first case the result says, ``if a number is between @mathit{-3} and
+3, its square is between 0 and 9.'' The second case says, ``the product
+of two numbers each between @mathit{-3} and 3 is between @mathit{-9} and 9.''
+
+An interval form is not a number; it is a symbol that can stand for
+many different numbers. Two identical-looking interval forms can stand
+for different numbers.
+
+The same issue arises when you try to square an error form.
+
address@hidden Types Answer 10, Types Answer 11, Types Answer 9, Answers to
Exercises
address@hidden Types Tutorial Exercise 10
+
address@hidden
+Testing the first number, we might arbitrarily choose 17 for @expr{x}.
+
address@hidden
address@hidden
+1: 17 mod 811749613 2: 17 mod 811749613 1: 533694123 mod 811749613
+ . 811749612 .
+ .
+
+ 17 M 811749613 @key{RET} 811749612 ^
address@hidden group
address@hidden smallexample
+
address@hidden
+Since 533694123 is (considerably) different from 1, the number 811749613
+must not be prime.
+
+It's awkward to type the number in twice as we did above. There are
+various ways to avoid this, and algebraic entry is one. In fact, using
+a vector mapping operation we can perform several tests at once. Let's
+use this method to test the second number.
+
address@hidden
address@hidden
+2: [17, 42, 100000] 1: [1 mod 15485863, 1 mod ... ]
+1: 15485863 .
+ .
+
+ [17 42 100000] 15485863 @key{RET} V M ' ($$ mod $)^($-1) @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+The result is three ones (modulo @expr{n}), so it's very probable that
+15485863 is prime. (In fact, this number is the millionth prime.)
+
+Note that the functions @samp{($$^($-1)) mod $} or @samp{$$^($-1) % $}
+would have been hopelessly inefficient, since they would have calculated
+the power using full integer arithmetic.
+
+Calc has a @kbd{k p} command that does primality testing. For small
+numbers it does an exact test; for large numbers it uses a variant
+of the Fermat test we used here. You can use @kbd{k p} repeatedly
+to prove that a large integer is prime with any desired probability.
+
address@hidden Types Answer 11, Types Answer 12, Types Answer 10, Answers to
Exercises
address@hidden Types Tutorial Exercise 11
+
address@hidden
+There are several ways to insert a calculated number into an HMS form.
+One way to convert a number of seconds to an HMS form is simply to
+multiply the number by an HMS form representing one second:
+
address@hidden
address@hidden
+1: 31415926.5359 2: 31415926.5359 1: 8726@@ 38' 46.5359"
+ . 1: 0@@ 0' 1" .
+ .
+
+ P 1e7 * 0@@ 0' 1" *
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+2: 8726@@ 38' 46.5359" 1: 6@@ 6' 2.5359" mod 24@@ 0' 0"
+1: 15@@ 27' 16" mod 24@@ 0' 0" .
+ .
+
+ x time @key{RET} +
address@hidden group
address@hidden smallexample
+
address@hidden
+It will be just after six in the morning.
+
+The algebraic @code{hms} function can also be used to build an
+HMS form:
+
address@hidden
address@hidden
+1: hms(0, 0, 10000000. pi) 1: 8726@@ 38' 46.5359"
+ . .
+
+ ' hms(0, 0, 1e7 pi) @key{RET} =
address@hidden group
address@hidden smallexample
+
address@hidden
+The @kbd{=} key is necessary to evaluate the symbol @samp{pi} to
+the actual number 3.14159...
+
address@hidden Types Answer 12, Types Answer 13, Types Answer 11, Answers to
Exercises
address@hidden Types Tutorial Exercise 12
+
address@hidden
+As we recall, there are 17 songs of about 2 minutes and 47 seconds
+each.
+
address@hidden
address@hidden
+2: 0@@ 2' 47" 1: [0@@ 3' 7" .. 0@@ 3' 47"]
+1: [0@@ 0' 20" .. 0@@ 1' 0"] .
+ .
+
+ [ 0@@ 20" .. 0@@ 1' ] +
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: [0@@ 52' 59." .. 1@@ 4' 19."]
+ .
+
+ 17 *
address@hidden group
address@hidden smallexample
+
address@hidden
+No matter how long it is, the album will fit nicely on one CD.
+
address@hidden Types Answer 13, Types Answer 14, Types Answer 12, Answers to
Exercises
address@hidden Types Tutorial Exercise 13
+
address@hidden
+Type @kbd{' 1 yr @key{RET} u c s @key{RET}}. The answer is 31557600 seconds.
+
address@hidden Types Answer 14, Types Answer 15, Types Answer 13, Answers to
Exercises
address@hidden Types Tutorial Exercise 14
+
address@hidden
+How long will it take for a signal to get from one end of the computer
+to the other?
+
address@hidden
address@hidden
+1: m / c 1: 3.3356 ns
+ . .
+
+ ' 1 m / c @key{RET} u c ns @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+(Recall, @samp{c} is a ``unit'' corresponding to the speed of light.)
+
address@hidden
address@hidden
+1: 3.3356 ns 1: 0.81356 ns / ns 1: 0.81356
+2: 4.1 ns . .
+ .
+
+ ' 4.1 ns @key{RET} / u s
address@hidden group
address@hidden smallexample
+
address@hidden
+Thus a signal could take up to 81 percent of a clock cycle just to
+go from one place to another inside the computer, assuming the signal
+could actually attain the full speed of light. Pretty tight!
+
address@hidden Types Answer 15, Algebra Answer 1, Types Answer 14, Answers to
Exercises
address@hidden Types Tutorial Exercise 15
+
address@hidden
+The speed limit is 55 miles per hour on most highways. We want to
+find the ratio of Sam's speed to the US speed limit.
+
address@hidden
address@hidden
+1: 55 mph 2: 55 mph 3: 11 hr mph / yd
+ . 1: 5 yd / hr .
+ .
+
+ ' 55 mph @key{RET} ' 5 yd/hr @key{RET} /
address@hidden group
address@hidden smallexample
+
+The @kbd{u s} command cancels out these units to get a plain
+number. Now we take the logarithm base two to find the final
+answer, assuming that each successive pill doubles his speed.
+
address@hidden
address@hidden
+1: 19360. 2: 19360. 1: 14.24
+ . 1: 2 .
+ .
+
+ u s 2 B
address@hidden group
address@hidden smallexample
+
address@hidden
+Thus Sam can take up to 14 pills without a worry.
+
address@hidden Algebra Answer 1, Algebra Answer 2, Types Answer 15, Answers to
Exercises
address@hidden Algebra Tutorial Exercise 1
+
address@hidden
address@hidden [fix-ref Declarations]
+The result @samp{sqrt(x)^2} is simplified back to @expr{x} by the
+Calculator, but @samp{sqrt(x^2)} is not. (Consider what happens
+if @address@hidden = -4}}.) If @expr{x} is real, this formula could be
+simplified to @samp{abs(x)}, but for general complex arguments even
+that is not safe. (@xref{Declarations}, for a way to tell Calc
+that @expr{x} is known to be real.)
+
address@hidden Algebra Answer 2, Algebra Answer 3, Algebra Answer 1, Answers to
Exercises
address@hidden Algebra Tutorial Exercise 2
+
address@hidden
+Suppose our roots are @expr{[a, b, c]}. We want a polynomial which
+is zero when @expr{x} is any of these values. The trivial polynomial
address@hidden is zero when @expr{x=a}, so the product @expr{(x-a)(x-b)(x-c)}
+will do the job. We can use @kbd{a c x} to write this in a more
+familiar form.
+
address@hidden
address@hidden
+1: 34 x - 24 x^3 1: [1.19023, -1.19023, 0]
+ . .
+
+ r 2 a P x @key{RET}
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: [x - 1.19023, x + 1.19023, x] 1: (x - 1.19023) (x + 1.19023) x
+ . .
+
+ V M ' x-$ @key{RET} V R *
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: x^3 - 1.41666 x 1: 34 x - 24 x^3
+ . .
+
+ a c x @key{RET} 24 n * a x
address@hidden group
address@hidden smallexample
+
address@hidden
+Sure enough, our answer (multiplied by a suitable constant) is the
+same as the original polynomial.
+
address@hidden Algebra Answer 3, Algebra Answer 4, Algebra Answer 2, Answers to
Exercises
address@hidden Algebra Tutorial Exercise 3
+
address@hidden
address@hidden
+1: x sin(pi x) 1: (sin(pi x) - pi x cos(pi x)) / pi^2
+ . .
+
+ ' x sin(pi x) @key{RET} m r a i x @key{RET}
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: [y, 1]
+2: (sin(pi x) - pi x cos(pi x)) / pi^2
+ .
+
+ ' [y,1] @key{RET} @key{TAB}
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: [(sin(pi y) - pi y cos(pi y)) / pi^2, (sin(pi) - pi cos(pi)) / pi^2]
+ .
+
+ V M $ @key{RET}
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: (sin(pi y) - pi y cos(pi y)) / pi^2 + (pi cos(pi) - sin(pi)) / pi^2
+ .
+
+ V R -
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: (sin(3.14159 y) - 3.14159 y cos(3.14159 y)) / 9.8696 - 0.3183
+ .
+
+ =
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: [0., -0.95493, 0.63662, -1.5915, 1.2732]
+ .
+
+ v x 5 @key{RET} @key{TAB} V M $ @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden Algebra Answer 4, Rewrites Answer 1, Algebra Answer 3, Answers
to Exercises
address@hidden Algebra Tutorial Exercise 4
+
address@hidden
+The hard part is that @kbd{V R +} is no longer sufficient to add up all
+the contributions from the slices, since the slices have varying
+coefficients. So first we must come up with a vector of these
+coefficients. Here's one way:
+
address@hidden
address@hidden
+2: -1 2: 3 1: [4, 2, ..., 4]
+1: [1, 2, ..., 9] 1: [-1, 1, ..., -1] .
+ . .
+
+ 1 n v x 9 @key{RET} V M ^ 3 @key{TAB} -
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: [4, 2, ..., 4, 1] 1: [1, 4, 2, ..., 4, 1]
+ . .
+
+ 1 | 1 @key{TAB} |
address@hidden group
address@hidden smallexample
+
address@hidden
+Now we compute the function values. Note that for this method we need
+eleven values, including both endpoints of the desired interval.
+
address@hidden
address@hidden
+2: [1, 4, 2, ..., 4, 1]
+1: [1, 1.1, 1.2, ... , 1.8, 1.9, 2.]
+ .
+
+ 11 @key{RET} 1 @key{RET} .1 @key{RET} C-u v x
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+2: [1, 4, 2, ..., 4, 1]
+1: [0., 0.084941, 0.16993, ... ]
+ .
+
+ ' sin(x) ln(x) @key{RET} m r p 5 @key{RET} V M $ @key{RET}
address@hidden group
address@hidden smallexample
+
address@hidden
+Once again this calls for @kbd{V M * V R +}; a simple @kbd{*} does the
+same thing.
+
address@hidden
address@hidden
+1: 11.22 1: 1.122 1: 0.374
+ . . .
+
+ * .1 * 3 /
address@hidden group
address@hidden smallexample
+
address@hidden
+Wow! That's even better than the result from the Taylor series method.
+
address@hidden Rewrites Answer 1, Rewrites Answer 2, Algebra Answer 4, Answers
to Exercises
address@hidden Rewrites Tutorial Exercise 1
+
address@hidden
+We'll use Big mode to make the formulas more readable.
+
address@hidden
address@hidden
+ ___
+ 2 + V 2
+1: (2 + sqrt(2)) / (1 + sqrt(2)) 1: --------
+ . ___
+ 1 + V 2
+
+ .
+
+ ' (2+sqrt(2)) / (1+sqrt(2)) @key{RET} d B
address@hidden group
address@hidden smallexample
+
address@hidden
+Multiplying by the conjugate helps because @expr{(a+b) (a-b) = a^2 - b^2}.
+
address@hidden
address@hidden
+ ___ ___
+1: (2 + V 2 ) (V 2 - 1)
+ .
+
+ a r a/(b+c) := a*(b-c) / (b^2-c^2) @key{RET}
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+ ___ ___
+1: 2 + V 2 - 2 1: V 2
+ . .
+
+ a r a*(b+c) := a*b + a*c a s
address@hidden group
address@hidden smallexample
+
address@hidden
+(We could have used @kbd{a x} instead of a rewrite rule for the
+second step.)
+
+The multiply-by-conjugate rule turns out to be useful in many
+different circumstances, such as when the denominator involves
+sines and cosines or the imaginary constant @code{i}.
+
address@hidden Rewrites Answer 2, Rewrites Answer 3, Rewrites Answer 1, Answers
to Exercises
address@hidden Rewrites Tutorial Exercise 2
+
address@hidden
+Here is the rule set:
+
address@hidden
address@hidden
+[ fib(n) := fib(n, 1, 1) :: integer(n) :: n >= 1,
+ fib(1, x, y) := x,
+ fib(n, x, y) := fib(n-1, y, x+y) ]
address@hidden group
address@hidden smallexample
+
address@hidden
+The first rule turns a one-argument @code{fib} that people like to write
+into a three-argument @code{fib} that makes computation easier. The
+second rule converts back from three-argument form once the computation
+is done. The third rule does the computation itself. It basically
+says that if @expr{x} and @expr{y} are two consecutive Fibonacci numbers,
+then @expr{y} and @expr{x+y} are the next (overlapping) pair of Fibonacci
+numbers.
+
+Notice that because the number @expr{n} was ``validated'' by the
+conditions on the first rule, there is no need to put conditions on
+the other rules because the rule set would never get that far unless
+the input were valid. That further speeds computation, since no
+extra conditions need to be checked at every step.
+
+Actually, a user with a nasty sense of humor could enter a bad
+three-argument @code{fib} call directly, say, @samp{fib(0, 1, 1)},
+which would get the rules into an infinite loop. One thing that would
+help keep this from happening by accident would be to use something like
address@hidden instead of @code{fib} as the name of the three-argument
+function.
+
address@hidden Rewrites Answer 3, Rewrites Answer 4, Rewrites Answer 2, Answers
to Exercises
address@hidden Rewrites Tutorial Exercise 3
+
address@hidden
+He got an infinite loop. First, Calc did as expected and rewrote
address@hidden@samp{2 + 3 x}} to @samp{f(2, 3, x)}. Then it looked for ways to
+apply the rule again, and found that @samp{f(2, 3, x)} looks like
address@hidden + b x} with @address@hidden = 0}} and @samp{b = 1}, so it
rewrote to
address@hidden(0, 1, f(2, 3, x))}. It then wrapped another @samp{f(0, 1, ...)}
+around that, and so on, ad infinitum. Joe should have used @kbd{M-1 a r}
+to make sure the rule applied only once.
+
+(Actually, even the first step didn't work as he expected. What Calc
+really gives for @kbd{M-1 a r} in this situation is @samp{f(3 x, 1, 2)},
+treating 2 as the ``variable,'' and @samp{3 x} as a constant being added
+to it. While this may seem odd, it's just as valid a solution as the
+``obvious'' one. One way to fix this would be to add the condition
address@hidden:: variable(x)} to the rule, to make sure the thing that matches
address@hidden is indeed a variable, or to change @samp{x} to @samp{quote(x)}
+on the lefthand side, so that the rule matches the actual variable
address@hidden rather than letting @samp{x} stand for something else.)
+
address@hidden Rewrites Answer 4, Rewrites Answer 5, Rewrites Answer 3, Answers
to Exercises
address@hidden Rewrites Tutorial Exercise 4
+
address@hidden
address@hidden
address@hidden
address@hidden ignore
address@hidden seq
+Here is a suitable set of rules to solve the first part of the problem:
+
address@hidden
address@hidden
+[ seq(n, c) := seq(n/2, c+1) :: n%2 = 0,
+ seq(n, c) := seq(3n+1, c+1) :: n%2 = 1 :: n > 1 ]
address@hidden group
address@hidden smallexample
+
+Given the initial formula @samp{seq(6, 0)}, application of these
+rules produces the following sequence of formulas:
+
address@hidden
+seq( 3, 1)
+seq(10, 2)
+seq( 5, 3)
+seq(16, 4)
+seq( 8, 5)
+seq( 4, 6)
+seq( 2, 7)
+seq( 1, 8)
address@hidden example
+
address@hidden
+whereupon neither of the rules match, and rewriting stops.
+
+We can pretty this up a bit with a couple more rules:
+
address@hidden
address@hidden
+[ seq(n) := seq(n, 0),
+ seq(1, c) := c,
+ ... ]
address@hidden group
address@hidden smallexample
+
address@hidden
+Now, given @samp{seq(6)} as the starting configuration, we get 8
+as the result.
+
+The change to return a vector is quite simple:
+
address@hidden
address@hidden
+[ seq(n) := seq(n, []) :: integer(n) :: n > 0,
+ seq(1, v) := v | 1,
+ seq(n, v) := seq(n/2, v | n) :: n%2 = 0,
+ seq(n, v) := seq(3n+1, v | n) :: n%2 = 1 ]
address@hidden group
address@hidden smallexample
+
address@hidden
+Given @samp{seq(6)}, the result is @samp{[6, 3, 10, 5, 16, 8, 4, 2, 1]}.
+
+Notice that the @expr{n > 1} guard is no longer necessary on the last
+rule since the @expr{n = 1} case is now detected by another rule.
+But a guard has been added to the initial rule to make sure the
+initial value is suitable before the computation begins.
+
+While still a good idea, this guard is not as vitally important as it
+was for the @code{fib} function, since calling, say, @samp{seq(x, [])}
+will not get into an infinite loop. Calc will not be able to prove
+the symbol @samp{x} is either even or odd, so none of the rules will
+apply and the rewrites will stop right away.
+
address@hidden Rewrites Answer 5, Rewrites Answer 6, Rewrites Answer 4, Answers
to Exercises
address@hidden Rewrites Tutorial Exercise 5
+
address@hidden
address@hidden
address@hidden
address@hidden ignore
address@hidden nterms
+If @expr{x} is the sum @expr{a + b}, then
address@hidden(address@hidden@tfn{)}' must
+be address@hidden(address@hidden@tfn{)}' plus
address@hidden(address@hidden@tfn{)}'. If @expr{x}
+is not a sum, then address@hidden(address@hidden@tfn{)}' = 1.
+
address@hidden
address@hidden
+[ nterms(a + b) := nterms(a) + nterms(b),
+ nterms(x) := 1 ]
address@hidden group
address@hidden smallexample
+
address@hidden
+Here we have taken advantage of the fact that earlier rules always
+match before later rules; @samp{nterms(x)} will only be tried if we
+already know that @samp{x} is not a sum.
+
address@hidden Rewrites Answer 6, Programming Answer 1, Rewrites Answer 5,
Answers to Exercises
address@hidden Rewrites Tutorial Exercise 6
+
address@hidden
+Here is a rule set that will do the job:
+
address@hidden
address@hidden
+[ a*(b + c) := a*b + a*c,
+ opt(a) O(x^n) + opt(b) O(x^m) := O(x^n) :: n <= m
+ :: constant(a) :: constant(b),
+ opt(a) O(x^n) + opt(b) x^m := O(x^n) :: n <= m
+ :: constant(a) :: constant(b),
+ a O(x^n) := O(x^n) :: constant(a),
+ x^opt(m) O(x^n) := O(x^(n+m)),
+ O(x^n) O(x^m) := O(x^(n+m)) ]
address@hidden group
address@hidden smallexample
+
+If we really want the @kbd{+} and @kbd{*} keys to operate naturally
+on power series, we should put these rules in @code{EvalRules}. For
+testing purposes, it is better to put them in a different variable,
+say, @code{O}, first.
+
+The first rule just expands products of sums so that the rest of the
+rules can assume they have an expanded-out polynomial to work with.
+Note that this rule does not mention @samp{O} at all, so it will
+apply to any product-of-sum it encounters---this rule may surprise
+you if you put it into @code{EvalRules}!
+
+In the second rule, the sum of two O's is changed to the smaller O.
+The optional constant coefficients are there mostly so that
address@hidden(x^2) - O(x^3)} and @samp{O(x^3) - O(x^2)} are handled
+as well as @samp{O(x^2) + O(x^3)}.
+
+The third rule absorbs higher powers of @samp{x} into O's.
+
+The fourth rule says that a constant times a negligible quantity
+is still negligible. (This rule will also match @samp{O(x^3) / 4},
+with @samp{a = 1/4}.)
+
+The fifth rule rewrites, for example, @samp{x^2 O(x^3)} to @samp{O(x^5)}.
+(It is easy to see that if one of these forms is negligible, the other
+is, too.) Notice the @samp{x^opt(m)} to pick up terms like
address@hidden@samp{x O(x^3)}}. Optional powers will match @samp{x} as
@samp{x^1}
+but not 1 as @samp{x^0}. This turns out to be exactly what we want here.
+
+The sixth rule is the corresponding rule for products of two O's.
+
+Another way to solve this problem would be to create a new ``data type''
+that represents truncated power series. We might represent these as
+function calls @samp{series(@var{coefs}, @var{x})} where @var{coefs} is
+a vector of coefficients for @expr{x^0}, @expr{x^1}, @expr{x^2}, and so
+on. Rules would exist for sums and products of such @code{series}
+objects, and as an optional convenience could also know how to combine a
address@hidden object with a normal polynomial. (With this, and with a
+rule that rewrites @samp{O(x^n)} to the equivalent @code{series} form,
+you could still enter power series in exactly the same notation as
+before.) Operations on such objects would probably be more efficient,
+although the objects would be a bit harder to read.
+
address@hidden [fix-ref Compositions]
+Some other symbolic math programs provide a power series data type
+similar to this. Mathematica, for example, has an object that looks
+like @address@hidden, @var{x0}, @var{coefs}, @var{nmin},
address@hidden, @var{den}]}, where @var{x0} is the point about which the
+power series is taken (we've been assuming this was always zero),
+and @var{nmin}, @var{nmax}, and @var{den} allow pseudo-power-series
+with fractional or negative powers. Also, the @code{PowerSeries}
+objects have a special display format that makes them look like
address@hidden x^2 + O(x^4)} when they are printed out. (@xref{Compositions},
+for a way to do this in Calc, although for something as involved as
+this it would probably be better to write the formatting routine
+in Lisp.)
+
address@hidden Programming Answer 1, Programming Answer 2, Rewrites Answer 6,
Answers to Exercises
address@hidden Programming Tutorial Exercise 1
+
address@hidden
+Just enter the formula @samp{ninteg(sin(t)/t, t, 0, x)}, type
address@hidden F}, and answer the questions. Since this formula contains two
+variables, the default argument list will be @samp{(t x)}. We want to
+change this to @samp{(x)} since @expr{t} is really a dummy variable
+to be used within @code{ninteg}.
+
+The exact keystrokes are @kbd{Z F s Si @key{RET} @key{RET} C-b C-b @key{DEL}
@key{DEL} @key{RET} y}.
+(The @kbd{C-b C-b @key{DEL} @key{DEL}} are what fix the argument list.)
+
address@hidden Programming Answer 2, Programming Answer 3, Programming Answer
1, Answers to Exercises
address@hidden Programming Tutorial Exercise 2
+
address@hidden
+One way is to move the number to the top of the stack, operate on
+it, then move it back: @kbd{C-x ( address@hidden n address@hidden
address@hidden C-x )}.
+
+Another way is to negate the top three stack entries, then negate
+again the top two stack entries: @kbd{C-x ( M-3 n M-2 n C-x )}.
+
+Finally, it turns out that a negative prefix argument causes a
+command like @kbd{n} to operate on the specified stack entry only,
+which is just what we want: @kbd{C-x ( M-- 3 n C-x )}.
+
+Just for kicks, let's also do it algebraically:
address@hidden@kbd{C-x ( ' -$$$, $$, $ @key{RET} C-x )}}.
+
address@hidden Programming Answer 3, Programming Answer 4, Programming Answer
2, Answers to Exercises
address@hidden Programming Tutorial Exercise 3
+
address@hidden
+Each of these functions can be computed using the stack, or using
+algebraic entry, whichever way you prefer:
+
address@hidden
+Computing
address@hidden @math{\displaystyle{\sin x \over x}}:
address@hidden @expr{sin(x) / x}:
+
+Using the stack: @kbd{C-x ( @key{RET} S @key{TAB} / C-x )}.
+
+Using algebraic entry: @kbd{C-x ( ' sin($)/$ @key{RET} C-x )}.
+
address@hidden
+Computing the logarithm:
+
+Using the stack: @kbd{C-x ( @key{TAB} B C-x )}
+
+Using algebraic entry: @kbd{C-x ( ' log($,$$) @key{RET} C-x )}.
+
address@hidden
+Computing the vector of integers:
+
+Using the stack: @kbd{C-x ( 1 @key{RET} 1 C-u v x C-x )}. (Recall that
address@hidden v x} takes the vector size, starting value, and increment
+from the stack.)
+
+Alternatively: @kbd{C-x ( ~ v x C-x )}. (The @kbd{~} key pops a
+number from the stack and uses it as the prefix argument for the
+next command.)
+
+Using algebraic entry: @kbd{C-x ( ' index($) @key{RET} C-x )}.
+
address@hidden Programming Answer 4, Programming Answer 5, Programming Answer
3, Answers to Exercises
address@hidden Programming Tutorial Exercise 4
+
address@hidden
+Here's one way: @kbd{C-x ( @key{RET} V R + @key{TAB} v l / C-x )}.
+
address@hidden Programming Answer 5, Programming Answer 6, Programming Answer
4, Answers to Exercises
address@hidden Programming Tutorial Exercise 5
+
address@hidden
address@hidden
+2: 1 1: 1.61803398502 2: 1.61803398502
+1: 20 . 1: 1.61803398875
+ . .
+
+ 1 @key{RET} 20 Z < & 1 + Z > I H P
address@hidden group
address@hidden smallexample
+
address@hidden
+This answer is quite accurate.
+
address@hidden Programming Answer 6, Programming Answer 7, Programming Answer
5, Answers to Exercises
address@hidden Programming Tutorial Exercise 6
+
address@hidden
+Here is the matrix:
+
address@hidden
+[ [ 0, 1 ] * [a, b] = [b, a + b]
+ [ 1, 1 ] ]
address@hidden example
+
address@hidden
+Thus @samp{[0, 1; 1, 1]^n * [1, 1]} computes Fibonacci numbers @expr{n+1}
+and @expr{n+2}. Here's one program that does the job:
+
address@hidden
+C-x ( ' [0, 1; 1, 1] ^ ($-1) * [1, 1] @key{RET} v u @key{DEL} C-x )
address@hidden example
+
address@hidden
+This program is quite efficient because Calc knows how to raise a
+matrix (or other value) to the power @expr{n} in only
address@hidden @math{\log_2 n}
address@hidden @expr{log(n,2)}
+steps. For example, this program can compute the 1000th Fibonacci
+number (a 209-digit integer!) in about 10 steps; even though the
address@hidden < ... Z >} solution had much simpler steps, it would have
+required so many steps that it would not have been practical.
+
address@hidden Programming Answer 7, Programming Answer 8, Programming Answer
6, Answers to Exercises
address@hidden Programming Tutorial Exercise 7
+
address@hidden
+The trick here is to compute the harmonic numbers differently, so that
+the loop counter itself accumulates the sum of reciprocals. We use
+a separate variable to hold the integer counter.
+
address@hidden
address@hidden
+1: 1 2: 1 1: .
+ . 1: 4
+ .
+
+ 1 t 1 1 @key{RET} 4 Z ( t 2 r 1 1 + s 1 & Z )
address@hidden group
address@hidden smallexample
+
address@hidden
+The body of the loop goes as follows: First save the harmonic sum
+so far in variable 2. Then delete it from the stack; the for loop
+itself will take care of remembering it for us. Next, recall the
+count from variable 1, add one to it, and feed its reciprocal to
+the for loop to use as the step value. The for loop will increase
+the ``loop counter'' by that amount and keep going until the
+loop counter exceeds 4.
+
address@hidden
address@hidden
+2: 31 3: 31
+1: 3.99498713092 2: 3.99498713092
+ . 1: 4.02724519544
+ .
+
+ r 1 r 2 @key{RET} 31 & +
address@hidden group
address@hidden smallexample
+
+Thus we find that the 30th harmonic number is 3.99, and the 31st
+harmonic number is 4.02.
+
address@hidden Programming Answer 8, Programming Answer 9, Programming Answer
7, Answers to Exercises
address@hidden Programming Tutorial Exercise 8
+
address@hidden
+The first step is to compute the derivative @expr{f'(x)} and thus
+the formula
address@hidden @math{\displaystyle{x - {f(x) \over f'(x)}}}.
address@hidden @expr{x - f(x)/f'(x)}.
+
+(Because this definition is long, it will be repeated in concise form
+below. You can use @address@hidden * m}} to load it from there. While you are
+entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
+keystrokes without executing them. In the following diagrams we'll
+pretend Calc actually executed the keystrokes as you typed them,
+just for purposes of illustration.)
+
address@hidden
address@hidden
+2: sin(cos(x)) - 0.5 3: 4.5
+1: 4.5 2: sin(cos(x)) - 0.5
+ . 1: -(sin(x) cos(cos(x)))
+ .
+
+' sin(cos(x))-0.5 @key{RET} 4.5 m r C-x ( Z ` @key{TAB} @key{RET} a d x
@key{RET}
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+2: 4.5
+1: x + (sin(cos(x)) - 0.5) / sin(x) cos(cos(x))
+ .
+
+ / ' x @key{RET} @key{TAB} - t 1
address@hidden group
address@hidden smallexample
+
+Now, we enter the loop. We'll use a repeat loop with a 20-repetition
+limit just in case the method fails to converge for some reason.
+(Normally, the @address@hidden /}} command will stop the loop before all 20
+repetitions are done.)
+
address@hidden
address@hidden
+1: 4.5 3: 4.5 2: 4.5
+ . 2: x + (sin(cos(x)) ... 1: 5.24196456928
+ 1: 4.5 .
+ .
+
+ 20 Z < @key{RET} r 1 @key{TAB} s l x @key{RET}
address@hidden group
address@hidden smallexample
+
+This is the new guess for @expr{x}. Now we compare it with the
+old one to see if we've converged.
+
address@hidden
address@hidden
+3: 5.24196 2: 5.24196 1: 5.24196 1: 5.26345856348
+2: 5.24196 1: 0 . .
+1: 4.5 .
+ .
+
+ @key{RET} address@hidden a = Z / Z > Z ' C-x
)
address@hidden group
address@hidden smallexample
+
+The loop converges in just a few steps to this value. To check
+the result, we can simply substitute it back into the equation.
+
address@hidden
address@hidden
+2: 5.26345856348
+1: 0.499999999997
+ .
+
+ @key{RET} ' sin(cos($)) @key{RET}
address@hidden group
address@hidden smallexample
+
+Let's test the new definition again:
+
address@hidden
address@hidden
+2: x^2 - 9 1: 3.
+1: 1 .
+ .
+
+ ' x^2-9 @key{RET} 1 X
address@hidden group
address@hidden smallexample
+
+Once again, here's the full Newton's Method definition:
+
address@hidden
address@hidden
+C-x ( Z ` @key{TAB} @key{RET} a d x @key{RET} / ' x @key{RET} @key{TAB} -
t 1
+ 20 Z < @key{RET} r 1 @key{TAB} s l x @key{RET}
+ @key{RET} address@hidden a = Z /
+ Z >
+ Z '
+C-x )
address@hidden group
address@hidden example
+
address@hidden [fix-ref Nesting and Fixed Points]
+It turns out that Calc has a built-in command for applying a formula
+repeatedly until it converges to a number. @xref{Nesting and Fixed Points},
+to see how to use it.
+
address@hidden [fix-ref Root Finding]
+Also, of course, @kbd{a R} is a built-in command that uses Newton's
+method (among others) to look for numerical solutions to any equation.
address@hidden Finding}.
+
address@hidden Programming Answer 9, Programming Answer 10, Programming Answer
8, Answers to Exercises
address@hidden Programming Tutorial Exercise 9
+
address@hidden
+The first step is to adjust @expr{z} to be greater than 5. A simple
+``for'' loop will do the job here. If @expr{z} is less than 5, we
+reduce the problem using
address@hidden @math{\psi(z) = \psi(z+1) - 1/z}.
address@hidden @expr{psi(z) = psi(z+1) - 1/z}. We go
+on to compute
address@hidden @math{\psi(z+1)},
address@hidden @expr{psi(z+1)},
+and remember to add back a factor of @expr{-1/z} when we're done. This
+step is repeated until @expr{z > 5}.
+
+(Because this definition is long, it will be repeated in concise form
+below. You can use @address@hidden * m}} to load it from there. While you are
+entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
+keystrokes without executing them. In the following diagrams we'll
+pretend Calc actually executed the keystrokes as you typed them,
+just for purposes of illustration.)
+
address@hidden
address@hidden
+1: 1. 1: 1.
+ . .
+
+ 1.0 @key{RET} C-x ( Z ` s 1 0 t 2
address@hidden group
address@hidden smallexample
+
+Here, variable 1 holds @expr{z} and variable 2 holds the adjustment
+factor. If @expr{z < 5}, we use a loop to increase it.
+
+(By the way, we started with @samp{1.0} instead of the integer 1 because
+otherwise the calculation below will try to do exact fractional arithmetic,
+and will never converge because fractions compare equal only if they
+are exactly equal, not just equal to within the current precision.)
+
address@hidden
address@hidden
+3: 1. 2: 1. 1: 6.
+2: 1. 1: 1 .
+1: 5 .
+ .
+
+ @key{RET} 5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ]
address@hidden group
address@hidden smallexample
+
+Now we compute the initial part of the sum:
address@hidden @math{\ln z - {1 \over 2z}}
address@hidden @expr{ln(z) - 1/2z}
+minus the adjustment factor.
+
address@hidden
address@hidden
+2: 1.79175946923 2: 1.7084261359 1: -0.57490719743
+1: 0.0833333333333 1: 2.28333333333 .
+ . .
+
+ L r 1 2 * & - r 2 -
address@hidden group
address@hidden smallexample
+
+Now we evaluate the series. We'll use another ``for'' loop counting
+up the value of @expr{2 n}. (Calc does have a summation command,
address@hidden +}, but we'll use loops just to get more practice with them.)
+
address@hidden
address@hidden
+3: -0.5749 3: -0.5749 4: -0.5749 2: -0.5749
+2: 2 2: 1:6 3: 1:6 1: 2.3148e-3
+1: 40 1: 2 2: 2 .
+ . . 1: 36.
+ .
+
+ 2 @key{RET} 40 Z ( @key{RET} k b @key{TAB} @key{RET} r 1
@key{TAB} ^ * /
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+3: -0.5749 3: -0.5772 2: -0.5772 1: -0.577215664892
+2: -0.5749 2: -0.5772 1: 0 .
+1: 2.3148e-3 1: -0.5749 .
+ . .
+
+ @key{TAB} @key{RET} address@hidden - @key{RET} address@hidden a =
Z / 2 Z ) Z ' C-x )
address@hidden group
address@hidden smallexample
+
+This is the value of
address@hidden @math{-\gamma},
address@hidden @expr{- gamma},
+with a slight bit of roundoff error. To get a full 12 digits, let's use
+a higher precision:
+
address@hidden
address@hidden
+2: -0.577215664892 2: -0.577215664892
+1: 1. 1: -0.577215664901532
+
+ 1. @key{RET} p 16 @key{RET} X
address@hidden group
address@hidden smallexample
+
+Here's the complete sequence of keystrokes:
+
address@hidden
address@hidden
+C-x ( Z ` s 1 0 t 2
+ @key{RET} 5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ]
+ L r 1 2 * & - r 2 -
+ 2 @key{RET} 40 Z ( @key{RET} k b @key{TAB} @key{RET} r 1
@key{TAB} ^ * /
+ @key{TAB} @key{RET} address@hidden - @key{RET}
address@hidden a = Z /
+ 2 Z )
+ Z '
+C-x )
address@hidden group
address@hidden example
+
address@hidden Programming Answer 10, Programming Answer 11, Programming Answer
9, Answers to Exercises
address@hidden Programming Tutorial Exercise 10
+
address@hidden
+Taking the derivative of a term of the form @expr{x^n} will produce
+a term like
address@hidden @math{n x^{n-1}}.
address@hidden @expr{n x^(n-1)}.
+Taking the derivative of a constant
+produces zero. From this it is easy to see that the @expr{n}th
+derivative of a polynomial, evaluated at @expr{x = 0}, will equal the
+coefficient on the @expr{x^n} term times @expr{n!}.
+
+(Because this definition is long, it will be repeated in concise form
+below. You can use @address@hidden * m}} to load it from there. While you are
+entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
+keystrokes without executing them. In the following diagrams we'll
+pretend Calc actually executed the keystrokes as you typed them,
+just for purposes of illustration.)
+
address@hidden
address@hidden
+2: 5 x^4 + (x + 1)^2 3: 5 x^4 + (x + 1)^2
+1: 6 2: 0
+ . 1: 6
+ .
+
+ ' 5 x^4 + (x+1)^2 @key{RET} 6 C-x ( Z ` [ ] t 1 0 @key{TAB}
address@hidden group
address@hidden smallexample
+
address@hidden
+Variable 1 will accumulate the vector of coefficients.
+
address@hidden
address@hidden
+2: 0 3: 0 2: 5 x^4 + ...
+1: 5 x^4 + ... 2: 5 x^4 + ... 1: 1
+ . 1: 1 .
+ .
+
+ Z ( @key{TAB} @key{RET} 0 s l x @key{RET} address@hidden
! / s | 1
address@hidden group
address@hidden smallexample
+
address@hidden
+Note that @kbd{s | 1} appends the top-of-stack value to the vector
+in a variable; it is completely analogous to @kbd{s + 1}. We could
+have written instead, @kbd{r 1 @key{TAB} | t 1}.
+
address@hidden
address@hidden
+1: 20 x^3 + 2 x + 2 1: 0 1: [1, 2, 1, 0, 5, 0, 0]
+ . . .
+
+ a d x @key{RET} 1 Z ) @key{DEL} r 1 Z ' C-x )
address@hidden group
address@hidden smallexample
+
+To convert back, a simple method is just to map the coefficients
+against a table of powers of @expr{x}.
+
address@hidden
address@hidden
+2: [1, 2, 1, 0, 5, 0, 0] 2: [1, 2, 1, 0, 5, 0, 0]
+1: 6 1: [0, 1, 2, 3, 4, 5, 6]
+ . .
+
+ 6 @key{RET} 1 + 0 @key{RET} 1 C-u v x
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+2: [1, 2, 1, 0, 5, 0, 0] 2: 1 + 2 x + x^2 + 5 x^4
+1: [1, x, x^2, x^3, ... ] .
+ .
+
+ ' x @key{RET} @key{TAB} V M ^ *
address@hidden group
address@hidden smallexample
+
+Once again, here are the whole polynomial to/from vector programs:
+
address@hidden
address@hidden
+C-x ( Z ` [ ] t 1 0 @key{TAB}
+ Z ( @key{TAB} @key{RET} 0 s l x @key{RET} address@hidden ! / s | 1
+ a d x @key{RET}
+ 1 Z ) r 1
+ Z '
+C-x )
+
+C-x ( 1 + 0 @key{RET} 1 C-u v x ' x @key{RET} @key{TAB} V M ^ * C-x )
address@hidden group
address@hidden example
+
address@hidden Programming Answer 11, Programming Answer 12, Programming Answer
10, Answers to Exercises
address@hidden Programming Tutorial Exercise 11
+
address@hidden
+First we define a dummy program to go on the @kbd{z s} key. The true
address@hidden@kbd{z s}} key is supposed to take two numbers from the stack and
+return one number, so @key{DEL} as a dummy definition will make
+sure the stack comes out right.
+
address@hidden
address@hidden
+2: 4 1: 4 2: 4
+1: 2 . 1: 2
+ . .
+
+ 4 @key{RET} 2 C-x ( @key{DEL} C-x ) Z K s @key{RET} 2
address@hidden group
address@hidden smallexample
+
+The last step replaces the 2 that was eaten during the creation
+of the dummy @kbd{z s} command. Now we move on to the real
+definition. The recurrence needs to be rewritten slightly,
+to the form @expr{s(n,m) = s(n-1,m-1) - (n-1) s(n-1,m)}.
+
+(Because this definition is long, it will be repeated in concise form
+below. You can use @kbd{C-x * m} to load it from there.)
+
address@hidden
address@hidden
+2: 4 4: 4 3: 4 2: 4
+1: 2 3: 2 2: 2 1: 2
+ . 2: 4 1: 0 .
+ 1: 2 .
+ .
+
+ C-x ( M-2 @key{RET} a = Z [ @key{DEL} @key{DEL} 1 Z :
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+4: 4 2: 4 2: 3 4: 3 4: 3 3: 3
+3: 2 1: 2 1: 2 3: 2 3: 2 2: 2
+2: 2 . . 2: 3 2: 3 1: 3
+1: 0 1: 2 1: 1 .
+ . . .
+
+ @key{RET} 0 a = Z [ @key{DEL} @key{DEL} 0 Z : @key{TAB} 1 - @key{TAB}
M-2 @key{RET} 1 - z s
address@hidden group
address@hidden smallexample
+
address@hidden
+(Note that the value 3 that our dummy @kbd{z s} produces is not correct;
+it is merely a placeholder that will do just as well for now.)
+
address@hidden
address@hidden
+3: 3 4: 3 3: 3 2: 3 1: -6
+2: 3 3: 3 2: 3 1: 9 .
+1: 2 2: 3 1: 3 .
+ . 1: 2 .
+ .
+
+ address@hidden address@hidden @key{TAB} @key{RET} address@hidden
z s * -
+
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
+1: -6 2: 4 1: 11 2: 11
+ . 1: 2 . 1: 11
+ . .
+
+ Z ] Z ] C-x ) Z K s @key{RET} @key{DEL} 4 @key{RET} 2 z s
address@hidden k s
address@hidden group
address@hidden smallexample
+
+Even though the result that we got during the definition was highly
+bogus, once the definition is complete the @kbd{z s} command gets
+the right answers.
+
+Here's the full program once again:
+
address@hidden
address@hidden
+C-x ( M-2 @key{RET} a =
+ Z [ @key{DEL} @key{DEL} 1
+ Z : @key{RET} 0 a =
+ Z [ @key{DEL} @key{DEL} 0
+ Z : @key{TAB} 1 - @key{TAB} M-2 @key{RET} 1 - z s
+ address@hidden address@hidden @key{TAB} @key{RET}
address@hidden z s * -
+ Z ]
+ Z ]
+C-x )
address@hidden group
address@hidden example
+
+You can read this definition using @kbd{C-x * m} (@code{read-kbd-macro})
+followed by @kbd{Z K s}, without having to make a dummy definition
+first, because @code{read-kbd-macro} doesn't need to execute the
+definition as it reads it in. For this reason, @code{C-x * m} is often
+the easiest way to create recursive programs in Calc.
+
address@hidden Programming Answer 12, , Programming Answer 11, Answers to
Exercises
address@hidden Programming Tutorial Exercise 12
+
address@hidden
+This turns out to be a much easier way to solve the problem. Let's
+denote Stirling numbers as calls of the function @samp{s}.
+
+First, we store the rewrite rules corresponding to the definition of
+Stirling numbers in a convenient variable:
+
address@hidden
+s e StirlingRules @key{RET}
+[ s(n,n) := 1 :: n >= 0,
+ s(n,0) := 0 :: n > 0,
+ s(n,m) := s(n-1,m-1) - (n-1) s(n-1,m) :: n >= m :: m >= 1 ]
+C-c C-c
address@hidden smallexample
+
+Now, it's just a matter of applying the rules:
+
address@hidden
address@hidden
+2: 4 1: s(4, 2) 1: 11
+1: 2 . .
+ .
+
+ 4 @key{RET} 2 C-x ( ' s($$,$) @key{RET} a r StirlingRules
@key{RET} C-x )
address@hidden group
address@hidden smallexample
+
+As in the case of the @code{fib} rules, it would be useful to put these
+rules in @code{EvalRules} and to add a @samp{:: remember} condition to
+the last rule.
+
address@hidden This ends the table-of-contents kludge from above:
address@hidden
+\global\let\chapternofonts=\oldchapternofonts
address@hidden tex
+
address@hidden [reference]
+
address@hidden Introduction, Data Types, Tutorial, Top
address@hidden Introduction
+
address@hidden
+This chapter is the beginning of the Calc reference manual.
+It covers basic concepts such as the stack, algebraic and
+numeric entry, undo, numeric prefix arguments, etc.
+
address@hidden [when-split]
address@hidden (Chapter 2, the Tutorial, has been printed in a separate volume.)
+
address@hidden
+* Basic Commands::
+* Help Commands::
+* Stack Basics::
+* Numeric Entry::
+* Algebraic Entry::
+* Quick Calculator::
+* Prefix Arguments::
+* Undo::
+* Error Messages::
+* Multiple Calculators::
+* Troubleshooting Commands::
address@hidden menu
+
address@hidden Basic Commands, Help Commands, Introduction, Introduction
address@hidden Basic Commands
+
address@hidden
address@hidden calc
address@hidden calc-mode
address@hidden Starting the Calculator
address@hidden Running the Calculator
+To start the Calculator in its standard interface, type @kbd{M-x calc}.
+By default this creates a pair of small windows, @samp{*Calculator*}
+and @samp{*Calc Trail*}. The former displays the contents of the
+Calculator stack and is manipulated exclusively through Calc commands.
+It is possible (though not usually necessary) to create several Calc
+mode buffers each of which has an independent stack, undo list, and
+mode settings. There is exactly one Calc Trail buffer; it records a
+list of the results of all calculations that have been done. The
+Calc Trail buffer uses a variant of Calc mode, so Calculator commands
+still work when the trail buffer's window is selected. It is possible
+to turn the trail window off, but the @samp{*Calc Trail*} buffer itself
+still exists and is updated silently. @xref{Trail Commands}.
+
address@hidden C-x * c
address@hidden C-x * *
address@hidden
address@hidden @null
address@hidden ignore
+In most installations, the @kbd{C-x * c} key sequence is a more
+convenient way to start the Calculator. Also, @kbd{C-x * *}
+is a synonym for @kbd{C-x * c} unless you last used Calc
+in its Keypad mode.
+
address@hidden x
address@hidden M-x
address@hidden calc-execute-extended-command
+Most Calc commands use one or two keystrokes. Lower- and upper-case
+letters are distinct. Commands may also be entered in full @kbd{M-x} form;
+for some commands this is the only form. As a convenience, the @kbd{x}
+key (@code{calc-execute-extended-command})
+is like @kbd{M-x} except that it enters the initial string @samp{calc-}
+for you. For example, the following key sequences are equivalent:
address@hidden, @kbd{M-x calc-sin @key{RET}}, @kbd{x sin @key{RET}}.
+
address@hidden Extensions module
address@hidden @file{calc-ext} module
+The Calculator exists in many parts. When you type @kbd{C-x * c}, the
+Emacs ``auto-load'' mechanism will bring in only the first part, which
+contains the basic arithmetic functions. The other parts will be
+auto-loaded the first time you use the more advanced commands like trig
+functions or matrix operations. This is done to improve the response time
+of the Calculator in the common case when all you need to do is a
+little arithmetic. If for some reason the Calculator fails to load an
+extension module automatically, you can force it to load all the
+extensions by using the @kbd{C-x * L} (@code{calc-load-everything})
+command. @xref{Mode Settings}.
+
+If you type @kbd{M-x calc} or @kbd{C-x * c} with any numeric prefix argument,
+the Calculator is loaded if necessary, but it is not actually started.
+If the argument is positive, the @file{calc-ext} extensions are also
+loaded if necessary. User-written Lisp code that wishes to make use
+of Calc's arithmetic routines can use @samp{(calc 0)} or @samp{(calc 1)}
+to auto-load the Calculator.
+
address@hidden C-x * b
address@hidden full-calc
+If you type @kbd{C-x * b}, then next time you use @kbd{C-x * c} you
+will get a Calculator that uses the full height of the Emacs screen.
+When full-screen mode is on, @kbd{C-x * c} runs the @code{full-calc}
+command instead of @code{calc}. From the Unix shell you can type
address@hidden -f full-calc} to start a new Emacs specifically for use
+as a calculator. When Calc is started from the Emacs command line
+like this, Calc's normal ``quit'' commands actually quit Emacs itself.
+
address@hidden C-x * o
address@hidden calc-other-window
+The @kbd{C-x * o} command is like @kbd{C-x * c} except that the Calc
+window is not actually selected. If you are already in the Calc
+window, @kbd{C-x * o} switches you out of it. (The regular Emacs
address@hidden o} command would also work for this, but it has a
+tendency to drop you into the Calc Trail window instead, which
address@hidden * o} takes care not to do.)
+
address@hidden
address@hidden C-x * q
address@hidden ignore
+For one quick calculation, you can type @kbd{C-x * q} (@code{quick-calc})
+which prompts you for a formula (like @samp{2+3/4}). The result is
+displayed at the bottom of the Emacs screen without ever creating
+any special Calculator windows. @xref{Quick Calculator}.
+
address@hidden
address@hidden C-x * k
address@hidden ignore
+Finally, if you are using the X window system you may want to try
address@hidden * k} (@code{calc-keypad}) which runs Calc with a
+``calculator keypad'' picture as well as a stack display. Click on
+the keys with the mouse to operate the calculator. @xref{Keypad Mode}.
+
address@hidden q
address@hidden calc-quit
address@hidden Quitting the Calculator
address@hidden Exiting the Calculator
+The @kbd{q} key (@code{calc-quit}) exits Calc mode and closes the
+Calculator's window(s). It does not delete the Calculator buffers.
+If you type @kbd{M-x calc} again, the Calculator will reappear with the
+contents of the stack intact. Typing @kbd{C-x * c} or @kbd{C-x * *}
+again from inside the Calculator buffer is equivalent to executing
address@hidden; you can think of @kbd{C-x * *} as toggling the
+Calculator on and off.
+
address@hidden C-x * x
+The @kbd{C-x * x} command also turns the Calculator off, no matter which
+user interface (standard, Keypad, or Embedded) is currently active.
+It also cancels @code{calc-edit} mode if used from there.
+
address@hidden d @key{SPC}
address@hidden calc-refresh
address@hidden Refreshing a garbled display
address@hidden Garbled displays, refreshing
+The @kbd{d @key{SPC}} key sequence (@code{calc-refresh}) redraws the contents
+of the Calculator buffer from memory. Use this if the contents of the
+buffer have been damaged somehow.
+
address@hidden
address@hidden o
address@hidden ignore
+The @kbd{o} key (@code{calc-realign}) moves the cursor back to its
+``home'' position at the bottom of the Calculator buffer.
+
address@hidden <
address@hidden >
address@hidden calc-scroll-left
address@hidden calc-scroll-right
address@hidden Horizontal scrolling
address@hidden Scrolling
address@hidden Wide text, scrolling
+The @kbd{<} and @kbd{>} keys are bound to @code{calc-scroll-left} and
address@hidden These are just like the normal horizontal
+scrolling commands except that they scroll one half-screen at a time by
+default. (Calc formats its output to fit within the bounds of the
+window whenever it can.)
+
address@hidden @{
address@hidden @}
address@hidden calc-scroll-down
address@hidden calc-scroll-up
address@hidden Vertical scrolling
+The @address@hidden and @address@hidden keys are bound to
@code{calc-scroll-down}
+and @code{calc-scroll-up}. They scroll up or down by one-half the
+height of the Calc window.
+
address@hidden C-x * 0
address@hidden calc-reset
+The @kbd{C-x * 0} command (@code{calc-reset}; that's @kbd{C-x *} followed
+by a zero) resets the Calculator to its initial state. This clears
+the stack, resets all the modes to their initial values (the values
+that were saved with @kbd{m m} (@code{calc-save-modes})), clears the
+caches (@pxref{Caches}), and so on. (It does @emph{not} erase the
+values of any variables.) With an argument of 0, Calc will be reset to
+its default state; namely, the modes will be given their default values.
+With a positive prefix argument, @kbd{C-x * 0} preserves the contents of
+the stack but resets everything else to its initial state; with a
+negative prefix argument, @kbd{C-x * 0} preserves the contents of the
+stack but resets everything else to its default state.
+
address@hidden calc-version
+The @kbd{M-x calc-version} command displays the current version number
+of Calc and the name of the person who installed it on your system.
+(This information is also present in the @samp{*Calc Trail*} buffer,
+and in the output of the @kbd{h h} command.)
+
address@hidden Help Commands, Stack Basics, Basic Commands, Introduction
address@hidden Help Commands
+
address@hidden
address@hidden Help commands
address@hidden ?
address@hidden calc-help
+The @kbd{?} key (@code{calc-help}) displays a series of brief help messages.
+Some keys (such as @kbd{b} and @kbd{d}) are prefix keys, like Emacs'
address@hidden and @kbd{C-x} prefixes. You can type
address@hidden after a prefix to see a list of commands beginning with that
+prefix. (If the message includes @samp{[MORE]}, press @kbd{?} again
+to see additional commands for that prefix.)
+
address@hidden h h
address@hidden calc-full-help
+The @kbd{h h} (@code{calc-full-help}) command displays all the @kbd{?}
+responses at once. When printed, this makes a nice, compact (three pages)
+summary of Calc keystrokes.
+
+In general, the @kbd{h} key prefix introduces various commands that
+provide help within Calc. Many of the @kbd{h} key functions are
+Calc-specific analogues to the @kbd{C-h} functions for Emacs help.
+
address@hidden h i
address@hidden C-x * i
address@hidden i
address@hidden calc-info
+The @kbd{h i} (@code{calc-info}) command runs the Emacs Info system
+to read this manual on-line. This is basically the same as typing
address@hidden i} (the regular way to run the Info system), then, if Info
+is not already in the Calc manual, selecting the beginning of the
+manual. The @kbd{C-x * i} command is another way to read the Calc
+manual; it is different from @kbd{h i} in that it works any time,
+not just inside Calc. The plain @kbd{i} key is also equivalent to
address@hidden i}, though this key is obsolete and may be replaced with a
+different command in a future version of Calc.
+
address@hidden h t
address@hidden C-x * t
address@hidden calc-tutorial
+The @kbd{h t} (@code{calc-tutorial}) command runs the Info system on
+the Tutorial section of the Calc manual. It is like @kbd{h i},
+except that it selects the starting node of the tutorial rather
+than the beginning of the whole manual. (It actually selects the
+node ``Interactive Tutorial'' which tells a few things about
+using the Info system before going on to the actual tutorial.)
+The @kbd{C-x * t} key is equivalent to @kbd{h t} (but it works at
+all times).
+
address@hidden h s
address@hidden C-x * s
address@hidden calc-info-summary
+The @kbd{h s} (@code{calc-info-summary}) command runs the Info system
+on the Summary node of the Calc manual. @xref{Summary}. The @kbd{C-x * s}
+key is equivalent to @kbd{h s}.
+
address@hidden h k
address@hidden calc-describe-key
+The @kbd{h k} (@code{calc-describe-key}) command looks up a key
+sequence in the Calc manual. For example, @kbd{h k H a S} looks
+up the documentation on the @kbd{H a S} (@code{calc-solve-for})
+command. This works by looking up the textual description of
+the key(s) in the Key Index of the manual, then jumping to the
+node indicated by the index.
+
+Most Calc commands do not have traditional Emacs documentation
+strings, since the @kbd{h k} command is both more convenient and
+more instructive. This means the regular Emacs @kbd{C-h k}
+(@code{describe-key}) command will not be useful for Calc keystrokes.
+
address@hidden h c
address@hidden calc-describe-key-briefly
+The @kbd{h c} (@code{calc-describe-key-briefly}) command reads a
+key sequence and displays a brief one-line description of it at
+the bottom of the screen. It looks for the key sequence in the
+Summary node of the Calc manual; if it doesn't find the sequence
+there, it acts just like its regular Emacs counterpart @kbd{C-h c}
+(@code{describe-key-briefly}). For example, @kbd{h c H a S}
+gives the description:
+
address@hidden
+H a S runs calc-solve-for: a `H a S' v => fsolve(a,v) (?=notes)
address@hidden smallexample
+
address@hidden
+which means the command @kbd{H a S} or @kbd{H M-x calc-solve-for}
+takes a value @expr{a} from the stack, prompts for a value @expr{v},
+then applies the algebraic function @code{fsolve} to these values.
+The @samp{?=notes} message means you can now type @kbd{?} to see
+additional notes from the summary that apply to this command.
+
address@hidden h f
address@hidden calc-describe-function
+The @kbd{h f} (@code{calc-describe-function}) command looks up an
+algebraic function or a command name in the Calc manual. Enter an
+algebraic function name to look up that function in the Function
+Index or enter a command name beginning with @samp{calc-} to look it
+up in the Command Index. This command will also look up operator
+symbols that can appear in algebraic formulas, like @samp{%} and
address@hidden>}.
+
address@hidden h v
address@hidden calc-describe-variable
+The @kbd{h v} (@code{calc-describe-variable}) command looks up a
+variable in the Calc manual. Enter a variable name like @code{pi} or
address@hidden
+
address@hidden h b
address@hidden describe-bindings
+The @kbd{h b} (@code{calc-describe-bindings}) command is just like
address@hidden b}, except that only local (Calc-related) key bindings are
+listed.
+
address@hidden h n
+The @kbd{h n} or @kbd{h C-n} (@code{calc-view-news}) command displays
+the ``news'' or change history of Calc. This is kept in the file
address@hidden, which Calc looks for in the same directory as the Calc
+source files.
+
address@hidden h C-c
address@hidden h C-d
address@hidden h C-w
+The @kbd{h C-c}, @kbd{h C-d}, and @kbd{h C-w} keys display copying,
+distribution, and warranty information about Calc. These work by
+pulling up the appropriate parts of the ``Copying'' or ``Reporting
+Bugs'' sections of the manual.
+
address@hidden Stack Basics, Numeric Entry, Help Commands, Introduction
address@hidden Stack Basics
+
address@hidden
address@hidden Stack basics
address@hidden [fix-tut RPN Calculations and the Stack]
+Calc uses RPN notation. If you are not familiar with RPN, @pxref{RPN
+Tutorial}.
+
+To add the numbers 1 and 2 in Calc you would type the keys:
address@hidden @key{RET} 2 +}.
+(@key{RET} corresponds to the @key{ENTER} key on most calculators.)
+The first three keystrokes ``push'' the numbers 1 and 2 onto the stack. The
address@hidden key always ``pops'' the top two numbers from the stack, adds
them,
+and pushes the result (3) back onto the stack. This number is ready for
+further calculations: @kbd{5 -} pushes 5 onto the stack, then pops the
+3 and 5, subtracts them, and pushes the result (@mathit{-2}).
+
+Note that the ``top'' of the stack actually appears at the @emph{bottom}
+of the buffer. A line containing a single @samp{.} character signifies
+the end of the buffer; Calculator commands operate on the number(s)
+directly above this line. The @kbd{d t} (@code{calc-truncate-stack})
+command allows you to move the @samp{.} marker up and down in the stack;
address@hidden the Stack}.
+
address@hidden d l
address@hidden calc-line-numbering
+Stack elements are numbered consecutively, with number 1 being the top of
+the stack. These line numbers are ordinarily displayed on the lefthand side
+of the window. The @kbd{d l} (@code{calc-line-numbering}) command controls
+whether these numbers appear. (Line numbers may be turned off since they
+slow the Calculator down a bit and also clutter the display.)
+
address@hidden o
address@hidden calc-realign
+The unshifted letter @kbd{o} (@code{calc-realign}) command repositions
+the cursor to its top-of-stack ``home'' position. It also undoes any
+horizontal scrolling in the window. If you give it a numeric prefix
+argument, it instead moves the cursor to the specified stack element.
+
+The @key{RET} (or equivalent @key{SPC}) key is only required to separate
+two consecutive numbers.
+(After all, if you typed @kbd{1 2} by themselves the Calculator
+would enter the number 12.) If you press @key{RET} or @key{SPC} @emph{not}
+right after typing a number, the key duplicates the number on the top of
+the stack. @address@hidden *} is thus a handy way to square a number.
+
+The @key{DEL} key pops and throws away the top number on the stack.
+The @key{TAB} key swaps the top two objects on the stack.
address@hidden and Trail}, for descriptions of these and other stack-related
+commands.
+
address@hidden Numeric Entry, Algebraic Entry, Stack Basics, Introduction
address@hidden Numeric Entry
+
address@hidden
address@hidden 0-9
address@hidden .
address@hidden e
address@hidden Numeric entry
address@hidden Entering numbers
+Pressing a digit or other numeric key begins numeric entry using the
+minibuffer. The number is pushed on the stack when you press the @key{RET}
+or @key{SPC} keys. If you press any other non-numeric key, the number is
+pushed onto the stack and the appropriate operation is performed. If
+you press a numeric key which is not valid, the key is ignored.
+
address@hidden Minus signs
address@hidden Negative numbers, entering
address@hidden _
+There are three different concepts corresponding to the word ``minus,''
+typified by @expr{a-b} (subtraction), @expr{-x}
+(change-sign), and @expr{-5} (negative number). Calc uses three
+different keys for these operations, respectively:
address@hidden, @kbd{n}, and @kbd{_} (the underscore). The @kbd{-} key
subtracts
+the two numbers on the top of the stack. The @kbd{n} key changes the sign
+of the number on the top of the stack or the number currently being entered.
+The @kbd{_} key begins entry of a negative number or changes the sign of
+the number currently being entered. The following sequences all enter the
+number @mathit{-5} onto the stack: @kbd{0 @key{RET} 5 -}, @kbd{5 n @key{RET}},
address@hidden @key{RET} n}, @kbd{_ 5 @key{RET}}, @kbd{5 _ @key{RET}}.
+
+Some other keys are active during numeric entry, such as @kbd{#} for
+non-decimal numbers, @kbd{:} for fractions, and @kbd{@@} for HMS forms.
+These notations are described later in this manual with the corresponding
+data types. @xref{Data Types}.
+
+During numeric entry, the only editing key available is @key{DEL}.
+
address@hidden Algebraic Entry, Quick Calculator, Numeric Entry, Introduction
address@hidden Algebraic Entry
+
address@hidden
address@hidden '
address@hidden calc-algebraic-entry
address@hidden Algebraic notation
address@hidden Formulas, entering
+Calculations can also be entered in algebraic form. This is accomplished
+by typing the apostrophe key, ', followed by the expression in
+standard format:
+
address@hidden
+' 2+3*4 @key{RET}.
address@hidden example
+
address@hidden
+This will compute
address@hidden @math{2+(3\times4) = 14}
address@hidden @expr{2+(3*4) = 14}
+and push it on the stack. If you wish you can
+ignore the RPN aspect of Calc altogether and simply enter algebraic
+expressions in this way. You may want to use @key{DEL} every so often to
+clear previous results off the stack.
+
+You can press the apostrophe key during normal numeric entry to switch
+the half-entered number into Algebraic entry mode. One reason to do this
+would be to use the full Emacs cursor motion and editing keys, which are
+available during algebraic entry but not during numeric entry.
+
+In the same vein, during either numeric or algebraic entry you can
+press @kbd{`} (backquote) to switch to @code{calc-edit} mode, where
+you complete your half-finished entry in a separate buffer.
address@hidden Stack Entries}.
+
address@hidden m a
address@hidden calc-algebraic-mode
address@hidden Algebraic Mode
+If you prefer algebraic entry, you can use the command @kbd{m a}
+(@code{calc-algebraic-mode}) to set Algebraic mode. In this mode,
+digits and other keys that would normally start numeric entry instead
+start full algebraic entry; as long as your formula begins with a digit
+you can omit the apostrophe. Open parentheses and square brackets also
+begin algebraic entry. You can still do RPN calculations in this mode,
+but you will have to press @key{RET} to terminate every number:
address@hidden @key{RET} 3 @key{RET} * 4 @key{RET} +} would accomplish the same
+thing as @kbd{2*3+4 @key{RET}}.
+
address@hidden Incomplete Algebraic Mode
+If you give a numeric prefix argument like @kbd{C-u} to the @kbd{m a}
+command, it enables Incomplete Algebraic mode; this is like regular
+Algebraic mode except that it applies to the @kbd{(} and @kbd{[} keys
+only. Numeric keys still begin a numeric entry in this mode.
+
address@hidden m t
address@hidden calc-total-algebraic-mode
address@hidden Total Algebraic Mode
+The @kbd{m t} (@code{calc-total-algebraic-mode}) gives you an even
+stronger algebraic-entry mode, in which @emph{all} regular letter and
+punctuation keys begin algebraic entry. Use this if you prefer typing
address@hidden@kbd{sqrt( )}} instead of @kbd{Q}, @address@hidden( )}} instead of
address@hidden f}, and so on. To type regular Calc commands when you are in
+Total Algebraic mode, hold down the @key{META} key. Thus @kbd{M-q}
+is the command to quit Calc, @kbd{M-p} sets the precision, and
address@hidden t} (or @kbd{M-m M-t}, if you prefer) turns Total Algebraic
+mode back off again. Meta keys also terminate algebraic entry, so
+that @kbd{2+3 M-S} is equivalent to @kbd{2+3 @key{RET} M-S}. The symbol
address@hidden will appear in the mode line whenever you are in this mode.
+
+Pressing @kbd{'} (the apostrophe) a second time re-enters the previous
+algebraic formula. You can then use the normal Emacs editing keys to
+modify this formula to your liking before pressing @key{RET}.
+
address@hidden $
address@hidden Formulas, referring to stack
+Within a formula entered from the keyboard, the symbol @kbd{$}
+represents the number on the top of the stack. If an entered formula
+contains any @kbd{$} characters, the Calculator replaces the top of
+stack with that formula rather than simply pushing the formula onto the
+stack. Thus, @kbd{' 1+2 @key{RET}} pushes 3 on the stack, and @kbd{$*2
address@hidden replaces it with 6. Note that the @kbd{$} key always
+initiates algebraic entry; the @kbd{'} is unnecessary if @kbd{$} is the
+first character in the new formula.
+
+Higher stack elements can be accessed from an entered formula with the
+symbols @kbd{$$}, @kbd{$$$}, and so on. The number of stack elements
+removed (to be replaced by the entered values) equals the number of dollar
+signs in the longest such symbol in the formula. For example, @samp{$$+$$$}
+adds the second and third stack elements, replacing the top three elements
+with the answer. (All information about the top stack element is thus lost
+since no single @samp{$} appears in this formula.)
+
+A slightly different way to refer to stack elements is with a dollar
+sign followed by a number: @samp{$1}, @samp{$2}, and so on are much
+like @samp{$}, @samp{$$}, etc., except that stack entries referred
+to numerically are not replaced by the algebraic entry. That is, while
address@hidden replaces 5 on the stack with 6, @samp{$1+1} leaves the 5
+on the stack and pushes an additional 6.
+
+If a sequence of formulas are entered separated by commas, each formula
+is pushed onto the stack in turn. For example, @samp{1,2,3} pushes
+those three numbers onto the stack (leaving the 3 at the top), and
address@hidden,$-1} replaces a 5 on the stack with 4 followed by 6. Also,
address@hidden,$$} exchanges the top two elements of the stack, just like the
address@hidden key.
+
+You can finish an algebraic entry with @kbd{M-=} or @address@hidden instead
+of @key{RET}. This uses @kbd{=} to evaluate the variables in each
+formula that goes onto the stack. (Thus @kbd{' pi @key{RET}} pushes
+the variable @samp{pi}, but @kbd{' pi address@hidden pushes 3.1415.)
+
+If you finish your algebraic entry by pressing @key{LFD} (or @kbd{C-j})
+instead of @key{RET}, Calc disables the default simplifications
+(as if by @kbd{m O}; @pxref{Simplification Modes}) while the entry
+is being pushed on the stack. Thus @kbd{' 1+2 @key{RET}} pushes 3
+on the stack, but @kbd{' 1+2 @key{LFD}} pushes the formula @expr{1+2};
+you might then press @kbd{=} when it is time to evaluate this formula.
+
address@hidden Quick Calculator, Prefix Arguments, Algebraic Entry, Introduction
address@hidden ``Quick Calculator'' Mode
+
address@hidden
address@hidden C-x * q
address@hidden quick-calc
address@hidden Quick Calculator
+There is another way to invoke the Calculator if all you need to do
+is make one or two quick calculations. Type @kbd{C-x * q} (or
address@hidden quick-calc}), then type any formula as an algebraic entry.
+The Calculator will compute the result and display it in the echo
+area, without ever actually putting up a Calc window.
+
+You can use the @kbd{$} character in a Quick Calculator formula to
+refer to the previous Quick Calculator result. Older results are
+not retained; the Quick Calculator has no effect on the full
+Calculator's stack or trail. If you compute a result and then
+forget what it was, just run @code{C-x * q} again and enter
address@hidden as the formula.
+
+If this is the first time you have used the Calculator in this Emacs
+session, the @kbd{C-x * q} command will create the @code{*Calculator*}
+buffer and perform all the usual initializations; it simply will
+refrain from putting that buffer up in a new window. The Quick
+Calculator refers to the @code{*Calculator*} buffer for all mode
+settings. Thus, for example, to set the precision that the Quick
+Calculator uses, simply run the full Calculator momentarily and use
+the regular @kbd{p} command.
+
+If you use @code{C-x * q} from inside the Calculator buffer, the
+effect is the same as pressing the apostrophe key (algebraic entry).
+
+The result of a Quick calculation is placed in the Emacs ``kill ring''
+as well as being displayed. A subsequent @kbd{C-y} command will
+yank the result into the editing buffer. You can also use this
+to yank the result into the next @kbd{C-x * q} input line as a more
+explicit alternative to @kbd{$} notation, or to yank the result
+into the Calculator stack after typing @kbd{C-x * c}.
+
+If you finish your formula by typing @key{LFD} (or @kbd{C-j}) instead
+of @key{RET}, the result is inserted immediately into the current
+buffer rather than going into the kill ring.
+
+Quick Calculator results are actually evaluated as if by the @kbd{=}
+key (which replaces variable names by their stored values, if any).
+If the formula you enter is an assignment to a variable using the
address@hidden:=} operator, say, @samp{foo := 2 + 3} or @samp{foo := foo + 1},
+then the result of the evaluation is stored in that Calc variable.
address@hidden and Recall}.
+
+If the result is an integer and the current display radix is decimal,
+the number will also be displayed in hex, octal and binary formats. If
+the integer is in the range from 1 to 126, it will also be displayed as
+an ASCII character.
+
+For example, the quoted character @samp{"x"} produces the vector
+result @samp{[120]} (because 120 is the ASCII code of the lower-case
+`x'; @pxref{Strings}). Since this is a vector, not an integer, it
+is displayed only according to the current mode settings. But
+running Quick Calc again and entering @samp{120} will produce the
+result @samp{120 (16#78, 8#170, x)} which shows the number in its
+decimal, hexadecimal, octal, and ASCII forms.
+
+Please note that the Quick Calculator is not any faster at loading
+or computing the answer than the full Calculator; the name ``quick''
+merely refers to the fact that it's much less hassle to use for
+small calculations.
+
address@hidden Prefix Arguments, Undo, Quick Calculator, Introduction
address@hidden Numeric Prefix Arguments
+
address@hidden
+Many Calculator commands use numeric prefix arguments. Some, such as
address@hidden s} (@code{calc-sci-notation}), set a parameter to the value of
+the prefix argument or use a default if you don't use a prefix.
+Others (like @kbd{d f} (@code{calc-fix-notation})) require an argument
+and prompt for a number if you don't give one as a prefix.
+
+As a rule, stack-manipulation commands accept a numeric prefix argument
+which is interpreted as an index into the stack. A positive argument
+operates on the top @var{n} stack entries; a negative argument operates
+on the @var{n}th stack entry in isolation; and a zero argument operates
+on the entire stack.
+
+Most commands that perform computations (such as the arithmetic and
+scientific functions) accept a numeric prefix argument that allows the
+operation to be applied across many stack elements. For unary operations
+(that is, functions of one argument like absolute value or complex
+conjugate), a positive prefix argument applies that function to the top
address@hidden stack entries simultaneously, and a negative argument applies it
+to the @var{n}th stack entry only. For binary operations (functions of
+two arguments like addition, GCD, and vector concatenation), a positive
+prefix argument ``reduces'' the function across the top @var{n}
+stack elements (for example, @kbd{C-u 5 +} sums the top 5 stack entries;
address@hidden and Mapping}), and a negative argument maps the next-to-top
address@hidden stack elements with the top stack element as a second argument
+(for example, @kbd{7 c-u -5 +} adds 7 to the top 5 stack elements).
+This feature is not available for operations which use the numeric prefix
+argument for some other purpose.
+
+Numeric prefixes are specified the same way as always in Emacs: Press
+a sequence of @key{META}-digits, or press @key{ESC} followed by digits,
+or press @kbd{C-u} followed by digits. Some commands treat plain
address@hidden (without any actual digits) specially.
+
address@hidden ~
address@hidden calc-num-prefix
+You can type @kbd{~} (@code{calc-num-prefix}) to pop an integer from the
+top of the stack and enter it as the numeric prefix for the next command.
+For example, @kbd{C-u 16 p} sets the precision to 16 digits; an alternate
+(silly) way to do this would be @kbd{2 @key{RET} 4 ^ ~ p}, i.e., compute 2
+to the fourth power and set the precision to that value.
+
+Conversely, if you have typed a numeric prefix argument the @kbd{~} key
+pushes it onto the stack in the form of an integer.
+
address@hidden Undo, Error Messages, Prefix Arguments, Introduction
address@hidden Undoing Mistakes
+
address@hidden
address@hidden U
address@hidden C-_
address@hidden calc-undo
address@hidden Mistakes, undoing
address@hidden Undoing mistakes
address@hidden Errors, undoing
+The address@hidden key (@code{calc-undo}) undoes the most recent operation.
+If that operation added or dropped objects from the stack, those objects
+are removed or restored. If it was a ``store'' operation, you are
+queried whether or not to restore the variable to its original value.
+The @kbd{U} key may be pressed any number of times to undo successively
+farther back in time; with a numeric prefix argument it undoes a
+specified number of operations. The undo history is cleared only by the
address@hidden (@code{calc-quit}) command. (Recall that @kbd{C-x * c} is
+synonymous with @code{calc-quit} while inside the Calculator; this
+also clears the undo history.)
+
+Currently the mode-setting commands (like @code{calc-precision}) are not
+undoable. You can undo past a point where you changed a mode, but you
+will need to reset the mode yourself.
+
address@hidden D
address@hidden calc-redo
address@hidden Redoing after an Undo
+The address@hidden key (@code{calc-redo}) redoes an operation that was
+mistakenly undone. Pressing @kbd{U} with a negative prefix argument is
+equivalent to executing @code{calc-redo}. You can redo any number of
+times, up to the number of recent consecutive undo commands. Redo
+information is cleared whenever you give any command that adds new undo
+information, i.e., if you undo, then enter a number on the stack or make
+any other change, then it will be too late to redo.
+
address@hidden address@hidden
address@hidden calc-last-args
address@hidden Last-arguments feature
address@hidden Arguments, restoring
+The @address@hidden key (@code{calc-last-args}) is like undo in that
+it restores the arguments of the most recent command onto the stack;
+however, it does not remove the result of that command. Given a numeric
+prefix argument, this command applies to the @expr{n}th most recent
+command which removed items from the stack; it pushes those items back
+onto the stack.
+
+The @kbd{K} (@code{calc-keep-args}) command provides a related function
+to @address@hidden @xref{Stack and Trail}.
+
+It is also possible to recall previous results or inputs using the trail.
address@hidden Commands}.
+
+The standard Emacs @kbd{C-_} undo key is recognized as a synonym for @kbd{U}.
+
address@hidden Error Messages, Multiple Calculators, Undo, Introduction
address@hidden Error Messages
+
address@hidden
address@hidden w
address@hidden calc-why
address@hidden Errors, messages
address@hidden Why did an error occur?
+Many situations that would produce an error message in other calculators
+simply create unsimplified formulas in the Emacs Calculator. For example,
address@hidden @key{RET} 0 /} pushes the formula @expr{1 / 0}; @address@hidden
L}} pushes
+the formula @samp{ln(0)}. Floating-point overflow and underflow are also
+reasons for this to happen.
+
+When a function call must be left in symbolic form, Calc usually
+produces a message explaining why. Messages that are probably
+surprising or indicative of user errors are displayed automatically.
+Other messages are simply kept in Calc's memory and are displayed only
+if you type @kbd{w} (@code{calc-why}). You can also press @kbd{w} if
+the same computation results in several messages. (The first message
+will end with @samp{[w=more]} in this case.)
+
address@hidden d w
address@hidden calc-auto-why
+The @kbd{d w} (@code{calc-auto-why}) command controls when error messages
+are displayed automatically. (Calc effectively presses @kbd{w} for you
+after your computation finishes.) By default, this occurs only for
+``important'' messages. The other possible modes are to report
address@hidden messages automatically, or to report none automatically (so
+that you must always press @kbd{w} yourself to see the messages).
+
address@hidden Multiple Calculators, Troubleshooting Commands, Error Messages,
Introduction
address@hidden Multiple Calculators
+
address@hidden
address@hidden another-calc
+It is possible to have any number of Calc mode buffers at once.
+Usually this is done by executing @kbd{M-x another-calc}, which
+is similar to @kbd{C-x * c} except that if a @samp{*Calculator*}
+buffer already exists, a new, independent one with a name of the
+form @samp{*Calculator*<@var{n}>} is created. You can also use the
+command @code{calc-mode} to put any buffer into Calculator mode, but
+this would ordinarily never be done.
+
+The @kbd{q} (@code{calc-quit}) command does not destroy a Calculator buffer;
+it only closes its window. Use @kbd{M-x kill-buffer} to destroy a
+Calculator buffer.
+
+Each Calculator buffer keeps its own stack, undo list, and mode settings
+such as precision, angular mode, and display formats. In Emacs terms,
+variables such as @code{calc-stack} are buffer-local variables. The
+global default values of these variables are used only when a new
+Calculator buffer is created. The @code{calc-quit} command saves
+the stack and mode settings of the buffer being quit as the new defaults.
+
+There is only one trail buffer, @samp{*Calc Trail*}, used by all
+Calculator buffers.
+
address@hidden Troubleshooting Commands, , Multiple Calculators, Introduction
address@hidden Troubleshooting Commands
+
address@hidden
+This section describes commands you can use in case a computation
+incorrectly fails or gives the wrong answer.
+
address@hidden Bugs}, if you find a problem that appears to be due
+to a bug or deficiency in Calc.
+
address@hidden
+* Autoloading Problems::
+* Recursion Depth::
+* Caches::
+* Debugging Calc::
address@hidden menu
+
address@hidden Autoloading Problems, Recursion Depth, Troubleshooting Commands,
Troubleshooting Commands
address@hidden Autoloading Problems
+
address@hidden
+The Calc program is split into many component files; components are
+loaded automatically as you use various commands that require them.
+Occasionally Calc may lose track of when a certain component is
+necessary; typically this means you will type a command and it won't
+work because some function you've never heard of was undefined.
+
address@hidden C-x * L
address@hidden calc-load-everything
+If this happens, the easiest workaround is to type @kbd{C-x * L}
+(@code{calc-load-everything}) to force all the parts of Calc to be
+loaded right away. This will cause Emacs to take up a lot more
+memory than it would otherwise, but it's guaranteed to fix the problem.
+
address@hidden Recursion Depth, Caches, Autoloading Problems, Troubleshooting
Commands
address@hidden Recursion Depth
+
address@hidden
address@hidden M
address@hidden I M
address@hidden calc-more-recursion-depth
address@hidden calc-less-recursion-depth
address@hidden Recursion depth
address@hidden ``Computation got stuck'' message
address@hidden @code{max-lisp-eval-depth}
address@hidden @code{max-specpdl-size}
+Calc uses recursion in many of its calculations. Emacs Lisp keeps a
+variable @code{max-lisp-eval-depth} which limits the amount of recursion
+possible in an attempt to recover from program bugs. If a calculation
+ever halts incorrectly with the message ``Computation got stuck or
+ran too long,'' use the @kbd{M} command (@code{calc-more-recursion-depth})
+to increase this limit. (Of course, this will not help if the
+calculation really did get stuck due to some problem inside Calc.)
+
+The limit is always increased (multiplied) by a factor of two. There
+is also an @kbd{I M} (@code{calc-less-recursion-depth}) command which
+decreases this limit by a factor of two, down to a minimum value of 200.
+The default value is 1000.
+
+These commands also double or halve @code{max-specpdl-size}, another
+internal Lisp recursion limit. The minimum value for this limit is 600.
+
address@hidden Caches, Debugging Calc, Recursion Depth, Troubleshooting Commands
address@hidden Caches
+
address@hidden
address@hidden Caches
address@hidden Flushing caches
+Calc saves certain values after they have been computed once. For
+example, the @kbd{P} (@code{calc-pi}) command initially ``knows'' the
+constant @cpi{} to about 20 decimal places; if the current precision
+is greater than this, it will recompute @cpi{} using a series
+approximation. This value will not need to be recomputed ever again
+unless you raise the precision still further. Many operations such as
+logarithms and sines make use of similarly cached values such as
address@hidden and
address@hidden @math{\ln 2}.
address@hidden @expr{ln(2)}.
+The visible effect of caching is that
+high-precision computations may seem to do extra work the first time.
+Other things cached include powers of two (for the binary arithmetic
+functions), matrix inverses and determinants, symbolic integrals, and
+data points computed by the graphing commands.
+
address@hidden calc-flush-caches
+If you suspect a Calculator cache has become corrupt, you can use the
address@hidden command to reset all caches to the empty state.
+(This should only be necessary in the event of bugs in the Calculator.)
+The @kbd{C-x * 0} (with the zero key) command also resets caches along
+with all other aspects of the Calculator's state.
+
address@hidden Debugging Calc, , Caches, Troubleshooting Commands
address@hidden Debugging Calc
+
address@hidden
+A few commands exist to help in the debugging of Calc commands.
address@hidden, to see the various ways that you can write
+your own Calc commands.
+
address@hidden Z T
address@hidden calc-timing
+The @kbd{Z T} (@code{calc-timing}) command turns on and off a mode
+in which the timing of slow commands is reported in the Trail.
+Any Calc command that takes two seconds or longer writes a line
+to the Trail showing how many seconds it took. This value is
+accurate only to within one second.
+
+All steps of executing a command are included; in particular, time
+taken to format the result for display in the stack and trail is
+counted. Some prompts also count time taken waiting for them to
+be answered, while others do not; this depends on the exact
+implementation of the command. For best results, if you are timing
+a sequence that includes prompts or multiple commands, define a
+keyboard macro to run the whole sequence at once. Calc's @kbd{X}
+command (@pxref{Keyboard Macros}) will then report the time taken
+to execute the whole macro.
+
+Another advantage of the @kbd{X} command is that while it is
+executing, the stack and trail are not updated from step to step.
+So if you expect the output of your test sequence to leave a result
+that may take a long time to format and you don't wish to count
+this formatting time, end your sequence with a @key{DEL} keystroke
+to clear the result from the stack. When you run the sequence with
address@hidden, Calc will never bother to format the large result.
+
+Another thing @kbd{Z T} does is to increase the Emacs variable
address@hidden to a much higher value (two million; the
+usual default in Calc is 250,000) for the duration of each command.
+This generally prevents garbage collection during the timing of
+the command, though it may cause your Emacs process to grow
+abnormally large. (Garbage collection time is a major unpredictable
+factor in the timing of Emacs operations.)
+
+Another command that is useful when debugging your own Lisp
+extensions to Calc is @kbd{M-x calc-pass-errors}, which disables
+the error handler that changes the address@hidden
+exceeded'' message to the much more friendly ``Computation got
+stuck or ran too long.'' This handler interferes with the Emacs
+Lisp debugger's @code{debug-on-error} mode. Errors are reported
+in the handler itself rather than at the true location of the
+error. After you have executed @code{calc-pass-errors}, Lisp
+errors will be reported correctly but the user-friendly message
+will be lost.
+
address@hidden Data Types, Stack and Trail, Introduction, Top
address@hidden Data Types
+
address@hidden
+This chapter discusses the various types of objects that can be placed
+on the Calculator stack, how they are displayed, and how they are
+entered. (@xref{Data Type Formats}, for information on how these data
+types are represented as underlying Lisp objects.)
+
+Integers, fractions, and floats are various ways of describing real
+numbers. HMS forms also for many purposes act as real numbers. These
+types can be combined to form complex numbers, modulo forms, error forms,
+or interval forms. (But these last four types cannot be combined
+arbitrarily:@: error forms may not contain modulo forms, for example.)
+Finally, all these types of numbers may be combined into vectors,
+matrices, or algebraic formulas.
+
address@hidden
+* Integers:: The most basic data type.
+* Fractions:: This and above are called @dfn{rationals}.
+* Floats:: This and above are called @dfn{reals}.
+* Complex Numbers:: This and above are called @dfn{numbers}.
+* Infinities::
+* Vectors and Matrices::
+* Strings::
+* HMS Forms::
+* Date Forms::
+* Modulo Forms::
+* Error Forms::
+* Interval Forms::
+* Incomplete Objects::
+* Variables::
+* Formulas::
address@hidden menu
+
address@hidden Integers, Fractions, Data Types, Data Types
address@hidden Integers
+
address@hidden
address@hidden Integers
+The Calculator stores integers to arbitrary precision. Addition,
+subtraction, and multiplication of integers always yields an exact
+integer result. (If the result of a division or exponentiation of
+integers is not an integer, it is expressed in fractional or
+floating-point form according to the current Fraction mode.
address@hidden Mode}.)
+
+A decimal integer is represented as an optional sign followed by a
+sequence of digits. Grouping (@pxref{Grouping Digits}) can be used to
+insert a comma at every third digit for display purposes, but you
+must not type commas during the entry of numbers.
+
address@hidden #
+A non-decimal integer is represented as an optional sign, a radix
+between 2 and 36, a @samp{#} symbol, and one or more digits. For radix 11
+and above, the letters A through Z (upper- or lower-case) count as
+digits and do not terminate numeric entry mode. @xref{Radix Modes}, for how
+to set the default radix for display of integers. Numbers of any radix
+may be entered at any time. If you press @kbd{#} at the beginning of a
+number, the current display radix is used.
+
address@hidden Fractions, Floats, Integers, Data Types
address@hidden Fractions
+
address@hidden
address@hidden Fractions
+A @dfn{fraction} is a ratio of two integers. Fractions are traditionally
+written ``2/3'' but Calc uses the notation @samp{2:3}. (The @kbd{/} key
+performs RPN division; the following two sequences push the number
address@hidden:3} on the stack: @kbd{2 :@: 3 @key{RET}}, or @kbd{2 @key{RET} 3
/}
+assuming Fraction mode has been enabled.)
+When the Calculator produces a fractional result it always reduces it to
+simplest form, which may in fact be an integer.
+
+Fractions may also be entered in a three-part form, where @samp{2:3:4}
+represents two-and-three-quarters. @xref{Fraction Formats}, for fraction
+display formats.
+
+Non-decimal fractions are entered and displayed as
address@hidden@address@hidden:@var{denom}} (or in the analogous three-part
+form). The numerator and denominator always use the same radix.
+
address@hidden Floats, Complex Numbers, Fractions, Data Types
address@hidden Floats
+
address@hidden
address@hidden Floating-point numbers
+A floating-point number or @dfn{float} is a number stored in scientific
+notation. The number of significant digits in the fractional part is
+governed by the current floating precision (@pxref{Precision}). The
+range of acceptable values is from
address@hidden @math{10^{-3999999}}
address@hidden @expr{10^-3999999}
+(inclusive) to
address@hidden @math{10^{4000000}}
address@hidden @expr{10^4000000}
+(exclusive), plus the corresponding negative values and zero.
+
+Calculations that would exceed the allowable range of values (such
+as @samp{exp(exp(20))}) are left in symbolic form by Calc. The
+messages ``floating-point overflow'' or ``floating-point underflow''
+indicate that during the calculation a number would have been produced
+that was too large or too close to zero, respectively, to be represented
+by Calc. This does not necessarily mean the final result would have
+overflowed, just that an overflow occurred while computing the result.
+(In fact, it could report an underflow even though the final result
+would have overflowed!)
+
+If a rational number and a float are mixed in a calculation, the result
+will in general be expressed as a float. Commands that require an integer
+value (such as @kbd{k g} address@hidden) will also accept integer-valued
+floats, i.e., floating-point numbers with nothing after the decimal point.
+
+Floats are identified by the presence of a decimal point and/or an
+exponent. In general a float consists of an optional sign, digits
+including an optional decimal point, and an optional exponent consisting
+of an @samp{e}, an optional sign, and up to seven exponent digits.
+For example, @samp{23.5e-2} is 23.5 times ten to the minus-second power,
+or 0.235.
+
+Floating-point numbers are normally displayed in decimal notation with
+all significant figures shown. Exceedingly large or small numbers are
+displayed in scientific notation. Various other display options are
+available. @xref{Float Formats}.
+
address@hidden Accuracy of calculations
+Floating-point numbers are stored in decimal, not binary. The result
+of each operation is rounded to the nearest value representable in the
+number of significant digits specified by the current precision,
+rounding away from zero in the case of a tie. Thus (in the default
+display mode) what you see is exactly what you get. Some operations such
+as square roots and transcendental functions are performed with several
+digits of extra precision and then rounded down, in an effort to make the
+final result accurate to the full requested precision. However,
+accuracy is not rigorously guaranteed. If you suspect the validity of a
+result, try doing the same calculation in a higher precision. The
+Calculator's arithmetic is not intended to be IEEE-conformant in any
+way.
+
+While floats are always @emph{stored} in decimal, they can be entered
+and displayed in any radix just like integers and fractions. Since a
+float that is entered in a radix other that 10 will be converted to
+decimal, the number that Calc stores may not be exactly the number that
+was entered, it will be the closest decimal approximation given the
+current precison. The notation @address@hidden@address@hidden
+is a floating-point number whose digits are in the specified radix.
+Note that the @samp{.} is more aptly referred to as a ``radix point''
+than as a decimal point in this case. The number @samp{8#123.4567} is
+defined as @samp{8#1234567 * 8^-4}. If the radix is 14 or less, you can
+use @samp{e} notation to write a non-decimal number in scientific
+notation. The exponent is written in decimal, and is considered to be a
+power of the radix: @samp{8#1234567e-4}. If the radix is 15 or above,
+the letter @samp{e} is a digit, so scientific notation must be written
+out, e.g., @samp{16#123.4567*16^2}. The first two exercises of the
+Modes Tutorial explore some of the properties of non-decimal floats.
+
address@hidden Complex Numbers, Infinities, Floats, Data Types
address@hidden Complex Numbers
+
address@hidden
address@hidden Complex numbers
+There are two supported formats for complex numbers: rectangular and
+polar. The default format is rectangular, displayed in the form
address@hidden(@var{real},@var{imag})} where @var{real} is the real part and
address@hidden is the imaginary part, each of which may be any real number.
+Rectangular complex numbers can also be displayed in @address@hidden@var{b}i}
+notation; @pxref{Complex Formats}.
+
+Polar complex numbers are displayed in the form
address@hidden address@hidden(address@hidden@tfn{;address@hidden@tfn{)}'
address@hidden address@hidden(address@hidden@tfn{;address@hidden@tfn{)}'
+where @var{r} is the nonnegative magnitude and
address@hidden @math{\theta}
address@hidden @var{theta}
+is the argument or phase angle. The range of
address@hidden @math{\theta}
address@hidden @var{theta}
+depends on the current angular mode (@pxref{Angular Modes}); it is
+generally between @mathit{-180} and @mathit{+180} degrees or the equivalent
range
+in radians.
+
+Complex numbers are entered in stages using incomplete objects.
address@hidden Objects}.
+
+Operations on rectangular complex numbers yield rectangular complex
+results, and similarly for polar complex numbers. Where the two types
+are mixed, or where new complex numbers arise (as for the square root of
+a negative real), the current @dfn{Polar mode} is used to determine the
+type. @xref{Polar Mode}.
+
+A complex result in which the imaginary part is zero (or the phase angle
+is 0 or 180 degrees or @cpi{} radians) is automatically converted to a real
+number.
+
address@hidden Infinities, Vectors and Matrices, Complex Numbers, Data Types
address@hidden Infinities
+
address@hidden
address@hidden Infinity
address@hidden @code{inf} variable
address@hidden @code{uinf} variable
address@hidden @code{nan} variable
address@hidden inf
address@hidden uinf
address@hidden nan
+The word @code{inf} represents the mathematical concept of @dfn{infinity}.
+Calc actually has three slightly different infinity-like values:
address@hidden, @code{uinf}, and @code{nan}. These are just regular
+variable names (@pxref{Variables}); you should avoid using these
+names for your own variables because Calc gives them special
+treatment. Infinities, like all variable names, are normally
+entered using algebraic entry.
+
+Mathematically speaking, it is not rigorously correct to treat
+``infinity'' as if it were a number, but mathematicians often do
+so informally. When they say that @samp{1 / inf = 0}, what they
+really mean is that @expr{1 / x}, as @expr{x} becomes larger and
+larger, becomes arbitrarily close to zero. So you can imagine
+that if @expr{x} got ``all the way to infinity,'' then @expr{1 / x}
+would go all the way to zero. Similarly, when they say that
address@hidden(inf) = inf}, they mean that
address@hidden @math{e^x}
address@hidden @expr{exp(x)}
+grows without bound as @expr{x} grows. The symbol @samp{-inf} likewise
+stands for an infinitely negative real value; for example, we say that
address@hidden(-inf) = 0}. You can have an infinity pointing in any
+direction on the complex plane: @samp{sqrt(-inf) = i inf}.
+
+The same concept of limits can be used to define @expr{1 / 0}. We
+really want the value that @expr{1 / x} approaches as @expr{x}
+approaches zero. But if all we have is @expr{1 / 0}, we can't
+tell which direction @expr{x} was coming from. If @expr{x} was
+positive and decreasing toward zero, then we should say that
address@hidden / 0 = inf}. But if @expr{x} was negative and increasing
+toward zero, the answer is @samp{1 / 0 = -inf}. In fact, @expr{x}
+could be an imaginary number, giving the answer @samp{i inf} or
address@hidden inf}. Calc uses the special symbol @samp{uinf} to mean
address@hidden infinity}, i.e., a value which is infinitely
+large but with an unknown sign (or direction on the complex plane).
+
+Calc actually has three modes that say how infinities are handled.
+Normally, infinities never arise from calculations that didn't
+already have them. Thus, @expr{1 / 0} is treated simply as an
+error and left unevaluated. The @kbd{m i} (@code{calc-infinite-mode})
+command (@pxref{Infinite Mode}) enables a mode in which
address@hidden / 0} evaluates to @code{uinf} instead. There is also
+an alternative type of infinite mode which says to treat zeros
+as if they were positive, so that @samp{1 / 0 = inf}. While this
+is less mathematically correct, it may be the answer you want in
+some cases.
+
+Since all infinities are ``as large'' as all others, Calc simplifies,
+e.g., @samp{5 inf} to @samp{inf}. Another example is
address@hidden - inf = -inf}, where the @samp{-inf} is so large that
+adding a finite number like five to it does not affect it.
+Note that @samp{a - inf} also results in @samp{-inf}; Calc assumes
+that variables like @code{a} always stand for finite quantities.
+Just to show that infinities really are all the same size,
+note that @samp{sqrt(inf) = inf^2 = exp(inf) = inf} in Calc's
+notation.
+
+It's not so easy to define certain formulas like @samp{0 * inf} and
address@hidden / inf}. Depending on where these zeros and infinities
+came from, the answer could be literally anything. The latter
+formula could be the limit of @expr{x / x} (giving a result of one),
+or @expr{2 x / x} (giving two), or @expr{x^2 / x} (giving @code{inf}),
+or @expr{x / x^2} (giving zero). Calc uses the symbol @code{nan}
+to represent such an @dfn{indeterminate} value. (The name ``nan''
+comes from analogy with the ``NAN'' concept of IEEE standard
+arithmetic; it stands for ``Not A Number.'' This is somewhat of a
+misnomer, since @code{nan} @emph{does} stand for some number or
+infinity, it's just that @emph{which} number it stands for
+cannot be determined.) In Calc's notation, @samp{0 * inf = nan}
+and @samp{inf / inf = nan}. A few other common indeterminate
+expressions are @samp{inf - inf} and @samp{inf ^ 0}. Also,
address@hidden / 0 = nan} if you have turned on Infinite mode
+(as described above).
+
+Infinities are especially useful as parts of @dfn{intervals}.
address@hidden Forms}.
+
address@hidden Vectors and Matrices, Strings, Infinities, Data Types
address@hidden Vectors and Matrices
+
address@hidden
address@hidden Vectors
address@hidden Plain vectors
address@hidden Matrices
+The @dfn{vector} data type is flexible and general. A vector is simply a
+list of zero or more data objects. When these objects are numbers, the
+whole is a vector in the mathematical sense. When these objects are
+themselves vectors of equal (nonzero) length, the whole is a @dfn{matrix}.
+A vector which is not a matrix is referred to here as a @dfn{plain vector}.
+
+A vector is displayed as a list of values separated by commas and enclosed
+in square brackets: @samp{[1, 2, 3]}. Thus the following is a 2 row by
+3 column matrix: @samp{[[1, 2, 3], [4, 5, 6]]}. Vectors, like complex
+numbers, are entered as incomplete objects. @xref{Incomplete Objects}.
+During algebraic entry, vectors are entered all at once in the usual
+brackets-and-commas form. Matrices may be entered algebraically as nested
+vectors, or using the shortcut notation @address@hidden, 2, 3; 4, 5, 6]}},
+with rows separated by semicolons. The commas may usually be omitted
+when entering vectors: @samp{[1 2 3]}. Curly braces may be used in
+place of brackets: @address@hidden, 2, address@hidden, but the commas are
required in
+this case.
+
+Traditional vector and matrix arithmetic is also supported;
address@hidden Arithmetic} and @pxref{Matrix Functions}.
+Many other operations are applied to vectors element-wise. For example,
+the complex conjugate of a vector is a vector of the complex conjugates
+of its elements.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden vec
+Algebraic functions for building vectors include @samp{vec(a, b, c)}
+to build @samp{[a, b, c]}, @samp{cvec(a, n, m)} to build an
address@hidden @math{n\times m}
address@hidden @address@hidden
+matrix of @samp{a}s, and @samp{index(n)} to build a vector of integers
+from 1 to @samp{n}.
+
address@hidden Strings, HMS Forms, Vectors and Matrices, Data Types
address@hidden Strings
+
address@hidden
address@hidden "
address@hidden Strings
address@hidden Character strings
+Character strings are not a special data type in the Calculator.
+Rather, a string is represented simply as a vector all of whose
+elements are integers in the range 0 to 255 (ASCII codes). You can
+enter a string at any time by pressing the @kbd{"} key. Quotation
+marks and backslashes are written @samp{\"} and @samp{\\}, respectively,
+inside strings. Other notations introduced by backslashes are:
+
address@hidden
address@hidden
+\a 7 \^@@ 0
+\b 8 \^a-z 1-26
+\e 27 \^[ 27
+\f 12 \^\\ 28
+\n 10 \^] 29
+\r 13 \^^ 30
+\t 9 \^_ 31
+ \^? 127
address@hidden group
address@hidden example
+
address@hidden
+Finally, a backslash followed by three octal digits produces any
+character from its ASCII code.
+
address@hidden d "
address@hidden calc-display-strings
+Strings are normally displayed in vector-of-integers form. The
address@hidden@kbd{d "}} (@code{calc-display-strings}) command toggles a mode in
+which any vectors of small integers are displayed as quoted strings
+instead.
+
+The backslash notations shown above are also used for displaying
+strings. Characters 128 and above are not translated by Calc; unless
+you have an Emacs modified for 8-bit fonts, these will show up in
+backslash-octal-digits notation. For characters below 32, and
+for character 127, Calc uses the backslash-letter combination if
+there is one, or otherwise uses a @samp{\^} sequence.
+
+The only Calc feature that uses strings is @dfn{compositions};
address@hidden Strings also provide a convenient
+way to do conversions between ASCII characters and integers.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden string
+There is a @code{string} function which provides a different display
+format for strings. Basically, @samp{string(@var{s})}, where @var{s}
+is a vector of integers in the proper range, is displayed as the
+corresponding string of characters with no surrounding quotation
+marks or other modifications. Thus @samp{string("ABC")} (or
address@hidden([65 66 67])}) will look like @samp{ABC} on the stack.
+This happens regardless of whether @address@hidden "}} has been used. The
+only way to turn it off is to use @kbd{d U} (unformatted language
+mode) which will display @samp{string("ABC")} instead.
+
+Control characters are displayed somewhat differently by @code{string}.
+Characters below 32, and character 127, are shown using @samp{^} notation
+(same as shown above, but without the backslash). The quote and
+backslash characters are left alone, as are characters 128 and above.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden bstring
+The @code{bstring} function is just like @code{string} except that
+the resulting string is breakable across multiple lines if it doesn't
+fit all on one line. Potential break points occur at every space
+character in the string.
+
address@hidden HMS Forms, Date Forms, Strings, Data Types
address@hidden HMS Forms
+
address@hidden
address@hidden Hours-minutes-seconds forms
address@hidden Degrees-minutes-seconds forms
address@hidden stands for Hours-Minutes-Seconds; when used as an angular
+argument, the interpretation is Degrees-Minutes-Seconds. All functions
+that operate on angles accept HMS forms. These are interpreted as
+degrees regardless of the current angular mode. It is also possible to
+use HMS as the angular mode so that calculated angles are expressed in
+degrees, minutes, and seconds.
+
address@hidden @@
address@hidden
address@hidden @null
address@hidden ignore
address@hidden ' (HMS forms)
address@hidden
address@hidden @null
address@hidden ignore
address@hidden " (HMS forms)
address@hidden
address@hidden @null
address@hidden ignore
address@hidden h (HMS forms)
address@hidden
address@hidden @null
address@hidden ignore
address@hidden o (HMS forms)
address@hidden
address@hidden @null
address@hidden ignore
address@hidden m (HMS forms)
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s (HMS forms)
+The default format for HMS values is
address@hidden@var{hours}@@ @var{mins}' @var{secs}"}. During entry, the letters
address@hidden (for ``hours'') or
address@hidden (approximating the ``degrees'' symbol) are accepted as well as
address@hidden@@}, @samp{m} is accepted in place of @samp{'}, and @samp{s} is
+accepted in place of @samp{"}.
+The @var{hours} value is an integer (or integer-valued float).
+The @var{mins} value is an integer or integer-valued float between 0 and 59.
+The @var{secs} value is a real number between 0 (inclusive) and 60
+(exclusive). A positive HMS form is interpreted as @var{hours} +
address@hidden/60 + @var{secs}/3600. A negative HMS form is interpreted
+as @mathit{- @var{hours}} @mathit{-} @var{mins}/60 @mathit{-} @var{secs}/3600.
+Display format for HMS forms is quite flexible. @xref{HMS Formats}.
+
+HMS forms can be added and subtracted. When they are added to numbers,
+the numbers are interpreted according to the current angular mode. HMS
+forms can also be multiplied and divided by real numbers. Dividing
+two HMS forms produces a real-valued ratio of the two angles.
+
address@hidden calc-time
address@hidden Time of day
+Just for kicks, @kbd{M-x calc-time} pushes the current time of day on
+the stack as an HMS form.
+
address@hidden Date Forms, Modulo Forms, HMS Forms, Data Types
address@hidden Date Forms
+
address@hidden
address@hidden Date forms
+A @dfn{date form} represents a date and possibly an associated time.
+Simple date arithmetic is supported: Adding a number to a date
+produces a new date shifted by that many days; adding an HMS form to
+a date shifts it by that many hours. Subtracting two date forms
+computes the number of days between them (represented as a simple
+number). Many other operations, such as multiplying two date forms,
+are nonsensical and are not allowed by Calc.
+
+Date forms are entered and displayed enclosed in @samp{< >} brackets.
+The default format is, e.g., @samp{<Wed Jan 9, 1991>} for dates,
+or @samp{<3:32:20pm Wed Jan 9, 1991>} for dates with times.
+Input is flexible; date forms can be entered in any of the usual
+notations for dates and times. @xref{Date Formats}.
+
+Date forms are stored internally as numbers, specifically the number
+of days since midnight on the morning of January 1 of the year 1 AD.
+If the internal number is an integer, the form represents a date only;
+if the internal number is a fraction or float, the form represents
+a date and time. For example, @samp{<6:00am Wed Jan 9, 1991>}
+is represented by the number 726842.25. The standard precision of
+12 decimal digits is enough to ensure that a (reasonable) date and
+time can be stored without roundoff error.
+
+If the current precision is greater than 12, date forms will keep
+additional digits in the seconds position. For example, if the
+precision is 15, the seconds will keep three digits after the
+decimal point. Decreasing the precision below 12 may cause the
+time part of a date form to become inaccurate. This can also happen
+if astronomically high years are used, though this will not be an
+issue in everyday (or even everymillennium) use. Note that date
+forms without times are stored as exact integers, so roundoff is
+never an issue for them.
+
+You can use the @kbd{v p} (@code{calc-pack}) and @kbd{v u}
+(@code{calc-unpack}) commands to get at the numerical representation
+of a date form. @xref{Packing and Unpacking}.
+
+Date forms can go arbitrarily far into the future or past. Negative
+year numbers represent years BC. Calc uses a combination of the
+Gregorian and Julian calendars, following the history of Great
+Britain and the British colonies. This is the same calendar that
+is used by the @code{cal} program in most Unix implementations.
+
address@hidden Julian calendar
address@hidden Gregorian calendar
+Some historical background: The Julian calendar was created by
+Julius Caesar in the year 46 BC as an attempt to fix the gradual
+drift caused by the lack of leap years in the calendar used
+until that time. The Julian calendar introduced an extra day in
+all years divisible by four. After some initial confusion, the
+calendar was adopted around the year we call 8 AD. Some centuries
+later it became apparent that the Julian year of 365.25 days was
+itself not quite right. In 1582 Pope Gregory XIII introduced the
+Gregorian calendar, which added the new rule that years divisible
+by 100, but not by 400, were not to be considered leap years
+despite being divisible by four. Many countries delayed adoption
+of the Gregorian calendar because of religious differences;
+in Britain it was put off until the year 1752, by which time
+the Julian calendar had fallen eleven days behind the true
+seasons. So the switch to the Gregorian calendar in early
+September 1752 introduced a discontinuity: The day after
+Sep 2, 1752 is Sep 14, 1752. Calc follows this convention.
+To take another example, Russia waited until 1918 before
+adopting the new calendar, and thus needed to remove thirteen
+days (between Feb 1, 1918 and Feb 14, 1918). This means that
+Calc's reckoning will be inconsistent with Russian history between
+1752 and 1918, and similarly for various other countries.
+
+Today's timekeepers introduce an occasional ``leap second'' as
+well, but Calc does not take these minor effects into account.
+(If it did, it would have to report a non-integer number of days
+between, say, @samp{<12:00am Mon Jan 1, 1900>} and
address@hidden<12:00am Sat Jan 1, 2000>}.)
+
+Calc uses the Julian calendar for all dates before the year 1752,
+including dates BC when the Julian calendar technically had not
+yet been invented. Thus the claim that day number @mathit{-10000} is
+called ``August 16, 28 BC'' should be taken with a grain of salt.
+
+Please note that there is no ``year 0''; the day before
address@hidden<Sat Jan 1, +1>} is @samp{<Fri Dec 31, -1>}. These are
+days 0 and @mathit{-1} respectively in Calc's internal numbering scheme.
+
address@hidden Julian day counting
+Another day counting system in common use is, confusingly, also called
+``Julian.'' The Julian day number is the numbers of days since
+12:00 noon (GMT) on Jan 1, 4713 BC, which in Calc's scheme (in GMT)
+is @mathit{-1721423.5} (recall that Calc starts at midnight instead
+of noon). Thus to convert a Calc date code obtained by unpacking a
+date form into a Julian day number, simply add 1721423.5 after
+compensating for the time zone difference. The built-in @kbd{t J}
+command performs this conversion for you.
+
+The Julian day number is based on the Julian cycle, which was invented
+in 1583 by Joseph Justus Scaliger. Scaliger named it the Julian cycle
+since it is involves the Julian calendar, but some have suggested that
+Scaliger named it in honor of his father, Julius Caesar Scaliger. The
+Julian cycle is based it on three other cycles: the indiction cycle,
+the Metonic cycle, and the solar cycle. The indiction cycle is a 15
+year cycle originally used by the Romans for tax purposes but later
+used to date medieval documents. The Metonic cycle is a 19 year
+cycle; 19 years is close to being a common multiple of a solar year
+and a lunar month, and so every 19 years the phases of the moon will
+occur on the same days of the year. The solar cycle is a 28 year
+cycle; the Julian calendar repeats itself every 28 years. The
+smallest time period which contains multiples of all three cycles is
+the least common multiple of 15 years, 19 years and 28 years, which
+(since they're pairwise relatively prime) is
address@hidden @math{15\times 19\times 28 = 7980} years.
address@hidden 15*19*28 = 7980 years.
+This is the length of a Julian cycle. Working backwards, the previous
+year in which all three cycles began was 4713 BC, and so Scalinger
+chose that year as the beginning of a Julian cycle. Since at the time
+there were no historical records from before 4713 BC, using this year
+as a starting point had the advantage of avoiding negative year
+numbers. In 1849, the astronomer John Herschel (son of William
+Herschel) suggested using the number of days since the beginning of
+the Julian cycle as an astronomical dating system; this idea was taken
+up by other astronomers. (At the time, noon was the start of the
+astronomical day. Herschel originally suggested counting the days
+since Jan 1, 4713 BC at noon Alexandria time; this was later amended to
+noon GMT.) Julian day numbering is largely used in astronomy.
+
address@hidden Unix time format
+The Unix operating system measures time as an integer number of
+seconds since midnight, Jan 1, 1970. To convert a Calc date
+value into a Unix time stamp, first subtract 719164 (the code
+for @samp{<Jan 1, 1970>}), then multiply by 86400 (the number of
+seconds in a day) and press @kbd{R} to round to the nearest
+integer. If you have a date form, you can simply subtract the
+day @samp{<Jan 1, 1970>} instead of unpacking and subtracting
+719164. Likewise, divide by 86400 and add @samp{<Jan 1, 1970>}
+to convert from Unix time to a Calc date form. (Note that
+Unix normally maintains the time in the GMT time zone; you may
+need to subtract five hours to get New York time, or eight hours
+for California time. The same is usually true of Julian day
+counts.) The built-in @kbd{t U} command performs these
+conversions.
+
address@hidden Modulo Forms, Error Forms, Date Forms, Data Types
address@hidden Modulo Forms
+
address@hidden
address@hidden Modulo forms
+A @dfn{modulo form} is a real number which is taken modulo (i.e., within
+an integer multiple of) some value @var{M}. Arithmetic modulo @var{M}
+often arises in number theory. Modulo forms are written
address@hidden @tfn{mod} @var{M}',
+where @var{a} and @var{M} are real numbers or HMS forms, and
address@hidden @math{0 \le a < M}.
address@hidden @expr{0 <= a < @var{M}}.
+In many applications @expr{a} and @expr{M} will be
+integers but this is not required.
+
address@hidden
address@hidden M
address@hidden ignore
address@hidden M (modulo forms)
address@hidden
address@hidden mod
address@hidden ignore
address@hidden mod (operator)
+To create a modulo form during numeric entry, press the address@hidden
+key to enter the word @samp{mod}. As a special convenience, pressing
address@hidden a second time automatically enters the value of @expr{M}
+that was most recently used before. During algebraic entry, either
+type @samp{mod} by hand or press @kbd{M-m} (that's @address@hidden).
+Once again, pressing this a second time enters the current modulo.
+
+Modulo forms are not to be confused with the modulo operator @samp{%}.
+The expression @samp{27 % 10} means to compute 27 modulo 10 to produce
+the result 7. Further computations treat this 7 as just a regular integer.
+The expression @samp{27 mod 10} produces the result @samp{7 mod 10};
+further computations with this value are again reduced modulo 10 so that
+the result always lies in the desired range.
+
+When two modulo forms with identical @expr{M}'s are added or multiplied,
+the Calculator simply adds or multiplies the values, then reduces modulo
address@hidden If one argument is a modulo form and the other a plain number,
+the plain number is treated like a compatible modulo form. It is also
+possible to raise modulo forms to powers; the result is the value raised
+to the power, then reduced modulo @expr{M}. (When all values involved
+are integers, this calculation is done much more efficiently than
+actually computing the power and then reducing.)
+
address@hidden Modulo division
+Two modulo forms address@hidden @tfn{mod} @var{M}' and address@hidden
@tfn{mod} @var{M}'
+can be divided if @expr{a}, @expr{b}, and @expr{M} are all
+integers. The result is the modulo form which, when multiplied by
address@hidden @tfn{mod} @var{M}', produces address@hidden @tfn{mod} @var{M}'.
If
+there is no solution to this equation (which can happen only when
address@hidden is non-prime), or if any of the arguments are non-integers, the
+division is left in symbolic form. Other operations, such as square
+roots, are not yet supported for modulo forms. (Note that, although
address@hidden@tfn{(address@hidden @tfn{mod} @address@hidden)^.5}'} will
compute a ``modulo square root''
+in the sense of reducing
address@hidden @math{\sqrt a}
address@hidden @expr{sqrt(a)}
+modulo @expr{M}, this is not a useful definition from the
+number-theoretical point of view.)
+
+It is possible to mix HMS forms and modulo forms. For example, an
+HMS form modulo 24 could be used to manipulate clock times; an HMS
+form modulo 360 would be suitable for angles. Making the modulo @expr{M}
+also be an HMS form eliminates troubles that would arise if the angular
+mode were inadvertently set to Radians, in which case
address@hidden@samp{2@@ 0' 0" mod 24}} would be interpreted as two degrees
modulo
+24 radians!
+
+Modulo forms cannot have variables or formulas for components. If you
+enter the formula @samp{(x + 2) mod 5}, Calc propagates the modulus
+to each of the coefficients: @samp{(1 mod 5) x + (2 mod 5)}.
+
+You can use @kbd{v p} and @kbd{%} to modify modulo forms.
address@hidden and Unpacking}. @xref{Basic Arithmetic}.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden makemod
+The algebraic function @samp{makemod(a, m)} builds the modulo form
address@hidden@samp{a mod m}}.
+
address@hidden Error Forms, Interval Forms, Modulo Forms, Data Types
address@hidden Error Forms
+
address@hidden
address@hidden Error forms
address@hidden Standard deviations
+An @dfn{error form} is a number with an associated standard
+deviation, as in @samp{2.3 +/- 0.12}. The notation
address@hidden address@hidden @tfn{+/-} @math{\sigma}'
address@hidden address@hidden @tfn{+/-} sigma'
+stands for an uncertain value which follows
+a normal or Gaussian distribution of mean @expr{x} and standard
+deviation or ``error''
address@hidden @math{\sigma}.
address@hidden @expr{sigma}.
+Both the mean and the error can be either numbers or
+formulas. Generally these are real numbers but the mean may also be
+complex. If the error is negative or complex, it is changed to its
+absolute value. An error form with zero error is converted to a
+regular number by the Calculator.
+
+All arithmetic and transcendental functions accept error forms as input.
+Operations on the mean-value part work just like operations on regular
+numbers. The error part for any function @expr{f(x)} (such as
address@hidden @math{\sin x}
address@hidden @expr{sin(x)})
+is defined by the error of @expr{x} times the derivative of @expr{f}
+evaluated at the mean value of @expr{x}. For a two-argument function
address@hidden(x,y)} (such as addition) the error is the square root of the sum
+of the squares of the errors due to @expr{x} and @expr{y}.
address@hidden
+$$ \eqalign{
+ f(x \hbox{\code{ +/- }} \sigma)
+ &= f(x) \hbox{\code{ +/- }} \sigma \left| {df(x) \over dx} \right| \cr
+ f(x \hbox{\code{ +/- }} \sigma_x, y \hbox{\code{ +/- }} \sigma_y)
+ &= f(x,y) \hbox{\code{ +/- }}
+ \sqrt{\left(\sigma_x \left| {\partial f(x,y) \over \partial x}
+ \right| \right)^2
+ +\left(\sigma_y \left| {\partial f(x,y) \over \partial y}
+ \right| \right)^2 } \cr
+} $$
address@hidden tex
+Note that this
+definition assumes the errors in @expr{x} and @expr{y} are uncorrelated.
+A side effect of this definition is that @samp{(2 +/- 1) * (2 +/- 1)}
+is not the same as @samp{(2 +/- 1)^2}; the former represents the product
+of two independent values which happen to have the same probability
+distributions, and the latter is the product of one random value with itself.
+The former will produce an answer with less error, since on the average
+the two independent errors can be expected to cancel out.
+
+Consult a good text on error analysis for a discussion of the proper use
+of standard deviations. Actual errors often are neither Gaussian-distributed
+nor uncorrelated, and the above formulas are valid only when errors
+are small. As an example, the error arising from
address@hidden address@hidden(address@hidden @tfn{+/-} @address@hidden)}'
address@hidden address@hidden(address@hidden @tfn{+/-} @address@hidden)}'
+is
address@hidden address@hidden @tfn{abs(cos(address@hidden@tfn{))}'.
address@hidden address@hidden @tfn{abs(cos(address@hidden@tfn{))}'.
+When @expr{x} is close to zero,
address@hidden @math{\cos x}
address@hidden @expr{cos(x)}
+is close to one so the error in the sine is close to
address@hidden @math{\sigma};
address@hidden @expr{sigma};
+this makes sense, since
address@hidden @math{\sin x}
address@hidden @expr{sin(x)}
+is approximately @expr{x} near zero, so a given error in @expr{x} will
+produce about the same error in the sine. Likewise, near 90 degrees
address@hidden @math{\cos x}
address@hidden @expr{cos(x)}
+is nearly zero and so the computed error is
+small: The sine curve is nearly flat in that region, so an error in @expr{x}
+has relatively little effect on the value of
address@hidden @math{\sin x}.
address@hidden @expr{sin(x)}.
+However, consider @samp{sin(90 +/- 1000)}. The cosine of 90 is zero, so
+Calc will report zero error! We get an obviously wrong result because
+we have violated the small-error approximation underlying the error
+analysis. If the error in @expr{x} had been small, the error in
address@hidden @math{\sin x}
address@hidden @expr{sin(x)}
+would indeed have been negligible.
+
address@hidden
address@hidden p
address@hidden ignore
address@hidden p (error forms)
address@hidden +/-
+To enter an error form during regular numeric entry, use the @kbd{p}
+(``plus-or-minus'') key to type the @samp{+/-} symbol. (If you try actually
+typing @samp{+/-} the @kbd{+} key will be interpreted as the Calculator's
address@hidden command!) Within an algebraic formula, you can press @kbd{M-+}
to
+type the @samp{+/-} symbol, or type it out by hand.
+
+Error forms and complex numbers can be mixed; the formulas shown above
+are used for complex numbers, too; note that if the error part evaluates
+to a complex number its absolute value (or the square root of the sum of
+the squares of the absolute values of the two error contributions) is
+used. Mathematically, this corresponds to a radially symmetric Gaussian
+distribution of numbers on the complex plane. However, note that Calc
+considers an error form with real components to represent a real number,
+not a complex distribution around a real mean.
+
+Error forms may also be composed of HMS forms. For best results, both
+the mean and the error should be HMS forms if either one is.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden sdev
+The algebraic function @samp{sdev(a, b)} builds the error form @samp{a +/- b}.
+
address@hidden Interval Forms, Incomplete Objects, Error Forms, Data Types
address@hidden Interval Forms
+
address@hidden
address@hidden Interval forms
+An @dfn{interval} is a subset of consecutive real numbers. For example,
+the interval @samp{[2 ..@: 4]} represents all the numbers from 2 to 4,
+inclusive. If you multiply it by the interval @samp{[0.5 ..@: 2]} you
+obtain @samp{[1 ..@: 8]}. This calculation represents the fact that if
+you multiply some number in the range @samp{[2 ..@: 4]} by some other
+number in the range @samp{[0.5 ..@: 2]}, your result will lie in the range
+from 1 to 8. Interval arithmetic is used to get a worst-case estimate
+of the possible range of values a computation will produce, given the
+set of possible values of the input.
+
address@hidden
+Calc supports several varieties of intervals, including @dfn{closed}
+intervals of the type shown above, @dfn{open} intervals such as
address@hidden(2 ..@: 4)}, which represents the range of numbers from 2 to 4
address@hidden, and @dfn{semi-open} intervals in which one end
+uses a round parenthesis and the other a square bracket. In mathematical
+terms,
address@hidden ..@: 4]} means @expr{2 <= x <= 4}, whereas
address@hidden ..@: 4)} represents @expr{2 <= x < 4},
address@hidden(2 ..@: 4]} represents @expr{2 < x <= 4}, and
address@hidden(2 ..@: 4)} represents @expr{2 < x < 4}.
address@hidden ifnottex
address@hidden
+Calc supports several varieties of intervals, including \dfn{closed}
+intervals of the type shown above, \dfn{open} intervals such as
+\samp{(2 ..\: 4)}, which represents the range of numbers from 2 to 4
+\emph{exclusive}, and \dfn{semi-open} intervals in which one end
+uses a round parenthesis and the other a square bracket. In mathematical
+terms,
+$$ \eqalign{
+ [2 \hbox{\cite{..}} 4] &\quad\hbox{means}\quad 2 \le x \le 4 \cr
+ [2 \hbox{\cite{..}} 4) &\quad\hbox{means}\quad 2 \le x < 4 \cr
+ (2 \hbox{\cite{..}} 4] &\quad\hbox{means}\quad 2 < x \le 4 \cr
+ (2 \hbox{\cite{..}} 4) &\quad\hbox{means}\quad 2 < x < 4 \cr
+} $$
address@hidden tex
+
+The lower and upper limits of an interval must be either real numbers
+(or HMS or date forms), or symbolic expressions which are assumed to be
+real-valued, or @samp{-inf} and @samp{inf}. In general the lower limit
+must be less than the upper limit. A closed interval containing only
+one value, @samp{[3 ..@: 3]}, is converted to a plain number (3)
+automatically. An interval containing no values at all (such as
address@hidden ..@: 2]} or @samp{[2 ..@: 2)}) can be represented but is not
+guaranteed to behave well when used in arithmetic. Note that the
+interval @samp{[3 .. inf)} represents all real numbers greater than
+or equal to 3, and @samp{(-inf .. inf)} represents all real numbers.
+In fact, @samp{[-inf .. inf]} represents all real numbers including
+the real infinities.
+
+Intervals are entered in the notation shown here, either as algebraic
+formulas, or using incomplete forms. (@xref{Incomplete Objects}.)
+In algebraic formulas, multiple periods in a row are collected from
+left to right, so that @samp{1...1e2} is interpreted as @samp{1.0 ..@: 1e2}
+rather than @samp{1 ..@: 0.1e2}. Add spaces or zeros if you want to
+get the other interpretation. If you omit the lower or upper limit,
+a default of @samp{-inf} or @samp{inf} (respectively) is furnished.
+
+Infinite mode also affects operations on intervals
+(@pxref{Infinities}). Calc will always introduce an open infinity,
+as in @samp{1 / (0 .. 2] = [0.5 .. inf)}. But closed infinities,
address@hidden@samp{1 / [0 .. 2] = [0.5 .. inf]}}, arise only in Infinite mode;
+otherwise they are left unevaluated. Note that the ``direction'' of
+a zero is not an issue in this case since the zero is always assumed
+to be continuous with the rest of the interval. For intervals that
+contain zero inside them Calc is forced to give the result,
address@hidden / (-2 .. 2) = [-inf .. inf]}.
+
+While it may seem that intervals and error forms are similar, they are
+based on entirely different concepts of inexact quantities. An error
+form
address@hidden address@hidden @tfn{+/-} @math{\sigma}'
address@hidden address@hidden @tfn{+/-} @var{sigma}'
+means a variable is random, and its value could
+be anything but is ``probably'' within one
address@hidden @math{\sigma}
address@hidden @var{sigma}
+of the mean value @expr{x}. An interval
address@hidden@var{a} @tfn{..@:} @address@hidden' means a
+variable's value is unknown, but guaranteed to lie in the specified
+range. Error forms are statistical or ``average case'' approximations;
+interval arithmetic tends to produce ``worst case'' bounds on an
+answer.
+
+Intervals may not contain complex numbers, but they may contain
+HMS forms or date forms.
+
address@hidden Operations}, for commands that interpret interval forms
+as subsets of the set of real numbers.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden intv
+The algebraic function @samp{intv(n, a, b)} builds an interval form
+from @samp{a} to @samp{b}; @samp{n} is an integer code which must
+be 0 for @samp{(..)}, 1 for @samp{(..]}, 2 for @samp{[..)}, or
+3 for @samp{[..]}.
+
+Please note that in fully rigorous interval arithmetic, care would be
+taken to make sure that the computation of the lower bound rounds toward
+minus infinity, while upper bound computations round toward plus
+infinity. Calc's arithmetic always uses a round-to-nearest mode,
+which means that roundoff errors could creep into an interval
+calculation to produce intervals slightly smaller than they ought to
+be. For example, entering @samp{[1..2]} and pressing @kbd{Q 2 ^}
+should yield the interval @samp{[1..2]} again, but in fact it yields the
+(slightly too small) interval @samp{[1..1.9999999]} due to roundoff
+error.
+
address@hidden Incomplete Objects, Variables, Interval Forms, Data Types
address@hidden Incomplete Objects
+
address@hidden
address@hidden
address@hidden [ ]
address@hidden ignore
address@hidden [
address@hidden
address@hidden ( )
address@hidden ignore
address@hidden (
address@hidden ,
address@hidden
address@hidden @null
address@hidden ignore
address@hidden ]
address@hidden
address@hidden @null
address@hidden ignore
address@hidden )
address@hidden Incomplete vectors
address@hidden Incomplete complex numbers
address@hidden Incomplete interval forms
+When @kbd{(} or @kbd{[} is typed to begin entering a complex number or
+vector, respectively, the effect is to push an @dfn{incomplete} complex
+number or vector onto the stack. The @kbd{,} key adds the value(s) at
+the top of the stack onto the current incomplete object. The @kbd{)}
+and @kbd{]} keys ``close'' the incomplete object after adding any values
+on the top of the stack in front of the incomplete object.
+
+As a result, the sequence of keystrokes @kbd{[ 2 , 3 @key{RET} 2 * , 9 ]}
+pushes the vector @samp{[2, 6, 9]} onto the stack. Likewise, @kbd{( 1 , 2 Q )}
+pushes the complex number @samp{(1, 1.414)} (approximately).
+
+If several values lie on the stack in front of the incomplete object,
+all are collected and appended to the object. Thus the @kbd{,} key
+is redundant: @kbd{[ 2 @key{RET} 3 @key{RET} 2 * 9 ]}. Some people
+prefer the equivalent @key{SPC} key to @key{RET}.
+
+As a special case, typing @kbd{,} immediately after @kbd{(}, @kbd{[}, or
address@hidden,} adds a zero or duplicates the preceding value in the list being
+formed. Typing @key{DEL} during incomplete entry removes the last item
+from the list.
+
address@hidden ;
+The @kbd{;} key is used in the same way as @kbd{,} to create polar complex
+numbers: @kbd{( 1 ; 2 )}. When entering a vector, @kbd{;} is useful for
+creating a matrix. In particular, @kbd{[ [ 1 , 2 ; 3 , 4 ; 5 , 6 ] ]} is
+equivalent to @kbd{[ [ 1 , 2 ] , [ 3 , 4 ] , [ 5 , 6 ] ]}.
+
address@hidden ..
address@hidden calc-dots
+Incomplete entry is also used to enter intervals. For example,
address@hidden 2 ..@: 4 )} enters a semi-open interval. Note that when you type
+the first period, it will be interpreted as a decimal point, but when
+you type a second period immediately afterward, it is re-interpreted as
+part of the interval symbol. Typing @kbd{..} corresponds to executing
+the @code{calc-dots} command.
+
+If you find incomplete entry distracting, you may wish to enter vectors
+and complex numbers as algebraic formulas by pressing the apostrophe key.
+
address@hidden Variables, Formulas, Incomplete Objects, Data Types
address@hidden Variables
+
address@hidden
address@hidden Variables, in formulas
+A @dfn{variable} is somewhere between a storage register on a conventional
+calculator, and a variable in a programming language. (In fact, a Calc
+variable is really just an Emacs Lisp variable that contains a Calc number
+or formula.) A variable's name is normally composed of letters and digits.
+Calc also allows apostrophes and @code{#} signs in variable names.
+(The Calc variable @code{foo} corresponds to the Emacs Lisp variable
address@hidden, but unless you access the variable from within Emacs
+Lisp, you don't need to worry about it. Variable names in algebraic
+formulas implicitly have @samp{var-} prefixed to their names. The
address@hidden character in variable names used in algebraic formulas
+corresponds to a dash @samp{-} in the Lisp variable name. If the name
+contains any dashes, the prefix @samp{var-} is @emph{not} automatically
+added. Thus the two formulas @samp{foo + 1} and @samp{var#foo + 1} both
+refer to the same variable.)
+
+In a command that takes a variable name, you can either type the full
+name of a variable, or type a single digit to use one of the special
+convenience variables @code{q0} through @code{q9}. For example,
address@hidden s s 2} stores the number 3 in variable @code{q2}, and
address@hidden@kbd{3 s s foo @key{RET}}} stores that number in variable
address@hidden
+
+To push a variable itself (as opposed to the variable's value) on the
+stack, enter its name as an algebraic expression using the apostrophe
+(@key{'}) key.
+
address@hidden =
address@hidden calc-evaluate
address@hidden Evaluation of variables in a formula
address@hidden Variables, evaluation
address@hidden Formulas, evaluation
+The @kbd{=} (@code{calc-evaluate}) key ``evaluates'' a formula by
+replacing all variables in the formula which have been given values by a
address@hidden or @code{calc-let} command by their stored values.
+Other variables are left alone. Thus a variable that has not been
+stored acts like an abstract variable in algebra; a variable that has
+been stored acts more like a register in a traditional calculator.
+With a positive numeric prefix argument, @kbd{=} evaluates the top
address@hidden stack entries; with a negative argument, @kbd{=} evaluates
+the @var{n}th stack entry.
+
address@hidden @code{e} variable
address@hidden @code{pi} variable
address@hidden @code{i} variable
address@hidden @code{phi} variable
address@hidden @code{gamma} variable
address@hidden e
address@hidden pi
address@hidden i
address@hidden phi
address@hidden gamma
+A few variables are called @dfn{special constants}. Their names are
address@hidden, @samp{pi}, @samp{i}, @samp{phi}, and @samp{gamma}.
+(@xref{Scientific Functions}.) When they are evaluated with @kbd{=},
+their values are calculated if necessary according to the current precision
+or complex polar mode. If you wish to use these symbols for other purposes,
+simply undefine or redefine them using @code{calc-store}.
+
+The variables @samp{inf}, @samp{uinf}, and @samp{nan} stand for
+infinite or indeterminate values. It's best not to use them as
+regular variables, since Calc uses special algebraic rules when
+it manipulates them. Calc displays a warning message if you store
+a value into any of these special variables.
+
address@hidden and Recall}, for a discussion of commands dealing with variables.
+
address@hidden Formulas, , Variables, Data Types
address@hidden Formulas
+
address@hidden
address@hidden Formulas
address@hidden Expressions
address@hidden Operators in formulas
address@hidden Precedence of operators
+When you press the apostrophe key you may enter any expression or formula
+in algebraic form. (Calc uses the terms ``expression'' and ``formula''
+interchangeably.) An expression is built up of numbers, variable names,
+and function calls, combined with various arithmetic operators.
+Parentheses may
+be used to indicate grouping. Spaces are ignored within formulas, except
+that spaces are not permitted within variable names or numbers.
+Arithmetic operators, in order from highest to lowest precedence, and
+with their equivalent function names, are:
+
address@hidden address@hidden (subscripts);
+
+postfix @samp{%} address@hidden (as in @samp{25% = 0.25});
+
+prefix @samp{+} and @samp{-} address@hidden (as in @samp{-x})
+and prefix @samp{!} address@hidden (logical ``not,'' as in @samp{!x});
+
address@hidden/-} address@hidden (the standard deviation symbol) and
address@hidden address@hidden (the symbol for modulo forms);
+
+postfix @samp{!} address@hidden (factorial, as in @samp{n!})
+and postfix @samp{!!} address@hidden (double factorial);
+
address@hidden address@hidden (raised-to-the-power-of);
+
address@hidden address@hidden;
+
address@hidden/} address@hidden, @samp{%} address@hidden (modulo), and
address@hidden address@hidden (integer division);
+
+infix @samp{+} address@hidden and @samp{-} address@hidden (as in @samp{x-y});
+
address@hidden|} address@hidden (vector concatenation);
+
+relations @samp{=} address@hidden, @samp{!=} address@hidden, @samp{<}
address@hidden,
address@hidden>} address@hidden, @samp{<=} address@hidden, and @samp{>=}
address@hidden;
+
address@hidden&&} address@hidden (logical ``and'');
+
address@hidden||} address@hidden (logical ``or'');
+
+the C-style ``if'' operator @samp{a?b:c} address@hidden;
+
address@hidden address@hidden (rewrite pattern ``not'');
+
address@hidden&&&} address@hidden (rewrite pattern ``and'');
+
address@hidden|||} address@hidden (rewrite pattern ``or'');
+
address@hidden:=} address@hidden (for assignments and rewrite rules);
+
address@hidden::} address@hidden (rewrite pattern condition);
+
address@hidden>} address@hidden
+
+Note that, unlike in usual computer notation, multiplication binds more
+strongly than division: @samp{a*b/c*d} is equivalent to
address@hidden @math{a b \over c d}.
address@hidden @expr{(a*b)/(c*d)}.
+
address@hidden Multiplication, implicit
address@hidden Implicit multiplication
+The multiplication sign @samp{*} may be omitted in many cases. In particular,
+if the righthand side is a number, variable name, or parenthesized
+expression, the @samp{*} may be omitted. Implicit multiplication has the
+same precedence as the explicit @samp{*} operator. The one exception to
+the rule is that a variable name followed by a parenthesized expression,
+as in @samp{f(x)},
+is interpreted as a function call, not an implicit @samp{*}. In many
+cases you must use a space if you omit the @samp{*}: @samp{2a} is the
+same as @samp{2*a}, and @samp{a b} is the same as @samp{a*b}, but @samp{ab}
+is a variable called @code{ab}, @emph{not} the product of @samp{a} and
address@hidden Also note that @samp{f (x)} is still a function call.
+
address@hidden Implicit comma in vectors
+The rules are slightly different for vectors written with square brackets.
+In vectors, the space character is interpreted (like the comma) as a
+separator of elements of the vector. Thus @address@hidden 2a b+c d ]}} is
+equivalent to @samp{[2*a, b+c, d]}, whereas @samp{2a b+c d} is equivalent
+to @samp{2*a*b + c*d}.
+Note that spaces around the brackets, and around explicit commas, are
+ignored. To force spaces to be interpreted as multiplication you can
+enclose a formula in parentheses as in @samp{[(a b) 2(c d)]}, which is
+interpreted as @samp{[a*b, 2*c*d]}. An implicit comma is also inserted
+between @samp{][}, as in the matrix @samp{[[1 2][3 4]]}.
+
+Vectors that contain commas (not embedded within nested parentheses or
+brackets) do not treat spaces specially: @samp{[a b, 2 c d]} is a vector
+of two elements. Also, if it would be an error to treat spaces as
+separators, but not otherwise, then Calc will ignore spaces:
address@hidden@samp{[a - b]}} is a vector of one element, but @address@hidden
-b]}} is
+a vector of two elements. Finally, vectors entered with curly braces
+instead of square brackets do not give spaces any special treatment.
+When Calc displays a vector that does not contain any commas, it will
+insert parentheses if necessary to make the meaning clear:
address@hidden@samp{[(a b)]}}.
+
+The expression @samp{5%-2} is ambiguous; is this five-percent minus two,
+or five modulo minus-two? Calc always interprets the leftmost symbol as
+an infix operator preferentially (modulo, in this case), so you would
+need to write @samp{(5%)-2} to get the former interpretation.
+
address@hidden Function call notation
+A function call is, e.g., @samp{sin(1+x)}. (The Calc algebraic function
address@hidden corresponds to the Emacs Lisp function @code{calcFunc-foo},
+but unless you access the function from within Emacs Lisp, you don't
+need to worry about it.) Most mathematical Calculator commands like
address@hidden have function equivalents like @code{sin}.
+If no Lisp function is defined for a function called by a formula, the
+call is left as it is during algebraic manipulation: @samp{f(x+y)} is
+left alone. Beware that many innocent-looking short names like @code{in}
+and @code{re} have predefined meanings which could surprise you; however,
+single letters or single letters followed by digits are always safe to
+use for your own function names. @xref{Function Index}.
+
+In the documentation for particular commands, the notation @kbd{H S}
+(@code{calc-sinh}) address@hidden means that the key sequence @kbd{H S}, the
+command @kbd{M-x calc-sinh}, and the algebraic function @code{sinh(x)} all
+represent the same operation.
+
+Commands that interpret (``parse'') text as algebraic formulas include
+algebraic entry (@kbd{'}), editing commands like @kbd{`} which parse
+the contents of the editing buffer when you finish, the @kbd{C-x * g}
+and @address@hidden * r}} commands, the @kbd{C-y} command, the X window system
+``paste'' mouse operation, and Embedded mode. All of these operations
+use the same rules for parsing formulas; in particular, language modes
+(@pxref{Language Modes}) affect them all in the same way.
+
+When you read a large amount of text into the Calculator (say a vector
+which represents a big set of rewrite rules; @pxref{Rewrite Rules}),
+you may wish to include comments in the text. Calc's formula parser
+ignores the symbol @samp{%%} and anything following it on a line:
+
address@hidden
+[ a + b, %% the sum of "a" and "b"
+ c + d,
+ %% last line is coming up:
+ e + f ]
address@hidden example
+
address@hidden
+This is parsed exactly the same as @samp{[ a + b, c + d, e + f ]}.
+
address@hidden Tables}, for a way to create your own operators and other
+input notations. @xref{Compositions}, for a way to create new display
+formats.
+
address@hidden, for commands for manipulating formulas symbolically.
+
address@hidden Stack and Trail, Mode Settings, Data Types, Top
address@hidden Stack and Trail Commands
+
address@hidden
+This chapter describes the Calc commands for manipulating objects on the
+stack and in the trail buffer. (These commands operate on objects of any
+type, such as numbers, vectors, formulas, and incomplete objects.)
+
address@hidden
+* Stack Manipulation::
+* Editing Stack Entries::
+* Trail Commands::
+* Keep Arguments::
address@hidden menu
+
address@hidden Stack Manipulation, Editing Stack Entries, Stack and Trail,
Stack and Trail
address@hidden Stack Manipulation Commands
+
address@hidden
address@hidden @key{RET}
address@hidden @key{SPC}
address@hidden calc-enter
address@hidden Duplicating stack entries
+To duplicate the top object on the stack, press @key{RET} or @key{SPC}
+(two equivalent keys for the @code{calc-enter} command).
+Given a positive numeric prefix argument, these commands duplicate
+several elements at the top of the stack.
+Given a negative argument,
+these commands duplicate the specified element of the stack.
+Given an argument of zero, they duplicate the entire stack.
+For example, with @samp{10 20 30} on the stack,
address@hidden creates @samp{10 20 30 30},
address@hidden 2 @key{RET}} creates @samp{10 20 30 20 30},
address@hidden - 2 @key{RET}} creates @samp{10 20 30 20}, and
address@hidden 0 @key{RET}} creates @samp{10 20 30 10 20 30}.
+
address@hidden @key{LFD}
address@hidden calc-over
+The @key{LFD} (@code{calc-over}) command (on a key marked Line-Feed if you
+have it, else on @kbd{C-j}) is like @code{calc-enter}
+except that the sign of the numeric prefix argument is interpreted
+oppositely. Also, with no prefix argument the default argument is 2.
+Thus with @samp{10 20 30} on the stack, @key{LFD} and @kbd{C-u 2 @key{LFD}}
+are both equivalent to @kbd{C-u - 2 @key{RET}}, producing
address@hidden 20 30 20}.
+
address@hidden @key{DEL}
address@hidden C-d
address@hidden calc-pop
address@hidden Removing stack entries
address@hidden Deleting stack entries
+To remove the top element from the stack, press @key{DEL} (@code{calc-pop}).
+The @kbd{C-d} key is a synonym for @key{DEL}.
+(If the top element is an incomplete object with at least one element, the
+last element is removed from it.) Given a positive numeric prefix argument,
+several elements are removed. Given a negative argument, the specified
+element of the stack is deleted. Given an argument of zero, the entire
+stack is emptied.
+For example, with @samp{10 20 30} on the stack,
address@hidden leaves @samp{10 20},
address@hidden 2 @key{DEL}} leaves @samp{10},
address@hidden - 2 @key{DEL}} leaves @samp{10 30}, and
address@hidden 0 @key{DEL}} leaves an empty stack.
+
address@hidden address@hidden
address@hidden calc-pop-above
+The @address@hidden (@code{calc-pop-above}) command is to @key{DEL} what
address@hidden is to @key{RET}: It interprets the sign of the numeric
+prefix argument in the opposite way, and the default argument is 2.
+Thus @address@hidden by itself removes the second-from-top stack element,
+leaving the first, third, fourth, and so on; @kbd{M-3 address@hidden deletes
+the third stack element.
+
address@hidden @key{TAB}
address@hidden calc-roll-down
+To exchange the top two elements of the stack, press @key{TAB}
+(@code{calc-roll-down}). Given a positive numeric prefix argument, the
+specified number of elements at the top of the stack are rotated downward.
+Given a negative argument, the entire stack is rotated downward the specified
+number of times. Given an argument of zero, the entire stack is reversed
+top-for-bottom.
+For example, with @samp{10 20 30 40 50} on the stack,
address@hidden creates @samp{10 20 30 50 40},
address@hidden 3 @key{TAB}} creates @samp{10 20 50 30 40},
address@hidden - 2 @key{TAB}} creates @samp{40 50 10 20 30}, and
address@hidden 0 @key{TAB}} creates @samp{50 40 30 20 10}.
+
address@hidden address@hidden
address@hidden calc-roll-up
+The command @address@hidden (@code{calc-roll-up}) is analogous to @key{TAB}
+except that it rotates upward instead of downward. Also, the default
+with no prefix argument is to rotate the top 3 elements.
+For example, with @samp{10 20 30 40 50} on the stack,
address@hidden@key{TAB}} creates @samp{10 20 40 50 30},
address@hidden 4 address@hidden creates @samp{10 30 40 50 20},
address@hidden - 2 address@hidden creates @samp{30 40 50 10 20}, and
address@hidden 0 address@hidden creates @samp{50 40 30 20 10}.
+
+A good way to view the operation of @key{TAB} and @address@hidden is in
+terms of moving a particular element to a new position in the stack.
+With a positive argument @var{n}, @key{TAB} moves the top stack
+element down to level @var{n}, making room for it by pulling all the
+intervening stack elements toward the top. @address@hidden moves the
+element at level @var{n} up to the top. (Compare with @key{LFD},
+which copies instead of moving the element in level @var{n}.)
+
+With a negative argument @address@hidden, @key{TAB} rotates the stack
+to move the object in level @var{n} to the deepest place in the
+stack, and the object in level @address@hidden to the top. @address@hidden
+rotates the deepest stack element to be in level @mathit{n}, also
+putting the top stack element in level @address@hidden
+
address@hidden Subformulas}, for a way to apply these commands to
+any portion of a vector or formula on the stack.
+
address@hidden Editing Stack Entries, Trail Commands, Stack Manipulation, Stack
and Trail
address@hidden Editing Stack Entries
+
address@hidden
address@hidden `
address@hidden calc-edit
address@hidden calc-edit-finish
address@hidden Editing the stack with Emacs
+The backquote, @kbd{`} (@code{calc-edit}) command creates a temporary
+buffer (@samp{*Calc Edit*}) for editing the top-of-stack value using
+regular Emacs commands. With a numeric prefix argument, it edits the
+specified number of stack entries at once. (An argument of zero edits
+the entire stack; a negative argument edits one specific stack entry.)
+
+When you are done editing, press @kbd{C-c C-c} to finish and return
+to Calc. The @key{RET} and @key{LFD} keys also work to finish most
+sorts of editing, though in some cases Calc leaves @key{RET} with its
+usual meaning (``insert a newline'') if it's a situation where you
+might want to insert new lines into the editing buffer.
+
+When you finish editing, the Calculator parses the lines of text in
+the @samp{*Calc Edit*} buffer as numbers or formulas, replaces the
+original stack elements in the original buffer with these new values,
+then kills the @samp{*Calc Edit*} buffer. The original Calculator buffer
+continues to exist during editing, but for best results you should be
+careful not to change it until you have finished the edit. You can
+also cancel the edit by killing the buffer with @kbd{C-x k}.
+
+The formula is normally reevaluated as it is put onto the stack.
+For example, editing @samp{a + 2} to @samp{3 + 2} and pressing
address@hidden C-c} will push 5 on the stack. If you use @key{LFD} to
+finish, Calc will put the result on the stack without evaluating it.
+
+If you give a prefix argument to @kbd{C-c C-c},
+Calc will not kill the @samp{*Calc Edit*} buffer. You can switch
+back to that buffer and continue editing if you wish. However, you
+should understand that if you initiated the edit with @kbd{`}, the
address@hidden C-c} operation will be programmed to replace the top of the
+stack with the new edited value, and it will do this even if you have
+rearranged the stack in the meanwhile. This is not so much of a problem
+with other editing commands, though, such as @kbd{s e}
+(@code{calc-edit-variable}; @pxref{Operations on Variables}).
+
+If the @code{calc-edit} command involves more than one stack entry,
+each line of the @samp{*Calc Edit*} buffer is interpreted as a
+separate formula. Otherwise, the entire buffer is interpreted as
+one formula, with line breaks ignored. (You can use @kbd{C-o} or
address@hidden C-j} to insert a newline in the buffer without pressing
@key{RET}.)
+
+The @kbd{`} key also works during numeric or algebraic entry. The
+text entered so far is moved to the @code{*Calc Edit*} buffer for
+more extensive editing than is convenient in the minibuffer.
+
address@hidden Trail Commands, Keep Arguments, Editing Stack Entries, Stack and
Trail
address@hidden Trail Commands
+
address@hidden
address@hidden Trail buffer
+The commands for manipulating the Calc Trail buffer are two-key sequences
+beginning with the @kbd{t} prefix.
+
address@hidden t d
address@hidden calc-trail-display
+The @kbd{t d} (@code{calc-trail-display}) command turns display of the
+trail on and off. Normally the trail display is toggled on if it was off,
+off if it was on. With a numeric prefix of zero, this command always
+turns the trail off; with a prefix of one, it always turns the trail on.
+The other trail-manipulation commands described here automatically turn
+the trail on. Note that when the trail is off values are still recorded
+there; they are simply not displayed. To set Emacs to turn the trail
+off by default, type @kbd{t d} and then save the mode settings with
address@hidden m} (@code{calc-save-modes}).
+
address@hidden t i
address@hidden calc-trail-in
address@hidden t o
address@hidden calc-trail-out
+The @kbd{t i} (@code{calc-trail-in}) and @kbd{t o}
+(@code{calc-trail-out}) commands switch the cursor into and out of the
+Calc Trail window. In practice they are rarely used, since the commands
+shown below are a more convenient way to move around in the
+trail, and they work ``by remote control'' when the cursor is still
+in the Calculator window.
+
address@hidden Trail pointer
+There is a @dfn{trail pointer} which selects some entry of the trail at
+any given time. The trail pointer looks like a @samp{>} symbol right
+before the selected number. The following commands operate on the
+trail pointer in various ways.
+
address@hidden t y
address@hidden calc-trail-yank
address@hidden Retrieving previous results
+The @kbd{t y} (@code{calc-trail-yank}) command reads the selected value in
+the trail and pushes it onto the Calculator stack. It allows you to
+re-use any previously computed value without retyping. With a numeric
+prefix argument @var{n}, it yanks the value @var{n} lines above the current
+trail pointer.
+
address@hidden t <
address@hidden calc-trail-scroll-left
address@hidden t >
address@hidden calc-trail-scroll-right
+The @kbd{t <} (@code{calc-trail-scroll-left}) and @kbd{t >}
+(@code{calc-trail-scroll-right}) commands horizontally scroll the trail
+window left or right by one half of its width.
+
address@hidden t n
address@hidden calc-trail-next
address@hidden t p
address@hidden calc-trail-previous
address@hidden t f
address@hidden calc-trail-forward
address@hidden t b
address@hidden calc-trail-backward
+The @kbd{t n} (@code{calc-trail-next}) and @kbd{t p}
+(@code{calc-trail-previous)} commands move the trail pointer down or up
+one line. The @kbd{t f} (@code{calc-trail-forward}) and @kbd{t b}
+(@code{calc-trail-backward}) commands move the trail pointer down or up
+one screenful at a time. All of these commands accept numeric prefix
+arguments to move several lines or screenfuls at a time.
+
address@hidden t [
address@hidden calc-trail-first
address@hidden t ]
address@hidden calc-trail-last
address@hidden t h
address@hidden calc-trail-here
+The @kbd{t [} (@code{calc-trail-first}) and @kbd{t ]}
+(@code{calc-trail-last}) commands move the trail pointer to the first or
+last line of the trail. The @kbd{t h} (@code{calc-trail-here}) command
+moves the trail pointer to the cursor position; unlike the other trail
+commands, @kbd{t h} works only when Calc Trail is the selected window.
+
address@hidden t s
address@hidden calc-trail-isearch-forward
address@hidden t r
address@hidden calc-trail-isearch-backward
address@hidden
+The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
+(@code{calc-trail-isearch-backward}) commands perform an incremental
+search forward or backward through the trail. You can press @key{RET}
+to terminate the search; the trail pointer moves to the current line.
+If you cancel the search with @kbd{C-g}, the trail pointer stays where
+it was when the search began.
address@hidden ifnottex
address@hidden
+The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
+(@code{calc-trail-isearch-backward}) com\-mands perform an incremental
+search forward or backward through the trail. You can press @key{RET}
+to terminate the search; the trail pointer moves to the current line.
+If you cancel the search with @kbd{C-g}, the trail pointer stays where
+it was when the search began.
address@hidden tex
+
address@hidden t m
address@hidden calc-trail-marker
+The @kbd{t m} (@code{calc-trail-marker}) command allows you to enter a
+line of text of your own choosing into the trail. The text is inserted
+after the line containing the trail pointer; this usually means it is
+added to the end of the trail. Trail markers are useful mainly as the
+targets for later incremental searches in the trail.
+
address@hidden t k
address@hidden calc-trail-kill
+The @kbd{t k} (@code{calc-trail-kill}) command removes the selected line
+from the trail. The line is saved in the Emacs kill ring suitable for
+yanking into another buffer, but it is not easy to yank the text back
+into the trail buffer. With a numeric prefix argument, this command
+kills the @var{n} lines below or above the selected one.
+
+The @kbd{t .} (@code{calc-full-trail-vectors}) command is described
+elsewhere; @pxref{Vector and Matrix Formats}.
+
address@hidden Keep Arguments, , Trail Commands, Stack and Trail
address@hidden Keep Arguments
+
address@hidden
address@hidden K
address@hidden calc-keep-args
+The @kbd{K} (@code{calc-keep-args}) command acts like a prefix for
+the following command. It prevents that command from removing its
+arguments from the stack. For example, after @kbd{2 @key{RET} 3 +},
+the stack contains the sole number 5, but after @kbd{2 @key{RET} 3 K +},
+the stack contains the arguments and the result: @samp{2 3 5}.
+
+With the exception of keyboard macros, this works for all commands that
+take arguments off the stack. (To avoid potentially unpleasant behavior,
+a @kbd{K} prefix before a keyboard macro will be ignored. A @kbd{K}
+prefix called @emph{within} the keyboard macro will still take effect.)
+As another example, @kbd{K a s} simplifies a formula, pushing the
+simplified version of the formula onto the stack after the original
+formula (rather than replacing the original formula). Note that you
+could get the same effect by typing @address@hidden a s}, copying the
+formula and then simplifying the copy. One difference is that for a very
+large formula the time taken to format the intermediate copy in
address@hidden@key{RET} a s} could be noticeable; @kbd{K a s} would avoid this
+extra work.
+
+Even stack manipulation commands are affected. @key{TAB} works by
+popping two values and pushing them back in the opposite order,
+so @kbd{2 @key{RET} 3 K @key{TAB}} produces @samp{2 3 3 2}.
+
+A few Calc commands provide other ways of doing the same thing.
+For example, @kbd{' sin($)} replaces the number on the stack with
+its sine using algebraic entry; to push the sine and keep the
+original argument you could use either @kbd{' sin($1)} or
address@hidden ' sin($)}. @xref{Algebraic Entry}. Also, the @kbd{s s}
+command is effectively the same as @kbd{K s t}. @xref{Storing Variables}.
+
+If you execute a command and then decide you really wanted to keep
+the argument, you can press @address@hidden (@code{calc-last-args}).
+This command pushes the last arguments that were popped by any command
+onto the stack. Note that the order of things on the stack will be
+different than with @kbd{K}: @kbd{2 @key{RET} 3 + address@hidden leaves
address@hidden 2 3} on the stack instead of @samp{2 3 5}. @xref{Undo}.
+
address@hidden Mode Settings, Arithmetic, Stack and Trail, Top
address@hidden Mode Settings
+
address@hidden
+This chapter describes commands that set modes in the Calculator.
+They do not affect the contents of the stack, although they may change
+the @emph{appearance} or @emph{interpretation} of the stack's contents.
+
address@hidden
+* General Mode Commands::
+* Precision::
+* Inverse and Hyperbolic::
+* Calculation Modes::
+* Simplification Modes::
+* Declarations::
+* Display Modes::
+* Language Modes::
+* Modes Variable::
+* Calc Mode Line::
address@hidden menu
+
address@hidden General Mode Commands, Precision, Mode Settings, Mode Settings
address@hidden General Mode Commands
+
address@hidden
address@hidden m m
address@hidden calc-save-modes
address@hidden Continuous memory
address@hidden Saving mode settings
address@hidden Permanent mode settings
address@hidden Calc init file, mode settings
+You can save all of the current mode settings in your Calc init file
+(the file given by the variable @code{calc-settings-file}, typically
address@hidden/.calc.el}) with the @kbd{m m} (@code{calc-save-modes}) command.
+This will cause Emacs to reestablish these modes each time it starts up.
+The modes saved in the file include everything controlled by the @kbd{m}
+and @kbd{d} prefix keys, the current precision and binary word size,
+whether or not the trail is displayed, the current height of the Calc
+window, and more. The current interface (used when you type @kbd{C-x * *})
+is also saved. If there were already saved mode settings in the
+file, they are replaced. Otherwise, the new mode information is
+appended to the end of the file.
+
address@hidden m R
address@hidden calc-mode-record-mode
+The @kbd{m R} (@code{calc-mode-record-mode}) command tells Calc to
+record all the mode settings (as if by pressing @kbd{m m}) every
+time a mode setting changes. If the modes are saved this way, then this
+``automatic mode recording'' mode is also saved.
+Type @kbd{m R} again to disable this method of recording the mode
+settings. To turn it off permanently, the @kbd{m m} command will also be
+necessary. (If Embedded mode is enabled, other options for recording
+the modes are available; @pxref{Mode Settings in Embedded Mode}.)
+
address@hidden m F
address@hidden calc-settings-file-name
+The @kbd{m F} (@code{calc-settings-file-name}) command allows you to
+choose a different file than the current value of @code{calc-settings-file}
+for @kbd{m m}, @kbd{Z P}, and similar commands to save permanent information.
+You are prompted for a file name. All Calc modes are then reset to
+their default values, then settings from the file you named are loaded
+if this file exists, and this file becomes the one that Calc will
+use in the future for commands like @kbd{m m}. The default settings
+file name is @file{~/.calc.el}. You can see the current file name by
+giving a blank response to the @kbd{m F} prompt. See also the
+discussion of the @code{calc-settings-file} variable; @pxref{Customizing Calc}.
+
+If the file name you give is your user init file (typically
address@hidden/.emacs}), @kbd{m F} will not automatically load the new file.
This
+is because your user init file may contain other things you don't want
+to reread. You can give
+a numeric prefix argument of 1 to @kbd{m F} to force it to read the
+file no matter what. Conversely, an argument of @mathit{-1} tells
address@hidden F} @emph{not} to read the new file. An argument of 2 or
@mathit{-2}
+tells @kbd{m F} not to reset the modes to their defaults beforehand,
+which is useful if you intend your new file to have a variant of the
+modes present in the file you were using before.
+
address@hidden m x
address@hidden calc-always-load-extensions
+The @kbd{m x} (@code{calc-always-load-extensions}) command enables a mode
+in which the first use of Calc loads the entire program, including all
+extensions modules. Otherwise, the extensions modules will not be loaded
+until the various advanced Calc features are used. Since this mode only
+has effect when Calc is first loaded, @kbd{m x} is usually followed by
address@hidden m} to make the mode-setting permanent. To load all of Calc just
+once, rather than always in the future, you can press @kbd{C-x * L}.
+
address@hidden m S
address@hidden calc-shift-prefix
+The @kbd{m S} (@code{calc-shift-prefix}) command enables a mode in which
+all of Calc's letter prefix keys may be typed shifted as well as unshifted.
+If you are typing, say, @kbd{a S} (@code{calc-solve-for}) quite often
+you might find it easier to turn this mode on so that you can type
address@hidden S} instead. When this mode is enabled, the commands that used to
+be on those single shifted letters (e.g., @kbd{A} (@code{calc-abs})) can
+now be invoked by pressing the shifted letter twice: @kbd{A A}. Note
+that the @kbd{v} prefix key always works both shifted and unshifted, and
+the @kbd{z} and @kbd{Z} prefix keys are always distinct. Also, the @kbd{h}
+prefix is not affected by this mode. Press @kbd{m S} again to disable
+shifted-prefix mode.
+
address@hidden Precision, Inverse and Hyperbolic, General Mode Commands, Mode
Settings
address@hidden Precision
+
address@hidden
address@hidden p
address@hidden calc-precision
address@hidden Precision of calculations
+The @kbd{p} (@code{calc-precision}) command controls the precision to
+which floating-point calculations are carried. The precision must be
+at least 3 digits and may be arbitrarily high, within the limits of
+memory and time. This affects only floats: Integer and rational
+calculations are always carried out with as many digits as necessary.
+
+The @kbd{p} key prompts for the current precision. If you wish you
+can instead give the precision as a numeric prefix argument.
+
+Many internal calculations are carried to one or two digits higher
+precision than normal. Results are rounded down afterward to the
+current precision. Unless a special display mode has been selected,
+floats are always displayed with their full stored precision, i.e.,
+what you see is what you get. Reducing the current precision does not
+round values already on the stack, but those values will be rounded
+down before being used in any calculation. The @kbd{c 0} through
address@hidden 9} commands (@pxref{Conversions}) can be used to round an
+existing value to a new precision.
+
address@hidden Accuracy of calculations
+It is important to distinguish the concepts of @dfn{precision} and
address@hidden In the normal usage of these words, the number
+123.4567 has a precision of 7 digits but an accuracy of 4 digits.
+The precision is the total number of digits not counting leading
+or trailing zeros (regardless of the position of the decimal point).
+The accuracy is simply the number of digits after the decimal point
+(again not counting trailing zeros). In Calc you control the precision,
+not the accuracy of computations. If you were to set the accuracy
+instead, then calculations like @samp{exp(100)} would generate many
+more digits than you would typically need, while @samp{exp(-100)} would
+probably round to zero! In Calc, both these computations give you
+exactly 12 (or the requested number of) significant digits.
+
+The only Calc features that deal with accuracy instead of precision
+are fixed-point display mode for floats (@kbd{d f}; @pxref{Float Formats}),
+and the rounding functions like @code{floor} and @code{round}
+(@pxref{Integer Truncation}). Also, @kbd{c 0} through @kbd{c 9}
+deal with both precision and accuracy depending on the magnitudes
+of the numbers involved.
+
+If you need to work with a particular fixed accuracy (say, dollars and
+cents with two digits after the decimal point), one solution is to work
+with integers and an ``implied'' decimal point. For example, $8.99
+divided by 6 would be entered @kbd{899 @key{RET} 6 /}, yielding 149.833
+(actually $1.49833 with our implied decimal point); pressing @kbd{R}
+would round this to 150 cents, i.e., $1.50.
+
address@hidden, for still more on floating-point precision and related
+issues.
+
address@hidden Inverse and Hyperbolic, Calculation Modes, Precision, Mode
Settings
address@hidden Inverse and Hyperbolic Flags
+
address@hidden
address@hidden I
address@hidden calc-inverse
+There is no single-key equivalent to the @code{calc-arcsin} function.
+Instead, you must first press @kbd{I} (@code{calc-inverse}) to set
+the @dfn{Inverse Flag}, then press @kbd{S} (@code{calc-sin}).
+The @kbd{I} key actually toggles the Inverse Flag. When this flag
+is set, the word @samp{Inv} appears in the mode line.
+
address@hidden H
address@hidden calc-hyperbolic
+Likewise, the @kbd{H} key (@code{calc-hyperbolic}) sets or clears the
+Hyperbolic Flag, which transforms @code{calc-sin} into @code{calc-sinh}.
+If both of these flags are set at once, the effect will be
address@hidden (The Hyperbolic flag is also used by some
+non-trigonometric commands; for example @kbd{H L} computes a base-10,
+instead of address@hidden, logarithm.)
+
+Command names like @code{calc-arcsin} are provided for completeness, and
+may be executed with @kbd{x} or @kbd{M-x}. Their effect is simply to
+toggle the Inverse and/or Hyperbolic flags and then execute the
+corresponding base command (@code{calc-sin} in this case).
+
+The Inverse and Hyperbolic flags apply only to the next Calculator
+command, after which they are automatically cleared. (They are also
+cleared if the next keystroke is not a Calc command.) Digits you
+type after @kbd{I} or @kbd{H} (or @kbd{K}) are treated as prefix
+arguments for the next command, not as numeric entries. The same
+is true of @kbd{C-u}, but not of the minus sign (@kbd{K -} means to
+subtract and keep arguments).
+
+The third Calc prefix flag, @kbd{K} (keep-arguments), is discussed
+elsewhere. @xref{Keep Arguments}.
+
address@hidden Calculation Modes, Simplification Modes, Inverse and Hyperbolic,
Mode Settings
address@hidden Calculation Modes
+
address@hidden
+The commands in this section are two-key sequences beginning with
+the @kbd{m} prefix. (That's the letter @kbd{m}, not the @key{META} key.)
+The @samp{m a} (@code{calc-algebraic-mode}) command is described elsewhere
+(@pxref{Algebraic Entry}).
+
address@hidden
+* Angular Modes::
+* Polar Mode::
+* Fraction Mode::
+* Infinite Mode::
+* Symbolic Mode::
+* Matrix Mode::
+* Automatic Recomputation::
+* Working Message::
address@hidden menu
+
address@hidden Angular Modes, Polar Mode, Calculation Modes, Calculation Modes
address@hidden Angular Modes
+
address@hidden
address@hidden Angular mode
+The Calculator supports three notations for angles: radians, degrees,
+and degrees-minutes-seconds. When a number is presented to a function
+like @code{sin} that requires an angle, the current angular mode is
+used to interpret the number as either radians or degrees. If an HMS
+form is presented to @code{sin}, it is always interpreted as
+degrees-minutes-seconds.
+
+Functions that compute angles produce a number in radians, a number in
+degrees, or an HMS form depending on the current angular mode. If the
+result is a complex number and the current mode is HMS, the number is
+instead expressed in degrees. (Complex-number calculations would
+normally be done in Radians mode, though. Complex numbers are converted
+to degrees by calculating the complex result in radians and then
+multiplying by 180 over @cpi{}.)
+
address@hidden m r
address@hidden calc-radians-mode
address@hidden m d
address@hidden calc-degrees-mode
address@hidden m h
address@hidden calc-hms-mode
+The @kbd{m r} (@code{calc-radians-mode}), @kbd{m d} (@code{calc-degrees-mode}),
+and @kbd{m h} (@code{calc-hms-mode}) commands control the angular mode.
+The current angular mode is displayed on the Emacs mode line.
+The default angular mode is Degrees.
+
address@hidden Polar Mode, Fraction Mode, Angular Modes, Calculation Modes
address@hidden Polar Mode
+
address@hidden
address@hidden Polar mode
+The Calculator normally ``prefers'' rectangular complex numbers in the
+sense that rectangular form is used when the proper form can not be
+decided from the input. This might happen by multiplying a rectangular
+number by a polar one, by taking the square root of a negative real
+number, or by entering @kbd{( 2 @key{SPC} 3 )}.
+
address@hidden m p
address@hidden calc-polar-mode
+The @kbd{m p} (@code{calc-polar-mode}) command toggles complex-number
+preference between rectangular and polar forms. In Polar mode, all
+of the above example situations would produce polar complex numbers.
+
address@hidden Fraction Mode, Infinite Mode, Polar Mode, Calculation Modes
address@hidden Fraction Mode
+
address@hidden
address@hidden Fraction mode
address@hidden Division of integers
+Division of two integers normally yields a floating-point number if the
+result cannot be expressed as an integer. In some cases you would
+rather get an exact fractional answer. One way to accomplish this is
+to use the @kbd{:} (@code{calc-fdiv}) address@hidden command, which
+divides the two integers on the top of the stack to produce a fraction:
address@hidden @key{RET} 4 :} produces @expr{3:2} even though
address@hidden @key{RET} 4 /} produces @expr{1.5}.
+
address@hidden m f
address@hidden calc-frac-mode
+To set the Calculator to produce fractional results for normal integer
+divisions, use the @kbd{m f} (@code{calc-frac-mode}) command.
+For example, @expr{8/4} produces @expr{2} in either mode,
+but @expr{6/4} produces @expr{3:2} in Fraction mode, @expr{1.5} in
+Float mode.
+
+At any time you can use @kbd{c f} (@code{calc-float}) to convert a
+fraction to a float, or @kbd{c F} (@code{calc-fraction}) to convert a
+float to a fraction. @xref{Conversions}.
+
address@hidden Infinite Mode, Symbolic Mode, Fraction Mode, Calculation Modes
address@hidden Infinite Mode
+
address@hidden
address@hidden Infinite mode
+The Calculator normally treats results like @expr{1 / 0} as errors;
+formulas like this are left in unsimplified form. But Calc can be
+put into a mode where such calculations instead produce ``infinite''
+results.
+
address@hidden m i
address@hidden calc-infinite-mode
+The @kbd{m i} (@code{calc-infinite-mode}) command turns this mode
+on and off. When the mode is off, infinities do not arise except
+in calculations that already had infinities as inputs. (One exception
+is that infinite open intervals like @samp{[0 .. inf)} can be
+generated; however, intervals closed at infinity (@samp{[0 .. inf]})
+will not be generated when Infinite mode is off.)
+
+With Infinite mode turned on, @samp{1 / 0} will generate @code{uinf},
+an undirected infinity. @xref{Infinities}, for a discussion of the
+difference between @code{inf} and @code{uinf}. Also, @expr{0 / 0}
+evaluates to @code{nan}, the ``indeterminate'' symbol. Various other
+functions can also return infinities in this mode; for example,
address@hidden(0) = -inf}, and @samp{gamma(-7) = uinf}. Once again,
+note that @samp{exp(inf) = inf} regardless of Infinite mode because
+this calculation has infinity as an input.
+
address@hidden Positive Infinite mode
+The @kbd{m i} command with a numeric prefix argument of zero,
+i.e., @kbd{C-u 0 m i}, turns on a Positive Infinite mode in
+which zero is treated as positive instead of being directionless.
+Thus, @samp{1 / 0 = inf} and @samp{-1 / 0 = -inf} in this mode.
+Note that zero never actually has a sign in Calc; there are no
+separate representations for @mathit{+0} and @mathit{-0}. Positive
+Infinite mode merely changes the interpretation given to the
+single symbol, @samp{0}. One consequence of this is that, while
+you might expect @samp{1 / -0 = -inf}, actually @samp{1 / -0}
+is equivalent to @samp{1 / 0}, which is equal to positive @code{inf}.
+
address@hidden Symbolic Mode, Matrix Mode, Infinite Mode, Calculation Modes
address@hidden Symbolic Mode
+
address@hidden
address@hidden Symbolic mode
address@hidden Inexact results
+Calculations are normally performed numerically wherever possible.
+For example, the @code{calc-sqrt} command, or @code{sqrt} function in an
+algebraic expression, produces a numeric answer if the argument is a
+number or a symbolic expression if the argument is an expression:
address@hidden Q} pushes 1.4142 but @address@hidden'} x+1 @key{RET} Q} pushes
@samp{sqrt(x+1)}.
+
address@hidden m s
address@hidden calc-symbolic-mode
+In @dfn{Symbolic mode}, controlled by the @kbd{m s} (@code{calc-symbolic-mode})
+command, functions which would produce inexact, irrational results are
+left in symbolic form. Thus @kbd{16 Q} pushes 4, but @kbd{2 Q} pushes
address@hidden(2)}.
+
address@hidden N
address@hidden calc-eval-num
+The address@hidden (@code{calc-eval-num}) command evaluates numerically
+the expression at the top of the stack, by temporarily disabling
address@hidden and executing @kbd{=} (@code{calc-evaluate}).
+Given a numeric prefix argument, it also
+sets the floating-point precision to the specified value for the duration
+of the command.
+
+To evaluate a formula numerically without expanding the variables it
+contains, you can use the key sequence @kbd{m s a v m s} (this uses
address@hidden, which resimplifies but doesn't evaluate
+variables.)
+
address@hidden Matrix Mode, Automatic Recomputation, Symbolic Mode, Calculation
Modes
address@hidden Matrix and Scalar Modes
+
address@hidden
address@hidden Matrix mode
address@hidden Scalar mode
+Calc sometimes makes assumptions during algebraic manipulation that
+are awkward or incorrect when vectors and matrices are involved.
+Calc has two modes, @dfn{Matrix mode} and @dfn{Scalar mode}, which
+modify its behavior around vectors in useful ways.
+
address@hidden m v
address@hidden calc-matrix-mode
+Press @kbd{m v} (@code{calc-matrix-mode}) once to enter Matrix mode.
+In this mode, all objects are assumed to be matrices unless provably
+otherwise. One major effect is that Calc will no longer consider
+multiplication to be commutative. (Recall that in matrix arithmetic,
address@hidden is not the same as @samp{B*A}.) This assumption affects
+rewrite rules and algebraic simplification. Another effect of this
+mode is that calculations that would normally produce constants like
+0 and 1 (e.g., @expr{a - a} and @expr{a / a}, respectively) will now
+produce function calls that represent ``generic'' zero or identity
+matrices: @samp{idn(0)}, @samp{idn(1)}. The @code{idn} function
address@hidden(@var{a},@var{n})} returns @var{a} times an @address@hidden
+identity matrix; if @var{n} is omitted, it doesn't know what
+dimension to use and so the @code{idn} call remains in symbolic
+form. However, if this generic identity matrix is later combined
+with a matrix whose size is known, it will be converted into
+a true identity matrix of the appropriate size. On the other hand,
+if it is combined with a scalar (as in @samp{idn(1) + 2}), Calc
+will assume it really was a scalar after all and produce, e.g., 3.
+
+Press @kbd{m v} a second time to get Scalar mode. Here, objects are
+assumed @emph{not} to be vectors or matrices unless provably so.
+For example, normally adding a variable to a vector, as in
address@hidden, y, z] + a}, will leave the sum in symbolic form because
+as far as Calc knows, @samp{a} could represent either a number or
+another 3-vector. In Scalar mode, @samp{a} is assumed to be a
+non-vector, and the addition is evaluated to @samp{[x+a, y+a, z+a]}.
+
+Press @kbd{m v} a third time to return to the normal mode of operation.
+
+If you press @kbd{m v} with a numeric prefix argument @var{n}, you
+get a special ``dimensioned'' Matrix mode in which matrices of
+unknown size are assumed to be @address@hidden square matrices.
+Then, the function call @samp{idn(1)} will expand into an actual
+matrix rather than representing a ``generic'' matrix. Simply typing
address@hidden m v} will get you a square Matrix mode, in which matrices of
+unknown size are assumed to be square matrices of unspecified size.
+
address@hidden Declaring scalar variables
+Of course these modes are approximations to the true state of
+affairs, which is probably that some quantities will be matrices
+and others will be scalars. One solution is to ``declare''
+certain variables or functions to be scalar-valued.
address@hidden, to see how to make declarations in Calc.
+
+There is nothing stopping you from declaring a variable to be
+scalar and then storing a matrix in it; however, if you do, the
+results you get from Calc may not be valid. Suppose you let Calc
+get the result @samp{[x+a, y+a, z+a]} shown above, and then stored
address@hidden, 2, 3]} in @samp{a}. The result would not be the same as
+for @samp{[x, y, z] + [1, 2, 3]}, but that's because you have broken
+your earlier promise to Calc that @samp{a} would be scalar.
+
+Another way to mix scalars and matrices is to use selections
+(@pxref{Selecting Subformulas}). Use Matrix mode when operating on
+your formula normally; then, to apply Scalar mode to a certain part
+of the formula without affecting the rest just select that part,
+change into Scalar mode and press @kbd{=} to resimplify the part
+under this mode, then change back to Matrix mode before deselecting.
+
address@hidden Automatic Recomputation, Working Message, Matrix Mode,
Calculation Modes
address@hidden Automatic Recomputation
+
address@hidden
+The @dfn{evaluates-to} operator, @samp{=>}, has the special
+property that any @samp{=>} formulas on the stack are recomputed
+whenever variable values or mode settings that might affect them
+are changed. @xref{Evaluates-To Operator}.
+
address@hidden m C
address@hidden calc-auto-recompute
+The @kbd{m C} (@code{calc-auto-recompute}) command turns this
+automatic recomputation on and off. If you turn it off, Calc will
+not update @samp{=>} operators on the stack (nor those in the
+attached Embedded mode buffer, if there is one). They will not
+be updated unless you explicitly do so by pressing @kbd{=} or until
+you press @kbd{m C} to turn recomputation back on. (While automatic
+recomputation is off, you can think of @kbd{m C m C} as a command
+to update all @samp{=>} operators while leaving recomputation off.)
+
+To update @samp{=>} operators in an Embedded buffer while
+automatic recomputation is off, use @address@hidden * u}}.
address@hidden Mode}.
+
address@hidden Working Message, , Automatic Recomputation, Calculation Modes
address@hidden Working Messages
+
address@hidden
address@hidden Performance
address@hidden Working messages
+Since the Calculator is written entirely in Emacs Lisp, which is not
+designed for heavy numerical work, many operations are quite slow.
+The Calculator normally displays the message @samp{Working...} in the
+echo area during any command that may be slow. In addition, iterative
+operations such as square roots and trigonometric functions display the
+intermediate result at each step. Both of these types of messages can
+be disabled if you find them distracting.
+
address@hidden m w
address@hidden calc-working
+Type @kbd{m w} (@code{calc-working}) with a numeric prefix of 0 to
+disable all ``working'' messages. Use a numeric prefix of 1 to enable
+only the plain @samp{Working...} message. Use a numeric prefix of 2 to
+see intermediate results as well. With no numeric prefix this displays
+the current mode.
+
+While it may seem that the ``working'' messages will slow Calc down
+considerably, experiments have shown that their impact is actually
+quite small. But if your terminal is slow you may find that it helps
+to turn the messages off.
+
address@hidden Simplification Modes, Declarations, Calculation Modes, Mode
Settings
address@hidden Simplification Modes
+
address@hidden
+The current @dfn{simplification mode} controls how numbers and formulas
+are ``normalized'' when being taken from or pushed onto the stack.
+Some normalizations are unavoidable, such as rounding floating-point
+results to the current precision, and reducing fractions to simplest
+form. Others, such as simplifying a formula like @expr{a+a} (or @expr{2+3}),
+are done by default but can be turned off when necessary.
+
+When you press a key like @kbd{+} when @expr{2} and @expr{3} are on the
+stack, Calc pops these numbers, normalizes them, creates the formula
address@hidden, normalizes it, and pushes the result. Of course the standard
+rules for normalizing @expr{2+3} will produce the result @expr{5}.
+
+Simplification mode commands consist of the lower-case @kbd{m} prefix key
+followed by a shifted letter.
+
address@hidden m O
address@hidden calc-no-simplify-mode
+The @kbd{m O} (@code{calc-no-simplify-mode}) command turns off all optional
+simplifications. These would leave a formula like @expr{2+3} alone. In
+fact, nothing except simple numbers are ever affected by normalization
+in this mode.
+
address@hidden m N
address@hidden calc-num-simplify-mode
+The @kbd{m N} (@code{calc-num-simplify-mode}) command turns off simplification
+of any formulas except those for which all arguments are constants. For
+example, @expr{1+2} is simplified to @expr{3}, and @expr{a+(2-2)} is
+simplified to @expr{a+0} but no further, since one argument of the sum
+is not a constant. Unfortunately, @expr{(a+2)-2} is @emph{not} simplified
+because the top-level @samp{-} operator's arguments are not both
+constant numbers (one of them is the formula @expr{a+2}).
+A constant is a number or other numeric object (such as a constant
+error form or modulo form), or a vector all of whose
+elements are constant.
+
address@hidden m D
address@hidden calc-default-simplify-mode
+The @kbd{m D} (@code{calc-default-simplify-mode}) command restores the
+default simplifications for all formulas. This includes many easy and
+fast algebraic simplifications such as @expr{a+0} to @expr{a}, and
address@hidden + 2 a} to @expr{3 a}, as well as evaluating functions like
address@hidden@tfn{deriv}(x^2, x)} to @expr{2 x}.
+
address@hidden m B
address@hidden calc-bin-simplify-mode
+The @kbd{m B} (@code{calc-bin-simplify-mode}) mode applies the default
+simplifications to a result and then, if the result is an integer,
+uses the @kbd{b c} (@code{calc-clip}) command to clip the integer according
+to the current binary word size. @xref{Binary Functions}. Real numbers
+are rounded to the nearest integer and then clipped; other kinds of
+results (after the default simplifications) are left alone.
+
address@hidden m A
address@hidden calc-alg-simplify-mode
+The @kbd{m A} (@code{calc-alg-simplify-mode}) mode does algebraic
+simplification; it applies all the default simplifications, and also
+the more powerful (and slower) simplifications made by @kbd{a s}
+(@code{calc-simplify}). @xref{Algebraic Simplifications}.
+
address@hidden m E
address@hidden calc-ext-simplify-mode
+The @kbd{m E} (@code{calc-ext-simplify-mode}) mode does ``extended''
+algebraic simplification, as by the @kbd{a e} (@code{calc-simplify-extended})
+command. @xref{Unsafe Simplifications}.
+
address@hidden m U
address@hidden calc-units-simplify-mode
+The @kbd{m U} (@code{calc-units-simplify-mode}) mode does units
+simplification; it applies the command @kbd{u s}
+(@code{calc-simplify-units}), which in turn
+is a superset of @kbd{a s}. In this mode, variable names which
+are identifiable as unit names (like @samp{mm} for ``millimeters'')
+are simplified with their unit definitions in mind.
+
+A common technique is to set the simplification mode down to the lowest
+amount of simplification you will allow to be applied automatically, then
+use manual commands like @kbd{a s} and @kbd{c c} (@code{calc-clean}) to
+perform higher types of simplifications on demand. @xref{Algebraic
+Definitions}, for another sample use of No-Simplification mode.
+
address@hidden Declarations, Display Modes, Simplification Modes, Mode Settings
address@hidden Declarations
+
address@hidden
+A @dfn{declaration} is a statement you make that promises you will
+use a certain variable or function in a restricted way. This may
+give Calc the freedom to do things that it couldn't do if it had to
+take the fully general situation into account.
+
address@hidden
+* Declaration Basics::
+* Kinds of Declarations::
+* Functions for Declarations::
address@hidden menu
+
address@hidden Declaration Basics, Kinds of Declarations, Declarations,
Declarations
address@hidden Declaration Basics
+
address@hidden
address@hidden s d
address@hidden calc-declare-variable
+The @kbd{s d} (@code{calc-declare-variable}) command is the easiest
+way to make a declaration for a variable. This command prompts for
+the variable name, then prompts for the declaration. The default
+at the declaration prompt is the previous declaration, if any.
+You can edit this declaration, or press @kbd{C-k} to erase it and
+type a new declaration. (Or, erase it and press @key{RET} to clear
+the declaration, effectively ``undeclaring'' the variable.)
+
+A declaration is in general a vector of @dfn{type symbols} and
address@hidden values. If there is only one type symbol or range value,
+you can write it directly rather than enclosing it in a vector.
+For example, @kbd{s d foo @key{RET} real @key{RET}} declares @code{foo} to
+be a real number, and @kbd{s d bar @key{RET} [int, const, [1..6]] @key{RET}}
+declares @code{bar} to be a constant integer between 1 and 6.
+(Actually, you can omit the outermost brackets and Calc will
+provide them for you: @kbd{s d bar @key{RET} int, const, [1..6] @key{RET}}.)
+
address@hidden @code{Decls} variable
address@hidden Decls
+Declarations in Calc are kept in a special variable called @code{Decls}.
+This variable encodes the set of all outstanding declarations in
+the form of a matrix. Each row has two elements: A variable or
+vector of variables declared by that row, and the declaration
+specifier as described above. You can use the @kbd{s D} command to
+edit this variable if you wish to see all the declarations at once.
address@hidden on Variables}, for a description of this command
+and the @kbd{s p} command that allows you to save your declarations
+permanently if you wish.
+
+Items being declared can also be function calls. The arguments in
+the call are ignored; the effect is to say that this function returns
+values of the declared type for any valid arguments. The @kbd{s d}
+command declares only variables, so if you wish to make a function
+declaration you will have to edit the @code{Decls} matrix yourself.
+
+For example, the declaration matrix
+
address@hidden
address@hidden
+[ [ foo, real ]
+ [ [j, k, n], int ]
+ [ f(1,2,3), [0 .. inf) ] ]
address@hidden group
address@hidden smallexample
+
address@hidden
+declares that @code{foo} represents a real number, @code{j}, @code{k}
+and @code{n} represent integers, and the function @code{f} always
+returns a real number in the interval shown.
+
address@hidden All
+If there is a declaration for the variable @code{All}, then that
+declaration applies to all variables that are not otherwise declared.
+It does not apply to function names. For example, using the row
address@hidden, real]} says that all your variables are real unless they
+are explicitly declared without @code{real} in some other row.
+The @kbd{s d} command declares @code{All} if you give a blank
+response to the variable-name prompt.
+
address@hidden Kinds of Declarations, Functions for Declarations, Declaration
Basics, Declarations
address@hidden Kinds of Declarations
+
address@hidden
+The type-specifier part of a declaration (that is, the second prompt
+in the @kbd{s d} command) can be a type symbol, an interval, or a
+vector consisting of zero or more type symbols followed by zero or
+more intervals or numbers that represent the set of possible values
+for the variable.
+
address@hidden
address@hidden
+[ [ a, [1, 2, 3, 4, 5] ]
+ [ b, [1 .. 5] ]
+ [ c, [int, 1 .. 5] ] ]
address@hidden group
address@hidden smallexample
+
+Here @code{a} is declared to contain one of the five integers shown;
address@hidden is any number in the interval from 1 to 5 (any real number
+since we haven't specified), and @code{c} is any integer in that
+interval. Thus the declarations for @code{a} and @code{c} are
+nearly equivalent (see below).
+
+The type-specifier can be the empty vector @samp{[]} to say that
+nothing is known about a given variable's value. This is the same
+as not declaring the variable at all except that it overrides any
address@hidden declaration which would otherwise apply.
+
+The initial value of @code{Decls} is the empty vector @samp{[]}.
+If @code{Decls} has no stored value or if the value stored in it
+is not valid, it is ignored and there are no declarations as far
+as Calc is concerned. (The @kbd{s d} command will replace such a
+malformed value with a fresh empty matrix, @samp{[]}, before recording
+the new declaration.) Unrecognized type symbols are ignored.
+
+The following type symbols describe what sorts of numbers will be
+stored in a variable:
+
address@hidden @code
address@hidden int
+Integers.
address@hidden numint
+Numerical integers. (Integers or integer-valued floats.)
address@hidden frac
+Fractions. (Rational numbers which are not integers.)
address@hidden rat
+Rational numbers. (Either integers or fractions.)
address@hidden float
+Floating-point numbers.
address@hidden real
+Real numbers. (Integers, fractions, or floats. Actually,
+intervals and error forms with real components also count as
+reals here.)
address@hidden pos
+Positive real numbers. (Strictly greater than zero.)
address@hidden nonneg
+Nonnegative real numbers. (Greater than or equal to zero.)
address@hidden number
+Numbers. (Real or complex.)
address@hidden table
+
+Calc uses this information to determine when certain simplifications
+of formulas are safe. For example, @samp{(x^y)^z} cannot be
+simplified to @samp{x^(y z)} in general; for example,
address@hidden((-3)^2)^1:2} is 3, but @samp{(-3)^(2*1:2) = (-3)^1} is
@mathit{-3}.
+However, this simplification @emph{is} safe if @code{z} is known
+to be an integer, or if @code{x} is known to be a nonnegative
+real number. If you have given declarations that allow Calc to
+deduce either of these facts, Calc will perform this simplification
+of the formula.
+
+Calc can apply a certain amount of logic when using declarations.
+For example, @samp{(x^y)^(2n+1)} will be simplified if @code{n}
+has been declared @code{int}; Calc knows that an integer times an
+integer, plus an integer, must always be an integer. (In fact,
+Calc would simplify @samp{(-x)^(2n+1)} to @samp{-(x^(2n+1))} since
+it is able to determine that @samp{2n+1} must be an odd integer.)
+
+Similarly, @samp{(abs(x)^y)^z} will be simplified to @samp{abs(x)^(y z)}
+because Calc knows that the @code{abs} function always returns a
+nonnegative real. If you had a @code{myabs} function that also had
+this property, you could get Calc to recognize it by adding the row
address@hidden(), nonneg]} to the @code{Decls} matrix.
+
+One instance of this simplification is @samp{sqrt(x^2)} (since the
address@hidden function is effectively a one-half power). Normally
+Calc leaves this formula alone. After the command
address@hidden d x @key{RET} real @key{RET}}, however, it can simplify the
formula to
address@hidden(x)}. And after @kbd{s d x @key{RET} nonneg @key{RET}}, Calc can
+simplify this formula all the way to @samp{x}.
+
+If there are any intervals or real numbers in the type specifier,
+they comprise the set of possible values that the variable or
+function being declared can have. In particular, the type symbol
address@hidden is effectively the same as the range @samp{[-inf .. inf]}
+(note that infinity is included in the range of possible values);
address@hidden is the same as @samp{(0 .. inf]}, and @code{nonneg} is
+the same as @samp{[0 .. inf]}. Saying @samp{[real, [-5 .. 5]]} is
+redundant because the fact that the variable is real can be
+deduced just from the interval, but @samp{[int, [-5 .. 5]]} and
address@hidden, [-5 .. 5]]} are useful combinations.
+
+Note that the vector of intervals or numbers is in the same format
+used by Calc's set-manipulation commands. @xref{Set Operations}.
+
+The type specifier @samp{[1, 2, 3]} is equivalent to
address@hidden, 1, 2, 3]}, @emph{not} to @samp{[int, 1, 2, 3]}.
+In other words, the range of possible values means only that
+the variable's value must be numerically equal to a number in
+that range, but not that it must be equal in type as well.
+Calc's set operations act the same way; @samp{in(2, [1., 2., 3.])}
+and @samp{in(1.5, [1:2, 3:2, 5:2])} both report ``true.''
+
+If you use a conflicting combination of type specifiers, the
+results are unpredictable. An example is @samp{[pos, [0 .. 5]]},
+where the interval does not lie in the range described by the
+type symbol.
+
+``Real'' declarations mostly affect simplifications involving powers
+like the one described above. Another case where they are used
+is in the @kbd{a P} command which returns a list of all roots of a
+polynomial; if the variable has been declared real, only the real
+roots (if any) will be included in the list.
+
+``Integer'' declarations are used for simplifications which are valid
+only when certain values are integers (such as @samp{(x^y)^z}
+shown above).
+
+Another command that makes use of declarations is @kbd{a s}, when
+simplifying equations and inequalities. It will cancel @code{x}
+from both sides of @samp{a x = b x} only if it is sure @code{x}
+is non-zero, say, because it has a @code{pos} declaration.
+To declare specifically that @code{x} is real and non-zero,
+use @samp{[[-inf .. 0), (0 .. inf]]}. (There is no way in the
+current notation to say that @code{x} is nonzero but not necessarily
+real.) The @kbd{a e} command does ``unsafe'' simplifications,
+including cancelling @samp{x} from the equation when @samp{x} is
+not known to be nonzero.
+
+Another set of type symbols distinguish between scalars and vectors.
+
address@hidden @code
address@hidden scalar
+The value is not a vector.
address@hidden vector
+The value is a vector.
address@hidden matrix
+The value is a matrix (a rectangular vector of vectors).
address@hidden sqmatrix
+The value is a square matrix.
address@hidden table
+
+These type symbols can be combined with the other type symbols
+described above; @samp{[int, matrix]} describes an object which
+is a matrix of integers.
+
+Scalar/vector declarations are used to determine whether certain
+algebraic operations are safe. For example, @samp{[a, b, c] + x}
+is normally not simplified to @samp{[a + x, b + x, c + x]}, but
+it will be if @code{x} has been declared @code{scalar}. On the
+other hand, multiplication is usually assumed to be commutative,
+but the terms in @samp{x y} will never be exchanged if both @code{x}
+and @code{y} are known to be vectors or matrices. (Calc currently
+never distinguishes between @code{vector} and @code{matrix}
+declarations.)
+
address@hidden Mode}, for a discussion of Matrix mode and
+Scalar mode, which are similar to declaring @samp{[All, matrix]}
+or @samp{[All, scalar]} but much more convenient.
+
+One more type symbol that is recognized is used with the @kbd{H a d}
+command for taking total derivatives of a formula. @xref{Calculus}.
+
address@hidden @code
address@hidden const
+The value is a constant with respect to other variables.
address@hidden table
+
+Calc does not check the declarations for a variable when you store
+a value in it. However, storing @mathit{-3.5} in a variable that has
+been declared @code{pos}, @code{int}, or @code{matrix} may have
+unexpected effects; Calc may evaluate @samp{sqrt(x^2)} to @expr{3.5}
+if it substitutes the value first, or to @expr{-3.5} if @code{x}
+was declared @code{pos} and the formula @samp{sqrt(x^2)} is
+simplified to @samp{x} before the value is substituted. Before
+using a variable for a new purpose, it is best to use @kbd{s d}
+or @kbd{s D} to check to make sure you don't still have an old
+declaration for the variable that will conflict with its new meaning.
+
address@hidden Functions for Declarations, , Kinds of Declarations,
Declarations
address@hidden Functions for Declarations
+
address@hidden
+Calc has a set of functions for accessing the current declarations
+in a convenient manner. These functions return 1 if the argument
+can be shown to have the specified property, or 0 if the argument
+can be shown @emph{not} to have that property; otherwise they are
+left unevaluated. These functions are suitable for use with rewrite
+rules (@pxref{Conditional Rewrite Rules}) or programming constructs
+(@pxref{Conditionals in Macros}). They can be entered only using
+algebraic notation. @xref{Logical Operations}, for functions
+that perform other tests not related to declarations.
+
+For example, @samp{dint(17)} returns 1 because 17 is an integer, as
+do @samp{dint(n)} and @samp{dint(2 n - 3)} if @code{n} has been declared
address@hidden, but @samp{dint(2.5)} and @samp{dint(n + 0.5)} return 0.
+Calc consults knowledge of its own built-in functions as well as your
+own declarations: @samp{dint(floor(x))} returns 1.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden dint
address@hidden
address@hidden
address@hidden ignore
address@hidden dnumint
address@hidden
address@hidden
address@hidden ignore
address@hidden dnatnum
+The @code{dint} function checks if its argument is an integer.
+The @code{dnatnum} function checks if its argument is a natural
+number, i.e., a nonnegative integer. The @code{dnumint} function
+checks if its argument is numerically an integer, i.e., either an
+integer or an integer-valued float. Note that these and the other
+data type functions also accept vectors or matrices composed of
+suitable elements, and that real infinities @samp{inf} and @samp{-inf}
+are considered to be integers for the purposes of these functions.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden drat
+The @code{drat} function checks if its argument is rational, i.e.,
+an integer or fraction. Infinities count as rational, but intervals
+and error forms do not.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden dreal
+The @code{dreal} function checks if its argument is real. This
+includes integers, fractions, floats, real error forms, and intervals.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden dimag
+The @code{dimag} function checks if its argument is imaginary,
+i.e., is mathematically equal to a real number times @expr{i}.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden dpos
address@hidden
address@hidden
address@hidden ignore
address@hidden dneg
address@hidden
address@hidden
address@hidden ignore
address@hidden dnonneg
+The @code{dpos} function checks for positive (but nonzero) reals.
+The @code{dneg} function checks for negative reals. The @code{dnonneg}
+function checks for nonnegative reals, i.e., reals greater than or
+equal to zero. Note that the @kbd{a s} command can simplify an
+expression like @expr{x > 0} to 1 or 0 using @code{dpos}, and that
address@hidden s} is effectively applied to all conditions in rewrite rules,
+so the actual functions @code{dpos}, @code{dneg}, and @code{dnonneg}
+are rarely necessary.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden dnonzero
+The @code{dnonzero} function checks that its argument is nonzero.
+This includes all nonzero real or complex numbers, all intervals that
+do not include zero, all nonzero modulo forms, vectors all of whose
+elements are nonzero, and variables or formulas whose values can be
+deduced to be nonzero. It does not include error forms, since they
+represent values which could be anything including zero. (This is
+also the set of objects considered ``true'' in conditional contexts.)
+
address@hidden
address@hidden
address@hidden ignore
address@hidden deven
address@hidden
address@hidden
address@hidden ignore
address@hidden dodd
+The @code{deven} function returns 1 if its argument is known to be
+an even integer (or integer-valued float); it returns 0 if its argument
+is known not to be even (because it is known to be odd or a non-integer).
+The @kbd{a s} command uses this to simplify a test of the form
address@hidden % 2 = 0}. There is also an analogous @code{dodd} function.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden drange
+The @code{drange} function returns a set (an interval or a vector
+of intervals and/or numbers; @pxref{Set Operations}) that describes
+the set of possible values of its argument. If the argument is
+a variable or a function with a declaration, the range is copied
+from the declaration. Otherwise, the possible signs of the
+expression are determined using a method similar to @code{dpos},
+etc., and a suitable set like @samp{[0 .. inf]} is returned. If
+the expression is not provably real, the @code{drange} function
+remains unevaluated.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden dscalar
+The @code{dscalar} function returns 1 if its argument is provably
+scalar, or 0 if its argument is provably non-scalar. It is left
+unevaluated if this cannot be determined. (If Matrix mode or Scalar
+mode is in effect, this function returns 1 or 0, respectively,
+if it has no other information.) When Calc interprets a condition
+(say, in a rewrite rule) it considers an unevaluated formula to be
+``false.'' Thus, @samp{dscalar(a)} is ``true'' only if @code{a} is
+provably scalar, and @samp{!dscalar(a)} is ``true'' only if @code{a}
+is provably non-scalar; both are ``false'' if there is insufficient
+information to tell.
+
address@hidden Display Modes, Language Modes, Declarations, Mode Settings
address@hidden Display Modes
+
address@hidden
+The commands in this section are two-key sequences beginning with the
address@hidden prefix. The @kbd{d l} (@code{calc-line-numbering}) and @kbd{d b}
+(@code{calc-line-breaking}) commands are described elsewhere;
address@hidden Basics} and @pxref{Normal Language Modes}, respectively.
+Display formats for vectors and matrices are also covered elsewhere;
address@hidden and Matrix Formats}.
+
+One thing all display modes have in common is their treatment of the
address@hidden prefix. This prefix causes any mode command that would normally
+refresh the stack to leave the stack display alone. The word ``Dirty''
+will appear in the mode line when Calc thinks the stack display may not
+reflect the latest mode settings.
+
address@hidden d @key{RET}
address@hidden calc-refresh-top
+The @kbd{d @key{RET}} (@code{calc-refresh-top}) command reformats the
+top stack entry according to all the current modes. Positive prefix
+arguments reformat the top @var{n} entries; negative prefix arguments
+reformat the specified entry, and a prefix of zero is equivalent to
address@hidden @key{SPC}} (@code{calc-refresh}), which reformats the entire
stack.
+For example, @kbd{H d s M-2 d @key{RET}} changes to scientific notation
+but reformats only the top two stack entries in the new mode.
+
+The @kbd{I} prefix has another effect on the display modes. The mode
+is set only temporarily; the top stack entry is reformatted according
+to that mode, then the original mode setting is restored. In other
+words, @kbd{I d s} is equivalent to @kbd{H d s d @key{RET} H d (@var{old
mode})}.
+
address@hidden
+* Radix Modes::
+* Grouping Digits::
+* Float Formats::
+* Complex Formats::
+* Fraction Formats::
+* HMS Formats::
+* Date Formats::
+* Truncating the Stack::
+* Justification::
+* Labels::
address@hidden menu
+
address@hidden Radix Modes, Grouping Digits, Display Modes, Display Modes
address@hidden Radix Modes
+
address@hidden
address@hidden Radix display
address@hidden Non-decimal numbers
address@hidden Decimal and non-decimal numbers
+Calc normally displays numbers in decimal (@dfn{base-10} or @dfn{radix-10})
+notation. Calc can actually display in any radix from two (binary) to 36.
+When the radix is above 10, the letters @code{A} to @code{Z} are used as
+digits. When entering such a number, letter keys are interpreted as
+potential digits rather than terminating numeric entry mode.
+
address@hidden d 2
address@hidden d 8
address@hidden d 6
address@hidden d 0
address@hidden Hexadecimal integers
address@hidden Octal integers
+The key sequences @kbd{d 2}, @kbd{d 8}, @kbd{d 6}, and @kbd{d 0} select
+binary, octal, hexadecimal, and decimal as the current display radix,
+respectively. Numbers can always be entered in any radix, though the
+current radix is used as a default if you press @kbd{#} without any initial
+digits. A number entered without a @kbd{#} is @emph{always} interpreted
+as decimal.
+
address@hidden d r
address@hidden calc-radix
+To set the radix generally, use @kbd{d r} (@code{calc-radix}) and enter
+an integer from 2 to 36. You can specify the radix as a numeric prefix
+argument; otherwise you will be prompted for it.
+
address@hidden d z
address@hidden calc-leading-zeros
address@hidden Leading zeros
+Integers normally are displayed with however many digits are necessary to
+represent the integer and no more. The @kbd{d z} (@code{calc-leading-zeros})
+command causes integers to be padded out with leading zeros according to the
+current binary word size. (@xref{Binary Functions}, for a discussion of
+word size.) If the absolute value of the word size is @expr{w}, all integers
+are displayed with at least enough digits to represent
address@hidden @math{2^w-1}
address@hidden @expr{(2^w)-1}
+in the current radix. (Larger integers will still be displayed in their
+entirety.)
+
address@hidden Grouping Digits, Float Formats, Radix Modes, Display Modes
address@hidden Grouping Digits
+
address@hidden
address@hidden d g
address@hidden calc-group-digits
address@hidden Grouping digits
address@hidden Digit grouping
+Long numbers can be hard to read if they have too many digits. For
+example, the factorial of 30 is 33 digits long! Press @kbd{d g}
+(@code{calc-group-digits}) to enable @dfn{Grouping} mode, in which digits
+are displayed in clumps of 3 or 4 (depending on the current radix)
+separated by commas.
+
+The @kbd{d g} command toggles grouping on and off.
+With a numeric prefix of 0, this command displays the current state of
+the grouping flag; with an argument of minus one it disables grouping;
+with a positive argument @expr{N} it enables grouping on every @expr{N}
+digits. For floating-point numbers, grouping normally occurs only
+before the decimal point. A negative prefix argument @expr{-N} enables
+grouping every @expr{N} digits both before and after the decimal point.
+
address@hidden d ,
address@hidden calc-group-char
+The @kbd{d ,} (@code{calc-group-char}) command allows you to choose any
+character as the grouping separator. The default is the comma character.
+If you find it difficult to read vectors of large integers grouped with
+commas, you may wish to use spaces or some other character instead.
+This command takes the next character you type, whatever it is, and
+uses it as the digit separator. As a special case, @kbd{d , \} selects
address@hidden,} (@TeX{}'s thin-space symbol) as the digit separator.
+
+Please note that grouped numbers will not generally be parsed correctly
+if re-read in textual form, say by the use of @kbd{C-x * y} and @kbd{C-x * g}.
+(@xref{Kill and Yank}, for details on these commands.) One exception is
+the @samp{\,} separator, which doesn't interfere with parsing because it
+is ignored by @TeX{} language mode.
+
address@hidden Float Formats, Complex Formats, Grouping Digits, Display Modes
address@hidden Float Formats
+
address@hidden
+Floating-point quantities are normally displayed in standard decimal
+form, with scientific notation used if the exponent is especially high
+or low. All significant digits are normally displayed. The commands
+in this section allow you to choose among several alternative display
+formats for floats.
+
address@hidden d n
address@hidden calc-normal-notation
+The @kbd{d n} (@code{calc-normal-notation}) command selects the normal
+display format. All significant figures in a number are displayed.
+With a positive numeric prefix, numbers are rounded if necessary to
+that number of significant digits. With a negative numerix prefix,
+the specified number of significant digits less than the current
+precision is used. (Thus @kbd{C-u -2 d n} displays 10 digits if the
+current precision is 12.)
+
address@hidden d f
address@hidden calc-fix-notation
+The @kbd{d f} (@code{calc-fix-notation}) command selects fixed-point
+notation. The numeric argument is the number of digits after the
+decimal point, zero or more. This format will relax into scientific
+notation if a nonzero number would otherwise have been rounded all the
+way to zero. Specifying a negative number of digits is the same as
+for a positive number, except that small nonzero numbers will be rounded
+to zero rather than switching to scientific notation.
+
address@hidden d s
address@hidden calc-sci-notation
address@hidden Scientific notation, display of
+The @kbd{d s} (@code{calc-sci-notation}) command selects scientific
+notation. A positive argument sets the number of significant figures
+displayed, of which one will be before and the rest after the decimal
+point. A negative argument works the same as for @kbd{d n} format.
+The default is to display all significant digits.
+
address@hidden d e
address@hidden calc-eng-notation
address@hidden Engineering notation, display of
+The @kbd{d e} (@code{calc-eng-notation}) command selects engineering
+notation. This is similar to scientific notation except that the
+exponent is rounded down to a multiple of three, with from one to three
+digits before the decimal point. An optional numeric prefix sets the
+number of significant digits to display, as for @kbd{d s}.
+
+It is important to distinguish between the current @emph{precision} and
+the current @emph{display format}. After the commands @kbd{C-u 10 p}
+and @kbd{C-u 6 d n} the Calculator computes all results to ten
+significant figures but displays only six. (In fact, intermediate
+calculations are often carried to one or two more significant figures,
+but values placed on the stack will be rounded down to ten figures.)
+Numbers are never actually rounded to the display precision for storage,
+except by commands like @kbd{C-k} and @kbd{C-x * y} which operate on the
+actual displayed text in the Calculator buffer.
+
address@hidden d .
address@hidden calc-point-char
+The @kbd{d .} (@code{calc-point-char}) command selects the character used
+as a decimal point. Normally this is a period; users in some countries
+may wish to change this to a comma. Note that this is only a display
+style; on entry, periods must always be used to denote floating-point
+numbers, and commas to separate elements in a list.
+
address@hidden Complex Formats, Fraction Formats, Float Formats, Display Modes
address@hidden Complex Formats
+
address@hidden
address@hidden d c
address@hidden calc-complex-notation
+There are three supported notations for complex numbers in rectangular
+form. The default is as a pair of real numbers enclosed in parentheses
+and separated by a comma: @samp{(a,b)}. The @kbd{d c}
+(@code{calc-complex-notation}) command selects this style.
+
address@hidden d i
address@hidden calc-i-notation
address@hidden d j
address@hidden calc-j-notation
+The other notations are @kbd{d i} (@code{calc-i-notation}), in which
+numbers are displayed in @samp{a+bi} form, and @kbd{d j}
+(@code{calc-j-notation}) which displays the form @samp{a+bj} preferred
+in some disciplines.
+
address@hidden @code{i} variable
address@hidden i
+Complex numbers are normally entered in @samp{(a,b)} format.
+If you enter @samp{2+3i} as an algebraic formula, it will be stored as
+the formula @samp{2 + 3 * i}. However, if you use @kbd{=} to evaluate
+this formula and you have not changed the variable @samp{i}, the @samp{i}
+will be interpreted as @samp{(0,1)} and the formula will be simplified
+to @samp{(2,3)}. Other commands (like @code{calc-sin}) will @emph{not}
+interpret the formula @samp{2 + 3 * i} as a complex number.
address@hidden, under ``special constants.''
+
address@hidden Fraction Formats, HMS Formats, Complex Formats, Display Modes
address@hidden Fraction Formats
+
address@hidden
address@hidden d o
address@hidden calc-over-notation
+Display of fractional numbers is controlled by the @kbd{d o}
+(@code{calc-over-notation}) command. By default, a number like
+eight thirds is displayed in the form @samp{8:3}. The @kbd{d o} command
+prompts for a one- or two-character format. If you give one character,
+that character is used as the fraction separator. Common separators are
address@hidden:} and @samp{/}. (During input of numbers, the @kbd{:} key must
be
+used regardless of the display format; in particular, the @kbd{/} is used
+for RPN-style division, @emph{not} for entering fractions.)
+
+If you give two characters, fractions use ``integer-plus-fractional-part''
+notation. For example, the format @samp{+/} would display eight thirds
+as @samp{2+2/3}. If two colons are present in a number being entered,
+the number is interpreted in this form (so that the entries @kbd{2:2:3}
+and @kbd{8:3} are equivalent).
+
+It is also possible to follow the one- or two-character format with
+a number. For example: @samp{:10} or @samp{+/3}. In this case,
+Calc adjusts all fractions that are displayed to have the specified
+denominator, if possible. Otherwise it adjusts the denominator to
+be a multiple of the specified value. For example, in @samp{:6} mode
+the fraction @expr{1:6} will be unaffected, but @expr{2:3} will be
+displayed as @expr{4:6}, @expr{1:2} will be displayed as @expr{3:6},
+and @expr{1:8} will be displayed as @expr{3:24}. Integers are also
+affected by this mode: 3 is displayed as @expr{18:6}. Note that the
+format @samp{:1} writes fractions the same as @samp{:}, but it writes
+integers as @expr{n:1}.
+
+The fraction format does not affect the way fractions or integers are
+stored, only the way they appear on the screen. The fraction format
+never affects floats.
+
address@hidden HMS Formats, Date Formats, Fraction Formats, Display Modes
address@hidden HMS Formats
+
address@hidden
address@hidden d h
address@hidden calc-hms-notation
+The @kbd{d h} (@code{calc-hms-notation}) command controls the display of
+HMS (hours-minutes-seconds) forms. It prompts for a string which
+consists basically of an ``hours'' marker, optional punctuation, a
+``minutes'' marker, more optional punctuation, and a ``seconds'' marker.
+Punctuation is zero or more spaces, commas, or semicolons. The hours
+marker is one or more non-punctuation characters. The minutes and
+seconds markers must be single non-punctuation characters.
+
+The default HMS format is @samp{@@ ' "}, producing HMS values of the form
address@hidden@@ 30' 15.75"}. The format @samp{deg, ms} would display this same
+value as @samp{23deg, 30m15.75s}. During numeric entry, the @kbd{h} or @kbd{o}
+keys are recognized as synonyms for @kbd{@@} regardless of display format.
+The @kbd{m} and @kbd{s} keys are recognized as synonyms for @kbd{'} and
address@hidden"}, respectively, but only if an @kbd{@@} (or @kbd{h} or @kbd{o})
has
+already been typed; otherwise, they have their usual meanings
+(@kbd{m-} prefix and @kbd{s-} prefix). Thus, @kbd{5 "}, @kbd{0 @@ 5 "}, and
address@hidden h 5 s} are some of the ways to enter the quantity ``five
seconds.''
+The @kbd{'} key is recognized as ``minutes'' only if @kbd{@@} (or @kbd{h} or
address@hidden) has already been pressed; otherwise it means to switch to
algebraic
+entry.
+
address@hidden Date Formats, Truncating the Stack, HMS Formats, Display Modes
address@hidden Date Formats
+
address@hidden
address@hidden d d
address@hidden calc-date-notation
+The @kbd{d d} (@code{calc-date-notation}) command controls the display
+of date forms (@pxref{Date Forms}). It prompts for a string which
+contains letters that represent the various parts of a date and time.
+To show which parts should be omitted when the form represents a pure
+date with no time, parts of the string can be enclosed in @samp{< >}
+marks. If you don't include @samp{< >} markers in the format, Calc
+guesses at which parts, if any, should be omitted when formatting
+pure dates.
+
+The default format is: @samp{<H:mm:SSpp >Www Mmm D, YYYY}.
+An example string in this format is @samp{3:32pm Wed Jan 9, 1991}.
+If you enter a blank format string, this default format is
+reestablished.
+
+Calc uses @samp{< >} notation for nameless functions as well as for
+dates. @xref{Specifying Operators}. To avoid confusion with nameless
+functions, your date formats should avoid using the @samp{#} character.
+
address@hidden
+* Date Formatting Codes::
+* Free-Form Dates::
+* Standard Date Formats::
address@hidden menu
+
address@hidden Date Formatting Codes, Free-Form Dates, Date Formats, Date
Formats
address@hidden Date Formatting Codes
+
address@hidden
+When displaying a date, the current date format is used. All
+characters except for letters and @samp{<} and @samp{>} are
+copied literally when dates are formatted. The portion between
address@hidden< >} markers is omitted for pure dates, or included for
+date/time forms. Letters are interpreted according to the table
+below.
+
+When dates are read in during algebraic entry, Calc first tries to
+match the input string to the current format either with or without
+the time part. The punctuation characters (including spaces) must
+match exactly; letter fields must correspond to suitable text in
+the input. If this doesn't work, Calc checks if the input is a
+simple number; if so, the number is interpreted as a number of days
+since Jan 1, 1 AD. Otherwise, Calc tries a much more relaxed and
+flexible algorithm which is described in the next section.
+
+Weekday names are ignored during reading.
+
+Two-digit year numbers are interpreted as lying in the range
+from 1941 to 2039. Years outside that range are always
+entered and displayed in full. Year numbers with a leading
address@hidden sign are always interpreted exactly, allowing the
+entry and display of the years 1 through 99 AD.
+
+Here is a complete list of the formatting codes for dates:
+
address@hidden @asis
address@hidden Y
+Year: ``91'' for 1991, ``7'' for 2007, ``+23'' for 23 AD.
address@hidden YY
+Year: ``91'' for 1991, ``07'' for 2007, ``+23'' for 23 AD.
address@hidden BY
+Year: ``91'' for 1991, `` 7'' for 2007, ``+23'' for 23 AD.
address@hidden YYY
+Year: ``1991'' for 1991, ``23'' for 23 AD.
address@hidden YYYY
+Year: ``1991'' for 1991, ``+23'' for 23 AD.
address@hidden aa
+Year: ``ad'' or blank.
address@hidden AA
+Year: ``AD'' or blank.
address@hidden aaa
+Year: ``ad '' or blank. (Note trailing space.)
address@hidden AAA
+Year: ``AD '' or blank.
address@hidden aaaa
+Year: ``a.d.'' or blank.
address@hidden AAAA
+Year: ``A.D.'' or blank.
address@hidden bb
+Year: ``bc'' or blank.
address@hidden BB
+Year: ``BC'' or blank.
address@hidden bbb
+Year: `` bc'' or blank. (Note leading space.)
address@hidden BBB
+Year: `` BC'' or blank.
address@hidden bbbb
+Year: ``b.c.'' or blank.
address@hidden BBBB
+Year: ``B.C.'' or blank.
address@hidden M
+Month: ``8'' for August.
address@hidden MM
+Month: ``08'' for August.
address@hidden BM
+Month: `` 8'' for August.
address@hidden MMM
+Month: ``AUG'' for August.
address@hidden Mmm
+Month: ``Aug'' for August.
address@hidden mmm
+Month: ``aug'' for August.
address@hidden MMMM
+Month: ``AUGUST'' for August.
address@hidden Mmmm
+Month: ``August'' for August.
address@hidden D
+Day: ``7'' for 7th day of month.
address@hidden DD
+Day: ``07'' for 7th day of month.
address@hidden BD
+Day: `` 7'' for 7th day of month.
address@hidden W
+Weekday: ``0'' for Sunday, ``6'' for Saturday.
address@hidden WWW
+Weekday: ``SUN'' for Sunday.
address@hidden Www
+Weekday: ``Sun'' for Sunday.
address@hidden www
+Weekday: ``sun'' for Sunday.
address@hidden WWWW
+Weekday: ``SUNDAY'' for Sunday.
address@hidden Wwww
+Weekday: ``Sunday'' for Sunday.
address@hidden d
+Day of year: ``34'' for Feb. 3.
address@hidden ddd
+Day of year: ``034'' for Feb. 3.
address@hidden bdd
+Day of year: `` 34'' for Feb. 3.
address@hidden h
+Hour: ``5'' for 5 AM; ``17'' for 5 PM.
address@hidden hh
+Hour: ``05'' for 5 AM; ``17'' for 5 PM.
address@hidden bh
+Hour: `` 5'' for 5 AM; ``17'' for 5 PM.
address@hidden H
+Hour: ``5'' for 5 AM and 5 PM.
address@hidden HH
+Hour: ``05'' for 5 AM and 5 PM.
address@hidden BH
+Hour: `` 5'' for 5 AM and 5 PM.
address@hidden p
+AM/PM: ``a'' or ``p''.
address@hidden P
+AM/PM: ``A'' or ``P''.
address@hidden pp
+AM/PM: ``am'' or ``pm''.
address@hidden PP
+AM/PM: ``AM'' or ``PM''.
address@hidden pppp
+AM/PM: ``a.m.'' or ``p.m.''.
address@hidden PPPP
+AM/PM: ``A.M.'' or ``P.M.''.
address@hidden m
+Minutes: ``7'' for 7.
address@hidden mm
+Minutes: ``07'' for 7.
address@hidden bm
+Minutes: `` 7'' for 7.
address@hidden s
+Seconds: ``7'' for 7; ``7.23'' for 7.23.
address@hidden ss
+Seconds: ``07'' for 7; ``07.23'' for 7.23.
address@hidden bs
+Seconds: `` 7'' for 7; `` 7.23'' for 7.23.
address@hidden SS
+Optional seconds: ``07'' for 7; blank for 0.
address@hidden BS
+Optional seconds: `` 7'' for 7; blank for 0.
address@hidden N
+Numeric date/time: ``726842.25'' for 6:00am Wed Jan 9, 1991.
address@hidden n
+Numeric date: ``726842'' for any time on Wed Jan 9, 1991.
address@hidden J
+Julian date/time: ``2448265.75'' for 6:00am Wed Jan 9, 1991.
address@hidden j
+Julian date: ``2448266'' for any time on Wed Jan 9, 1991.
address@hidden U
+Unix time: ``663400800'' for 6:00am Wed Jan 9, 1991.
address@hidden X
+Brackets suppression. An ``X'' at the front of the format
+causes the surrounding @address@hidden< >}} delimiters to be omitted
+when formatting dates. Note that the brackets are still
+required for algebraic entry.
address@hidden table
+
+If ``SS'' or ``BS'' (optional seconds) is preceded by a colon, the
+colon is also omitted if the seconds part is zero.
+
+If ``bb,'' ``bbb'' or ``bbbb'' or their upper-case equivalents
+appear in the format, then negative year numbers are displayed
+without a minus sign. Note that ``aa'' and ``bb'' are mutually
+exclusive. Some typical usages would be @samp{YYYY AABB};
address@hidden; @samp{YYYYBBB}.
+
+The formats ``YY,'' ``YYYY,'' ``MM,'' ``DD,'' ``ddd,'' ``hh,'' ``HH,''
+``mm,'' ``ss,'' and ``SS'' actually match any number of digits during
+reading unless several of these codes are strung together with no
+punctuation in between, in which case the input must have exactly as
+many digits as there are letters in the format.
+
+The ``j,'' ``J,'' and ``U'' formats do not make any time zone
+adjustment. They effectively use @samp{julian(x,0)} and
address@hidden(x,0)} to make the conversion; @pxref{Date Arithmetic}.
+
address@hidden Free-Form Dates, Standard Date Formats, Date Formatting Codes,
Date Formats
address@hidden Free-Form Dates
+
address@hidden
+When reading a date form during algebraic entry, Calc falls back
+on the algorithm described here if the input does not exactly
+match the current date format. This algorithm generally
+``does the right thing'' and you don't have to worry about it,
+but it is described here in full detail for the curious.
+
+Calc does not distinguish between upper- and lower-case letters
+while interpreting dates.
+
+First, the time portion, if present, is located somewhere in the
+text and then removed. The remaining text is then interpreted as
+the date.
+
+A time is of the form @samp{hh:mm:ss}, possibly with the seconds
+part omitted and possibly with an AM/PM indicator added to indicate
+12-hour time. If the AM/PM is present, the minutes may also be
+omitted. The AM/PM part may be any of the words @samp{am},
address@hidden, @samp{noon}, or @samp{midnight}; each of these may be
+abbreviated to one letter, and the alternate forms @samp{a.m.},
address@hidden, and @samp{mid} are also understood. Obviously
address@hidden and @samp{midnight} are allowed only on 12:00:00.
+The words @samp{noon}, @samp{mid}, and @samp{midnight} are also
+recognized with no number attached.
+
+If there is no AM/PM indicator, the time is interpreted in 24-hour
+format.
+
+To read the date portion, all words and numbers are isolated
+from the string; other characters are ignored. All words must
+be either month names or day-of-week names (the latter of which
+are ignored). Names can be written in full or as three-letter
+abbreviations.
+
+Large numbers, or numbers with @samp{+} or @samp{-} signs,
+are interpreted as years. If one of the other numbers is
+greater than 12, then that must be the day and the remaining
+number in the input is therefore the month. Otherwise, Calc
+assumes the month, day and year are in the same order that they
+appear in the current date format. If the year is omitted, the
+current year is taken from the system clock.
+
+If there are too many or too few numbers, or any unrecognizable
+words, then the input is rejected.
+
+If there are any large numbers (of five digits or more) other than
+the year, they are ignored on the assumption that they are something
+like Julian dates that were included along with the traditional
+date components when the date was formatted.
+
+One of the words @samp{ad}, @samp{a.d.}, @samp{bc}, or @samp{b.c.}
+may optionally be used; the latter two are equivalent to a
+minus sign on the year value.
+
+If you always enter a four-digit year, and use a name instead
+of a number for the month, there is no danger of ambiguity.
+
address@hidden Standard Date Formats, , Free-Form Dates, Date Formats
address@hidden Standard Date Formats
+
address@hidden
+There are actually ten standard date formats, numbered 0 through 9.
+Entering a blank line at the @kbd{d d} command's prompt gives
+you format number 1, Calc's usual format. You can enter any digit
+to select the other formats.
+
+To create your own standard date formats, give a numeric prefix
+argument from 0 to 9 to the @address@hidden d}} command. The format you
+enter will be recorded as the new standard format of that
+number, as well as becoming the new current date format.
+You can save your formats permanently with the @address@hidden m}}
+command (@pxref{Mode Settings}).
+
address@hidden @asis
address@hidden 0
address@hidden (Numerical format)
address@hidden 1
address@hidden<H:mm:SSpp >Www Mmm D, YYYY} (American format)
address@hidden 2
address@hidden Mmm YYYY<, h:mm:SS>} (European format)
address@hidden 3
address@hidden Mmm BD< hh:mm:ss> YYYY} (Unix written date format)
address@hidden 4
address@hidden/D/Y< H:mm:SSpp>} (American slashed format)
address@hidden 5
address@hidden< h:mm:SS>} (European dotted format)
address@hidden 6
address@hidden< H:mm:SSpp>} (American dashed format)
address@hidden 7
address@hidden< h:mm:SS>} (European dashed format)
address@hidden 8
address@hidden<, h:mm:ss>} (Julian day plus time)
address@hidden 9
address@hidden< hh:mm:ss>} (Year-day format)
address@hidden table
+
address@hidden Truncating the Stack, Justification, Date Formats, Display Modes
address@hidden Truncating the Stack
+
address@hidden
address@hidden d t
address@hidden calc-truncate-stack
address@hidden Truncating the stack
address@hidden Narrowing the stack
+The @kbd{d t} (@code{calc-truncate-stack}) command moves the @samp{.}@:
+line that marks the top-of-stack up or down in the Calculator buffer.
+The number right above that line is considered to the be at the top of
+the stack. Any numbers below that line are ``hidden'' from all stack
+operations (although still visible to the user). This is similar to the
+Emacs ``narrowing'' feature, except that the values below the @samp{.}
+are @emph{visible}, just temporarily frozen. This feature allows you to
+keep several independent calculations running at once in different parts
+of the stack, or to apply a certain command to an element buried deep in
+the stack.
+
+Pressing @kbd{d t} by itself moves the @samp{.} to the line the cursor
+is on. Thus, this line and all those below it become hidden. To un-hide
+these lines, move down to the end of the buffer and press @address@hidden t}}.
+With a positive numeric prefix argument @expr{n}, @kbd{d t} hides the
+bottom @expr{n} values in the buffer. With a negative argument, it hides
+all but the top @expr{n} values. With an argument of zero, it hides zero
+values, i.e., moves the @samp{.} all the way down to the bottom.
+
address@hidden d [
address@hidden calc-truncate-up
address@hidden d ]
address@hidden calc-truncate-down
+The @kbd{d [} (@code{calc-truncate-up}) and @kbd{d ]}
+(@code{calc-truncate-down}) commands move the @samp{.} up or down one
+line at a time (or several lines with a prefix argument).
+
address@hidden Justification, Labels, Truncating the Stack, Display Modes
address@hidden Justification
+
address@hidden
address@hidden d <
address@hidden calc-left-justify
address@hidden d =
address@hidden calc-center-justify
address@hidden d >
address@hidden calc-right-justify
+Values on the stack are normally left-justified in the window. You can
+control this arrangement by typing @kbd{d <} (@code{calc-left-justify}),
address@hidden >} (@code{calc-right-justify}), or @kbd{d =}
+(@code{calc-center-justify}). For example, in Right-Justification mode,
+stack entries are displayed flush-right against the right edge of the
+window.
+
+If you change the width of the Calculator window you may have to type
address@hidden @key{SPC}} (@code{calc-refresh}) to re-align right-justified or
centered
+text.
+
+Right-justification is especially useful together with fixed-point
+notation (see @code{d f}; @code{calc-fix-notation}). With these modes
+together, the decimal points on numbers will always line up.
+
+With a numeric prefix argument, the justification commands give you
+a little extra control over the display. The argument specifies the
+horizontal ``origin'' of a display line. It is also possible to
+specify a maximum line width using the @kbd{d b} command (@pxref{Normal
+Language Modes}). For reference, the precise rules for formatting and
+breaking lines are given below. Notice that the interaction between
+origin and line width is slightly different in each justification
+mode.
+
+In Left-Justified mode, the line is indented by a number of spaces
+given by the origin (default zero). If the result is longer than the
+maximum line width, if given, or too wide to fit in the Calc window
+otherwise, then it is broken into lines which will fit; each broken
+line is indented to the origin.
+
+In Right-Justified mode, lines are shifted right so that the rightmost
+character is just before the origin, or just before the current
+window width if no origin was specified. If the line is too long
+for this, then it is broken; the current line width is used, if
+specified, or else the origin is used as a width if that is
+specified, or else the line is broken to fit in the window.
+
+In Centering mode, the origin is the column number of the center of
+each stack entry. If a line width is specified, lines will not be
+allowed to go past that width; Calc will either indent less or
+break the lines if necessary. If no origin is specified, half the
+line width or Calc window width is used.
+
+Note that, in each case, if line numbering is enabled the display
+is indented an additional four spaces to make room for the line
+number. The width of the line number is taken into account when
+positioning according to the current Calc window width, but not
+when positioning by explicit origins and widths. In the latter
+case, the display is formatted as specified, and then uniformly
+shifted over four spaces to fit the line numbers.
+
address@hidden Labels, , Justification, Display Modes
address@hidden Labels
+
address@hidden
address@hidden d @{
address@hidden calc-left-label
+The @kbd{d @{} (@code{calc-left-label}) command prompts for a string,
+then displays that string to the left of every stack entry. If the
+entries are left-justified (@pxref{Justification}), then they will
+appear immediately after the label (unless you specified an origin
+greater than the length of the label). If the entries are centered
+or right-justified, the label appears on the far left and does not
+affect the horizontal position of the stack entry.
+
+Give a blank string (with @kbd{d @{ @key{RET}}) to turn the label off.
+
address@hidden d @}
address@hidden calc-right-label
+The @kbd{d @}} (@code{calc-right-label}) command similarly adds a
+label on the righthand side. It does not affect positioning of
+the stack entries unless they are right-justified. Also, if both
+a line width and an origin are given in Right-Justified mode, the
+stack entry is justified to the origin and the righthand label is
+justified to the line width.
+
+One application of labels would be to add equation numbers to
+formulas you are manipulating in Calc and then copying into a
+document (possibly using Embedded mode). The equations would
+typically be centered, and the equation numbers would be on the
+left or right as you prefer.
+
address@hidden Language Modes, Modes Variable, Display Modes, Mode Settings
address@hidden Language Modes
+
address@hidden
+The commands in this section change Calc to use a different notation for
+entry and display of formulas, corresponding to the conventions of some
+other common language such as Pascal or address@hidden Objects displayed on
the
+stack or yanked from the Calculator to an editing buffer will be formatted
+in the current language; objects entered in algebraic entry or yanked from
+another buffer will be interpreted according to the current language.
+
+The current language has no effect on things written to or read from the
+trail buffer, nor does it affect numeric entry. Only algebraic entry is
+affected. You can make even algebraic entry ignore the current language
+and use the standard notation by giving a numeric prefix, e.g., @kbd{C-u '}.
+
+For example, suppose the formula @samp{2*a[1] + atan(a[2])} occurs in a C
+program; elsewhere in the program you need the derivatives of this formula
+with respect to @samp{a[1]} and @samp{a[2]}. First, type @kbd{d C}
+to switch to C notation. Now use @code{C-u C-x * g} to grab the formula
+into the Calculator, @kbd{a d a[1] @key{RET}} to differentiate with respect
+to the first variable, and @kbd{C-x * y} to yank the formula for the derivative
+back into your C program. Press @kbd{U} to undo the differentiation and
+repeat with @kbd{a d a[2] @key{RET}} for the other derivative.
+
+Without being switched into C mode first, Calc would have misinterpreted
+the brackets in @samp{a[1]} and @samp{a[2]}, would not have known that
address@hidden was equivalent to Calc's built-in @code{arctan} function,
+and would have written the formula back with notations (like implicit
+multiplication) which would not have been valid for a C program.
+
+As another example, suppose you are maintaining a C program and a
address@hidden
+document, each of which needs a copy of the same formula. You can grab the
+formula from the program in C mode, switch to address@hidden mode, and yank the
+formula into the document in address@hidden math-mode format.
+
+Language modes are selected by typing the letter @kbd{d} followed by a
+shifted letter key.
+
address@hidden
+* Normal Language Modes::
+* C FORTRAN Pascal::
+* TeX and LaTeX Language Modes::
+* Eqn Language Mode::
+* Mathematica Language Mode::
+* Maple Language Mode::
+* Compositions::
+* Syntax Tables::
address@hidden menu
+
address@hidden Normal Language Modes, C FORTRAN Pascal, Language Modes,
Language Modes
address@hidden Normal Language Modes
+
address@hidden
address@hidden d N
address@hidden calc-normal-language
+The @kbd{d N} (@code{calc-normal-language}) command selects the usual
+notation for Calc formulas, as described in the rest of this manual.
+Matrices are displayed in a multi-line tabular format, but all other
+objects are written in linear form, as they would be typed from the
+keyboard.
+
address@hidden d O
address@hidden calc-flat-language
address@hidden Matrix display
+The @kbd{d O} (@code{calc-flat-language}) command selects a language
+identical with the normal one, except that matrices are written in
+one-line form along with everything else. In some applications this
+form may be more suitable for yanking data into other buffers.
+
address@hidden d b
address@hidden calc-line-breaking
address@hidden Line breaking
address@hidden Breaking up long lines
+Even in one-line mode, long formulas or vectors will still be split
+across multiple lines if they exceed the width of the Calculator window.
+The @kbd{d b} (@code{calc-line-breaking}) command turns this line-breaking
+feature on and off. (It works independently of the current language.)
+If you give a numeric prefix argument of five or greater to the @kbd{d b}
+command, that argument will specify the line width used when breaking
+long lines.
+
address@hidden d B
address@hidden calc-big-language
+The @kbd{d B} (@code{calc-big-language}) command selects a language
+which uses textual approximations to various mathematical notations,
+such as powers, quotients, and square roots:
+
address@hidden
+ ____________
+ | a + 1 2
+ | ----- + c
+\| b
address@hidden example
+
address@hidden
+in place of @samp{sqrt((a+1)/b + c^2)}.
+
+Subscripts like @samp{a_i} are displayed as actual subscripts in Big
+mode. Double subscripts, @samp{a_i_j} (@samp{subscr(subscr(a, i), j)})
+are displayed as @samp{a} with subscripts separated by commas:
address@hidden, j}. They must still be entered in the usual underscore
+notation.
+
+One slight ambiguity of Big notation is that
+
address@hidden
+ 3
+- -
+ 4
address@hidden example
+
address@hidden
+can represent either the negative rational number @expr{-3:4}, or the
+actual expression @samp{-(3/4)}; but the latter formula would normally
+never be displayed because it would immediately be evaluated to
address@hidden:4} or @expr{-0.75}, so this ambiguity is not a problem in
+typical use.
+
+Non-decimal numbers are displayed with subscripts. Thus there is no
+way to tell the difference between @samp{16#C2} and @samp{C2_16},
+though generally you will know which interpretation is correct.
+Logarithms @samp{log(x,b)} and @samp{log10(x)} also use subscripts
+in Big mode.
+
+In Big mode, stack entries often take up several lines. To aid
+readability, stack entries are separated by a blank line in this mode.
+You may find it useful to expand the Calc window's height using
address@hidden ^} (@code{enlarge-window}) or to make the Calc window the only
+one on the screen with @kbd{C-x 1} (@code{delete-other-windows}).
+
+Long lines are currently not rearranged to fit the window width in
+Big mode, so you may need to use the @kbd{<} and @kbd{>} keys
+to scroll across a wide formula. For really big formulas, you may
+even need to use @address@hidden and @address@hidden to scroll up and down.
+
address@hidden d U
address@hidden calc-unformatted-language
+The @kbd{d U} (@code{calc-unformatted-language}) command altogether disables
+the use of operator notation in formulas. In this mode, the formula
+shown above would be displayed:
+
address@hidden
+sqrt(add(div(add(a, 1), b), pow(c, 2)))
address@hidden example
+
+These four modes differ only in display format, not in the format
+expected for algebraic entry. The standard Calc operators work in
+all four modes, and unformatted notation works in any language mode
+(except that Mathematica mode expects square brackets instead of
+parentheses).
+
address@hidden C FORTRAN Pascal, TeX and LaTeX Language Modes, Normal Language
Modes, Language Modes
address@hidden C, FORTRAN, and Pascal Modes
+
address@hidden
address@hidden d C
address@hidden calc-c-language
address@hidden C language
+The @kbd{d C} (@code{calc-c-language}) command selects the conventions
+of the C language for display and entry of formulas. This differs from
+the normal language mode in a variety of (mostly minor) ways. In
+particular, C language operators and operator precedences are used in
+place of Calc's usual ones. For example, @samp{a^b} means @samp{xor(a,b)}
+in C mode; a value raised to a power is written as a function call,
address@hidden(a,b)}.
+
+In C mode, vectors and matrices use curly braces instead of brackets.
+Octal and hexadecimal values are written with leading @samp{0} or @samp{0x}
+rather than using the @samp{#} symbol. Array subscripting is
+translated into @code{subscr} calls, so that @samp{a[i]} in C
+mode is the same as @samp{a_i} in Normal mode. Assignments
+turn into the @code{assign} function, which Calc normally displays
+using the @samp{:=} symbol.
+
+The variables @code{pi} and @code{e} would be displayed @samp{pi}
+and @samp{e} in Normal mode, but in C mode they are displayed as
address@hidden and @samp{M_E}, corresponding to the names of constants
+typically provided in the @file{<math.h>} header. Functions whose
+names are different in C are translated automatically for entry and
+display purposes. For example, entering @samp{asin(x)} will push the
+formula @samp{arcsin(x)} onto the stack; this formula will be displayed
+as @samp{asin(x)} as long as C mode is in effect.
+
address@hidden d P
address@hidden calc-pascal-language
address@hidden Pascal language
+The @kbd{d P} (@code{calc-pascal-language}) command selects Pascal
+conventions. Like C mode, Pascal mode interprets array brackets and uses
+a different table of operators. Hexadecimal numbers are entered and
+displayed with a preceding dollar sign. (Thus the regular meaning of
address@hidden during algebraic entry does not work in Pascal mode, though
address@hidden (and @kbd{$$}, etc.) not followed by digits works the same as
+always.) No special provisions are made for other non-decimal numbers,
+vectors, and so on, since there is no universally accepted standard way
+of handling these in Pascal.
+
address@hidden d F
address@hidden calc-fortran-language
address@hidden FORTRAN language
+The @kbd{d F} (@code{calc-fortran-language}) command selects FORTRAN
+conventions. Various function names are transformed into FORTRAN
+equivalents. Vectors are written as @samp{/1, 2, 3/}, and may be
+entered this way or using square brackets. Since FORTRAN uses round
+parentheses for both function calls and array subscripts, Calc displays
+both in the same way; @samp{a(i)} is interpreted as a function call
+upon reading, and subscripts must be entered as @samp{subscr(a, i)}.
+Also, if the variable @code{a} has been declared to have type
address@hidden or @code{matrix} then @samp{a(i)} will be parsed as a
+subscript. (@xref{Declarations}.) Usually it doesn't matter, though;
+if you enter the subscript expression @samp{a(i)} and Calc interprets
+it as a function call, you'll never know the difference unless you
+switch to another language mode or replace @code{a} with an actual
+vector (or unless @code{a} happens to be the name of a built-in
+function!).
+
+Underscores are allowed in variable and function names in all of these
+language modes. The underscore here is equivalent to the @samp{#} in
+Normal mode, or to hyphens in the underlying Emacs Lisp variable names.
+
+FORTRAN and Pascal modes normally do not adjust the case of letters in
+formulas. Most built-in Calc names use lower-case letters. If you use a
+positive numeric prefix argument with @kbd{d P} or @kbd{d F}, these
+modes will use upper-case letters exclusively for display, and will
+convert to lower-case on input. With a negative prefix, these modes
+convert to lower-case for display and input.
+
address@hidden TeX and LaTeX Language Modes, Eqn Language Mode, C FORTRAN
Pascal, Language Modes
address@hidden @TeX{} and address@hidden Language Modes
+
address@hidden
address@hidden d T
address@hidden calc-tex-language
address@hidden TeX language
address@hidden d L
address@hidden calc-latex-language
address@hidden LaTeX language
+The @kbd{d T} (@code{calc-tex-language}) command selects the conventions
+of ``math mode'' in Donald Knuth's @TeX{} typesetting language,
+and the @kbd{d L} (@code{calc-latex-language}) command selects the
+conventions of ``math mode'' in address@hidden, a typesetting language that
+uses @TeX{} as its formatting engine. Calc's address@hidden language mode can
+read any formula that the @TeX{} language mode can, although address@hidden
+mode may display it differently.
+
+Formulas are entered and displayed in the appropriate notation;
address@hidden @math{\sin(a/b)}
address@hidden @expr{sin(a/b)}
+will appear as @samp{\sin\left( a \over b \right)} in @TeX{} mode and
address@hidden(address@hidden@address@hidden@}\right)} in address@hidden mode.
+Math formulas are often enclosed by @samp{$ $} signs in @TeX{} and
address@hidden; these should be omitted when interfacing with Calc. To Calc,
+the @samp{$} sign has the same meaning it always does in algebraic
+formulas (a reference to an existing entry on the stack).
+
+Complex numbers are displayed as in @samp{3 + 4i}. Fractions and
+quotients are written using @code{\over} in @TeX{} mode (as in
address@hidden@{a \over address@hidden) and @code{\frac} in address@hidden mode
(as in
address@hidden@address@hidden@address@hidden); binomial coefficients are
written with
address@hidden in @TeX{} mode (as in @address@hidden \choose address@hidden) and
address@hidden in address@hidden mode (as in @address@hidden@address@hidden@}}).
+Interval forms are written with @code{\ldots}, and error forms are
+written with @code{\pm}. Absolute values are written as in
address@hidden|x + 1|}, and the floor and ceiling functions are written with
address@hidden, @code{\rfloor}, etc. The words @code{\left} and
address@hidden are ignored when reading formulas in @TeX{} and address@hidden
+modes. Both @code{inf} and @code{uinf} are written as @code{\infty};
+when read, @code{\infty} always translates to @code{inf}.
+
+Function calls are written the usual way, with the function name followed
+by the arguments in parentheses. However, functions for which @TeX{}
+and address@hidden have special names (like @code{\sin}) will use curly braces
+instead of parentheses for very simple arguments. During input, curly
+braces and parentheses work equally well for grouping, but when the
+document is formatted the curly braces will be invisible. Thus the
+printed result is
address@hidden @math{\sin{2 x}}
address@hidden @expr{sin 2x}
+but
address@hidden @math{\sin(2 + x)}.
address@hidden @expr{sin(2 + x)}.
+
+Function and variable names not treated specially by @TeX{} and address@hidden
+are simply written out as-is, which will cause them to come out in
+italic letters in the printed document. If you invoke @kbd{d T} or
address@hidden L} with a positive numeric prefix argument, names of more than
+one character will instead be enclosed in a protective commands that
+will prevent them from being typeset in the math italics; they will be
+written @address@hidden@address@hidden in @TeX{} mode and
address@hidden@address@hidden@}} in address@hidden mode. The
address@hidden@{ @}} and @address@hidden @}} notations are ignored during
+reading. If you use a negative prefix argument, such function names are
+written @address@hidden, and function names that begin with @code{\} during
+reading have the @code{\} removed. (Note that in this mode, long
+variable names are still written with @code{\hbox} or @code{\text}.
+However, you can always make an actual variable name like @code{\bar} in
+any @TeX{} mode.)
+
+During reading, text of the form @address@hidden ...@: @}} is replaced
+by @samp{[ ...@: ]}. The same also applies to @code{\pmatrix} and
address@hidden In address@hidden mode this also applies to
address@hidden@address@hidden ... address@hidden@}},
address@hidden@address@hidden ... address@hidden@}},
address@hidden@address@hidden ... address@hidden@}}, as well as
address@hidden@address@hidden ... address@hidden@}}.
+The symbol @samp{&} is interpreted as a comma,
+and the symbols @samp{\cr} and @samp{\\} are interpreted as semicolons.
+During output, matrices are displayed in @address@hidden a & b \\ c &
address@hidden
+format in @TeX{} mode and in
address@hidden@address@hidden a & b \\ c & d address@hidden@}} format in
address@hidden mode; you may need to edit this afterwards to change to your
+preferred matrix form. If you invoke @kbd{d T} or @kbd{d L} with an
+argument of 2 or -2, then matrices will be displayed in two-dimensional
+form, such as
+
address@hidden
address@hidden@}
+a & b \\
+c & d
address@hidden@}
address@hidden example
+
address@hidden
+This may be convenient for isolated matrices, but could lead to
+expressions being displayed like
+
address@hidden
address@hidden@} \times x
+a & b \\
+c & d
address@hidden@}
address@hidden example
+
address@hidden
+While this wouldn't bother Calc, it is incorrect address@hidden
+(Similarly for @TeX{}.)
+
+Accents like @code{\tilde} and @code{\bar} translate into function
+calls internally (@samp{tilde(x)}, @samp{bar(x)}). The @code{\underline}
+sequence is treated as an accent. The @code{\vec} accent corresponds
+to the function name @code{Vec}, because @code{vec} is the name of
+a built-in Calc function. The following table shows the accents
+in Calc, @TeX{}, address@hidden and @dfn{eqn} (described in the next section):
+
address@hidden
address@hidden
address@hidden@address@hidden @c Suppress marginal notes
address@hidden@address@hidden
address@hidden iftex
address@hidden
address@hidden
address@hidden ignore
address@hidden acute
address@hidden
address@hidden
address@hidden ignore
address@hidden Acute
address@hidden
address@hidden
address@hidden ignore
address@hidden bar
address@hidden
address@hidden
address@hidden ignore
address@hidden Bar
address@hidden
address@hidden
address@hidden ignore
address@hidden breve
address@hidden
address@hidden
address@hidden ignore
address@hidden Breve
address@hidden
address@hidden
address@hidden ignore
address@hidden check
address@hidden
address@hidden
address@hidden ignore
address@hidden Check
address@hidden
address@hidden
address@hidden ignore
address@hidden dddot
address@hidden
address@hidden
address@hidden ignore
address@hidden ddddot
address@hidden
address@hidden
address@hidden ignore
address@hidden dot
address@hidden
address@hidden
address@hidden ignore
address@hidden Dot
address@hidden
address@hidden
address@hidden ignore
address@hidden dotdot
address@hidden
address@hidden
address@hidden ignore
address@hidden DotDot
address@hidden
address@hidden
address@hidden ignore
address@hidden dyad
address@hidden
address@hidden
address@hidden ignore
address@hidden grave
address@hidden
address@hidden
address@hidden ignore
address@hidden Grave
address@hidden
address@hidden
address@hidden ignore
address@hidden hat
address@hidden
address@hidden
address@hidden ignore
address@hidden Hat
address@hidden
address@hidden
address@hidden ignore
address@hidden Prime
address@hidden
address@hidden
address@hidden ignore
address@hidden tilde
address@hidden
address@hidden
address@hidden ignore
address@hidden Tilde
address@hidden
address@hidden
address@hidden ignore
address@hidden under
address@hidden
address@hidden
address@hidden ignore
address@hidden Vec
address@hidden
address@hidden
address@hidden ignore
address@hidden VEC
address@hidden
address@hidden
address@hidden iftex
address@hidden
+Calc TeX LaTeX eqn
+---- --- ----- ---
+acute \acute \acute
+Acute \Acute
+bar \bar \bar bar
+Bar \Bar
+breve \breve \breve
+Breve \Breve
+check \check \check
+Check \Check
+dddot \dddot
+ddddot \ddddot
+dot \dot \dot dot
+Dot \Dot
+dotdot \ddot \ddot dotdot
+DotDot \Ddot
+dyad dyad
+grave \grave \grave
+Grave \Grave
+hat \hat \hat hat
+Hat \Hat
+Prime prime
+tilde \tilde \tilde tilde
+Tilde \Tilde
+under \underline \underline under
+Vec \vec \vec vec
+VEC \Vec
address@hidden example
+
+The @samp{=>} (evaluates-to) operator appears as a @code{\to} symbol:
address@hidden@address@hidden \to @address@hidden @TeX{} defines @code{\to} as
an
+alias for @code{\rightarrow}. However, if the @samp{=>} is the
+top-level expression being formatted, a slightly different notation
+is used: @samp{\evalto @var{a} \to @var{b}}. The @code{\evalto}
+word is ignored by Calc's input routines, and is undefined in @TeX{}.
+You will typically want to include one of the following definitions
+at the top of a @TeX{} file that uses @code{\evalto}:
+
address@hidden
address@hidden@}
address@hidden@}
address@hidden example
+
+The first definition formats evaluates-to operators in the usual
+way. The second causes only the @var{b} part to appear in the
+printed document; the @var{a} part and the arrow are hidden.
+Another definition you may wish to use is @samp{\let\to=\Rightarrow}
+which causes @code{\to} to appear more like Calc's @samp{=>} symbol.
address@hidden Operator}, for a discussion of @code{evalto}.
+
+The complete set of @TeX{} control sequences that are ignored during
+reading is:
+
address@hidden
+\hbox \mbox \text \left \right
+\, \> \: \; \! \quad \qquad \hfil \hfill
+\displaystyle \textstyle \dsize \tsize
+\scriptstyle \scriptscriptstyle \ssize \ssize
+\rm \bf \it \sl \roman \bold \italic \slanted
+\cal \mit \Cal \Bbb \frak \goth
+\evalto
address@hidden example
+
+Note that, because these symbols are ignored, reading a @TeX{} or
address@hidden formula into Calc and writing it back out may lose spacing and
+font information.
+
+Also, the ``discretionary multiplication sign'' @samp{\*} is read
+the same as @samp{*}.
+
address@hidden
+The @TeX{} version of this manual includes some printed examples at the
+end of this section.
address@hidden ifnottex
address@hidden
+Here are some examples of how various Calc formulas are formatted in @TeX{}:
+
address@hidden
address@hidden
+sin(a^2 / b_i)
+\sin\left( {a^2 \over b_i} \right)
address@hidden group
address@hidden example
address@hidden
+$$ \sin\left( a^2 \over b_i \right) $$
address@hidden tex
address@hidden 1
+
address@hidden
address@hidden
+[(3, 4), 3:4, 3 +/- 4, [3 .. inf)]
+[3 + 4i, @{3 \over address@hidden, 3 \pm 4, [3 \ldots \infty)]
address@hidden group
address@hidden example
address@hidden
+\turnoffactive
+$$ [3 + 4i, {3 \over 4}, 3 \pm 4, [ 3 \ldots \infty)] $$
address@hidden tex
address@hidden 1
+
address@hidden
address@hidden
+[abs(a), abs(a / b), floor(a), ceil(a / b)]
+[|a|, \left| a \over b \right|,
+ \lfloor a \rfloor, \left\lceil a \over b \right\rceil]
address@hidden group
address@hidden example
address@hidden
+$$ [|a|, \left| a \over b \right|,
+ \lfloor a \rfloor, \left\lceil a \over b \right\rceil] $$
address@hidden tex
address@hidden 1
+
address@hidden
address@hidden
+[sin(a), sin(2 a), sin(2 + a), sin(a / b)]
address@hidden@}, address@hidden address@hidden, \sin(2 + a),
+ \sin\left( @{a \over address@hidden \right)]
address@hidden group
address@hidden example
address@hidden
+\turnoffactive
+$$ [\sin{a}, \sin{2 a}, \sin(2 + a), \sin\left( {a \over b} \right)] $$
address@hidden tex
address@hidden 2
+
+First with plain @kbd{d T}, then with @kbd{C-u d T}, then finally with
address@hidden - d T} (using the example definition
address@hidden@{\tilde F(#1)@}}:
+
address@hidden
address@hidden
+[f(a), foo(bar), sin(pi)]
+[f(a), foo(bar), \sin{\pi}]
+[f(a), address@hidden@}(address@hidden@}), address@hidden@}]
+[f(a), address@hidden@address@hidden@}, address@hidden@}]
address@hidden group
address@hidden example
address@hidden
+$$ [f(a), foo(bar), \sin{\pi}] $$
+$$ [f(a), \hbox{foo}(\hbox{bar}), \sin{\pi}] $$
+$$ [f(a), \tilde F(\hbox{bar}), \sin{\pi}] $$
address@hidden tex
address@hidden 2
+
+First with @address@hidden@}}, then with @address@hidden@}}:
+
address@hidden
address@hidden
+2 + 3 => 5
+\evalto 2 + 3 \to 5
address@hidden group
address@hidden example
address@hidden
+\turnoffactive
+$$ 2 + 3 \to 5 $$
+$$ 5 $$
address@hidden tex
address@hidden 2
+
+First with standard @code{\to}, then with @samp{\let\to\Rightarrow}:
+
address@hidden
address@hidden
+[2 + 3 => 5, a / 2 => (b + c) / 2]
address@hidden + 3 \to address@hidden, @address@hidden \over address@hidden \to
@{b + c \over address@hidden@}]
address@hidden group
address@hidden example
address@hidden
+\turnoffactive
+$$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$
+{\let\to\Rightarrow
+$$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$}
address@hidden tex
address@hidden 2
+
+Matrices normally, then changing @code{\matrix} to @code{\pmatrix}:
+
address@hidden
address@hidden
+[ [ a / b, 0 ], [ 0, 2^(x + 1) ] ]
address@hidden @{a \over address@hidden & 0 \\ 0 & address@hidden(x + 1)@} @}
address@hidden @{a \over address@hidden & 0 \\ 0 & address@hidden(x + 1)@} @}
address@hidden group
address@hidden example
address@hidden
+\turnoffactive
+$$ \matrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
+$$ \pmatrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
address@hidden tex
address@hidden 2
address@hidden iftex
+
address@hidden Eqn Language Mode, Mathematica Language Mode, TeX and LaTeX
Language Modes, Language Modes
address@hidden Eqn Language Mode
+
address@hidden
address@hidden d E
address@hidden calc-eqn-language
address@hidden is another popular formatter for math formulas. It is
+designed for use with the TROFF text formatter, and comes standard
+with many versions of Unix. The @kbd{d E} (@code{calc-eqn-language})
+command selects @dfn{eqn} notation.
+
+The @dfn{eqn} language's main idiosyncrasy is that whitespace plays
+a significant part in the parsing of the language. For example,
address@hidden x+1 + y} treats @samp{x+1} as the argument of the
address@hidden operator. @dfn{Eqn} also understands more conventional
+grouping using curly braces: @address@hidden@} + y}. Braces are
+required only when the argument contains spaces.
+
+In Calc's @dfn{eqn} mode, however, curly braces are required to
+delimit arguments of operators like @code{sqrt}. The first of the
+above examples would treat only the @samp{x} as the argument of
address@hidden, and in fact @samp{sin x+1} would be interpreted as
address@hidden * x + 1}, because @code{sin} is not a special operator
+in the @dfn{eqn} language. If you always surround the argument
+with curly braces, Calc will never misunderstand.
+
+Calc also understands parentheses as grouping characters. Another
+peculiarity of @dfn{eqn}'s syntax makes it advisable to separate
+words with spaces from any surrounding characters that aren't curly
+braces, so Calc writes @samp{sin ( x + y )} in @dfn{eqn} mode.
+(The spaces around @code{sin} are important to make @dfn{eqn}
+recognize that @code{sin} should be typeset in a roman font, and
+the spaces around @code{x} and @code{y} are a good idea just in
+case the @dfn{eqn} document has defined special meanings for these
+names, too.)
+
+Powers and subscripts are written with the @code{sub} and @code{sup}
+operators, respectively. Note that the caret symbol @samp{^} is
+treated the same as a space in @dfn{eqn} mode, as is the @samp{~}
+symbol (these are used to introduce spaces of various widths into
+the typeset output of @dfn{eqn}).
+
+As in address@hidden mode, Calc's formatter omits parentheses around the
+arguments of functions like @code{ln} and @code{sin} if they are
+``simple-looking''; in this case Calc surrounds the argument with
+braces, separated by a @samp{~} from the function name: @address@hidden@}}.
+
+Font change codes (like @samp{roman @var{x}}) and positioning codes
+(like @samp{~} and @samp{down @var{n} @var{x}}) are ignored by the
address@hidden reader. Also ignored are the words @code{left}, @code{right},
address@hidden, and @code{lineup}. Quotation marks in @dfn{eqn} mode input
+are treated the same as curly braces: @samp{sqrt "1+x"} is equivalent to
address@hidden @address@hidden; this is only an approximation to the true
meaning
+of quotes in @dfn{eqn}, but it is good enough for most uses.
+
+Accent codes (@address@hidden dot}) are handled by treating them as
+function calls (@samp{dot(@var{x})}) internally.
address@hidden and LaTeX Language Modes}, for a table of these accent
+functions. The @code{prime} accent is treated specially if it occurs on
+a variable or function name: @samp{f prime prime @w{( x prime )}} is
+stored internally as @samp{f'@w{'}(x')}. For example, taking the
+derivative of @samp{f(2 x)} with @kbd{a d x} will produce @samp{2 f'(2
+x)}, which @dfn{eqn} mode will display as @samp{2 f prime ( 2 x )}.
+
+Assignments are written with the @samp{<-} (left-arrow) symbol,
+and @code{evalto} operators are written with @samp{->} or
address@hidden ... ->} (@pxref{TeX and LaTeX Language Modes}, for a discussion
+of this). The regular Calc symbols @samp{:=} and @samp{=>} are also
+recognized for these operators during reading.
+
+Vectors in @dfn{eqn} mode use regular Calc square brackets, but
+matrices are formatted as @samp{matrix @{ ccol @{ a above b @} ... @}}.
+The words @code{lcol} and @code{rcol} are recognized as synonyms
+for @code{ccol} during input, and are generated instead of @code{ccol}
+if the matrix justification mode so specifies.
+
address@hidden Mathematica Language Mode, Maple Language Mode, Eqn Language
Mode, Language Modes
address@hidden Mathematica Language Mode
+
address@hidden
address@hidden d M
address@hidden calc-mathematica-language
address@hidden Mathematica language
+The @kbd{d M} (@code{calc-mathematica-language}) command selects the
+conventions of Mathematica. Notable differences in Mathematica mode
+are that the names of built-in functions are capitalized, and function
+calls use square brackets instead of parentheses. Thus the Calc
+formula @samp{sin(2 x)} is entered and displayed @address@hidden x]}} in
+Mathematica mode.
+
+Vectors and matrices use curly braces in Mathematica. Complex numbers
+are written @samp{3 + 4 I}. The standard special constants in Calc are
+written @code{Pi}, @code{E}, @code{I}, @code{GoldenRatio}, @code{EulerGamma},
address@hidden, @code{ComplexInfinity}, and @code{Indeterminate} in
+Mathematica mode.
+Non-decimal numbers are written, e.g., @samp{16^^7fff}. Floating-point
+numbers in scientific notation are written @samp{1.23*10.^3}.
+Subscripts use double square brackets: @samp{a[[i]]}.
+
address@hidden Maple Language Mode, Compositions, Mathematica Language Mode,
Language Modes
address@hidden Maple Language Mode
+
address@hidden
address@hidden d W
address@hidden calc-maple-language
address@hidden Maple language
+The @kbd{d W} (@code{calc-maple-language}) command selects the
+conventions of Maple.
+
+Maple's language is much like C. Underscores are allowed in symbol
+names; square brackets are used for subscripts; explicit @samp{*}s for
+multiplications are required. Use either @samp{^} or @samp{**} to
+denote powers.
+
+Maple uses square brackets for lists and curly braces for sets. Calc
+interprets both notations as vectors, and displays vectors with square
+brackets. This means Maple sets will be converted to lists when they
+pass through Calc. As a special case, matrices are written as calls
+to the function @code{matrix}, given a list of lists as the argument,
+and can be read in this form or with all-capitals @code{MATRIX}.
+
+The Maple interval notation @samp{2 .. 3} has no surrounding brackets;
+Calc reads @samp{2 .. 3} as the closed interval @samp{[2 .. 3]}, and
+writes any kind of interval as @samp{2 .. 3}. This means you cannot
+see the difference between an open and a closed interval while in
+Maple display mode.
+
+Maple writes complex numbers as @samp{3 + 4*I}. Its special constants
+are @code{Pi}, @code{E}, @code{I}, and @code{infinity} (all three of
address@hidden, @code{uinf}, and @code{nan} display as @code{infinity}).
+Floating-point numbers are written @samp{1.23*10.^3}.
+
+Among things not currently handled by Calc's Maple mode are the
+various quote symbols, procedures and functional operators, and
+inert (@samp{&}) operators.
+
address@hidden Compositions, Syntax Tables, Maple Language Mode, Language Modes
address@hidden Compositions
+
address@hidden
address@hidden Compositions
+There are several @dfn{composition functions} which allow you to get
+displays in a variety of formats similar to those in Big language
+mode. Most of these functions do not evaluate to anything; they are
+placeholders which are left in symbolic form by Calc's evaluator but
+are recognized by Calc's display formatting routines.
+
+Two of these, @code{string} and @code{bstring}, are described elsewhere.
address@hidden For example, @samp{string("ABC")} is displayed as
address@hidden When viewed on the stack it will be indistinguishable from
+the variable @code{ABC}, but internally it will be stored as
address@hidden([65, 66, 67])} and can still be manipulated this way; for
+example, the selection and vector commands @kbd{j 1 v v j u} would
+select the vector portion of this object and reverse the elements, then
+deselect to reveal a string whose characters had been reversed.
+
+The composition functions do the same thing in all language modes
+(although their components will of course be formatted in the current
+language mode). The one exception is Unformatted mode (@kbd{d U}),
+which does not give the composition functions any special treatment.
+The functions are discussed here because of their relationship to
+the language modes.
+
address@hidden
+* Composition Basics::
+* Horizontal Compositions::
+* Vertical Compositions::
+* Other Compositions::
+* Information about Compositions::
+* User-Defined Compositions::
address@hidden menu
+
address@hidden Composition Basics, Horizontal Compositions, Compositions,
Compositions
address@hidden Composition Basics
+
address@hidden
+Compositions are generally formed by stacking formulas together
+horizontally or vertically in various ways. Those formulas are
+themselves compositions. @TeX{} users will find this analogous
+to @TeX{}'s ``boxes.'' Each multi-line composition has a
address@hidden; horizontal compositions use the baselines to
+decide how formulas should be positioned relative to one another.
+For example, in the Big mode formula
+
address@hidden
address@hidden
+ 2
+ a + b
+17 + ------
+ c
address@hidden group
address@hidden example
+
address@hidden
+the second term of the sum is four lines tall and has line three as
+its baseline. Thus when the term is combined with 17, line three
+is placed on the same level as the baseline of 17.
+
address@hidden
+\bigskip
address@hidden tex
+
+Another important composition concept is @dfn{precedence}. This is
+an integer that represents the binding strength of various operators.
+For example, @samp{*} has higher precedence (195) than @samp{+} (180),
+which means that @samp{(a * b) + c} will be formatted without the
+parentheses, but @samp{a * (b + c)} will keep the parentheses.
+
+The operator table used by normal and Big language modes has the
+following precedences:
+
address@hidden
+_ 1200 @r{(subscripts)}
+% 1100 @r{(as in address@hidden)}
+- 1000 @r{(as in address@hidden)}
+! 1000 @r{(as in address@hidden)}
+mod 400
++/- 300
+!! 210 @r{(as in address@hidden)}
+! 210 @r{(as in address@hidden)}
+^ 200
+* 195 @r{(or implicit multiplication)}
+/ % \ 190
++ - 180 @r{(as in address@hidden)}
+| 170
+< = 160 @r{(and other relations)}
+&& 110
+|| 100
+? : 90
+!!! 85
+&&& 80
+||| 75
+:= 50
+:: 45
+=> 40
address@hidden example
+
+The general rule is that if an operator with precedence @expr{n}
+occurs as an argument to an operator with precedence @expr{m}, then
+the argument is enclosed in parentheses if @expr{n < m}. Top-level
+expressions and expressions which are function arguments, vector
+components, etc., are formatted with precedence zero (so that they
+normally never get additional parentheses).
+
+For binary left-associative operators like @samp{+}, the righthand
+argument is actually formatted with one-higher precedence than shown
+in the table. This makes sure @samp{(a + b) + c} omits the parentheses,
+but the unnatural form @samp{a + (b + c)} keeps its parentheses.
+Right-associative operators like @samp{^} format the lefthand argument
+with one-higher precedence.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden cprec
+The @code{cprec} function formats an expression with an arbitrary
+precedence. For example, @samp{cprec(abc, 185)} will combine into
+sums and products as follows: @samp{7 + abc}, @samp{7 (abc)} (because
+this @code{cprec} form has higher precedence than addition, but lower
+precedence than multiplication).
+
address@hidden
+\bigskip
address@hidden tex
+
+A final composition issue is @dfn{line breaking}. Calc uses two
+different strategies for ``flat'' and ``non-flat'' compositions.
+A non-flat composition is anything that appears on multiple lines
+(not counting line breaking). Examples would be matrices and Big
+mode powers and quotients. Non-flat compositions are displayed
+exactly as specified. If they come out wider than the current
+window, you must use horizontal scrolling (@kbd{<} and @kbd{>}) to
+view them.
+
+Flat compositions, on the other hand, will be broken across several
+lines if they are too wide to fit the window. Certain points in a
+composition are noted internally as @dfn{break points}. Calc's
+general strategy is to fill each line as much as possible, then to
+move down to the next line starting at the first break point that
+didn't fit. However, the line breaker understands the hierarchical
+structure of formulas. It will not break an ``inner'' formula if
+it can use an earlier break point from an ``outer'' formula instead.
+For example, a vector of sums might be formatted as:
+
address@hidden
address@hidden
+[ a + b + c, d + e + f,
+ g + h + i, j + k + l, m ]
address@hidden group
address@hidden example
+
address@hidden
+If the @samp{m} can fit, then so, it seems, could the @samp{g}.
+But Calc prefers to break at the comma since the comma is part
+of a ``more outer'' formula. Calc would break at a plus sign
+only if it had to, say, if the very first sum in the vector had
+itself been too large to fit.
+
+Of the composition functions described below, only @code{choriz}
+generates break points. The @code{bstring} function (@pxref{Strings})
+also generates breakable items: A break point is added after every
+space (or group of spaces) except for spaces at the very beginning or
+end of the string.
+
+Composition functions themselves count as levels in the formula
+hierarchy, so a @code{choriz} that is a component of a larger
address@hidden will be less likely to be broken. As a special case,
+if a @code{bstring} occurs as a component of a @code{choriz} or
address@hidden object (such as a vector or a list of arguments
+in a function call), then the break points in that @code{bstring}
+will be on the same level as the break points of the surrounding
+object.
+
address@hidden Horizontal Compositions, Vertical Compositions, Composition
Basics, Compositions
address@hidden Horizontal Compositions
+
address@hidden
address@hidden
address@hidden
address@hidden ignore
address@hidden choriz
+The @code{choriz} function takes a vector of objects and composes
+them horizontally. For example, @samp{choriz([17, a b/c, d])} formats
+as @address@hidden b / cd}} in Normal language mode, or as
+
address@hidden
address@hidden
+ a b
+17---d
+ c
address@hidden group
address@hidden example
+
address@hidden
+in Big language mode. This is actually one case of the general
+function @samp{choriz(@var{vec}, @var{sep}, @var{prec})}, where
+either or both of @var{sep} and @var{prec} may be omitted.
address@hidden gives the @dfn{precedence} to use when formatting
+each of the components of @var{vec}. The default precedence is
+the precedence from the surrounding environment.
+
address@hidden is a string (i.e., a vector of character codes as might
+be entered with @code{" "} notation) which should separate components
+of the composition. Also, if @var{sep} is given, the line breaker
+will allow lines to be broken after each occurrence of @var{sep}.
+If @var{sep} is omitted, the composition will not be breakable
+(unless any of its component compositions are breakable).
+
+For example, @samp{2 choriz([a, b c, d = e], " + ", 180)} is
+formatted as @samp{2 a + b c + (d = e)}. To get the @code{choriz}
+to have precedence 180 ``outwards'' as well as ``inwards,''
+enclose it in a @code{cprec} form: @samp{2 cprec(choriz(...), 180)}
+formats as @samp{2 (a + b c + (d = e))}.
+
+The baseline of a horizontal composition is the same as the
+baselines of the component compositions, which are all aligned.
+
address@hidden Vertical Compositions, Other Compositions, Horizontal
Compositions, Compositions
address@hidden Vertical Compositions
+
address@hidden
address@hidden
address@hidden
address@hidden ignore
address@hidden cvert
+The @code{cvert} function makes a vertical composition. Each
+component of the vector is centered in a column. The baseline of
+the result is by default the top line of the resulting composition.
+For example, @samp{f(cvert([a, bb, ccc]), cvert([a^2 + 1, b^2]))}
+formats in Big mode as
+
address@hidden
address@hidden
+f( a , 2 )
+ bb a + 1
+ ccc 2
+ b
address@hidden group
address@hidden example
+
address@hidden
address@hidden
address@hidden ignore
address@hidden cbase
+There are several special composition functions that work only as
+components of a vertical composition. The @code{cbase} function
+controls the baseline of the vertical composition; the baseline
+will be the same as the baseline of whatever component is enclosed
+in @code{cbase}. Thus @samp{f(cvert([a, cbase(bb), ccc]),
+cvert([a^2 + 1, cbase(b^2)]))} displays as
+
address@hidden
address@hidden
+ 2
+ a + 1
+ a 2
+f(bb , b )
+ ccc
address@hidden group
address@hidden example
+
address@hidden
address@hidden
address@hidden ignore
address@hidden ctbase
address@hidden
address@hidden
address@hidden ignore
address@hidden cbbase
+There are also @code{ctbase} and @code{cbbase} functions which
+make the baseline of the vertical composition equal to the top
+or bottom line (rather than the baseline) of that component.
+Thus @samp{cvert([cbase(a / b)]) + cvert([ctbase(a / b)]) +
+cvert([cbbase(a / b)])} gives
+
address@hidden
address@hidden
+ a
+a -
+- + a + b
+b -
+ b
address@hidden group
address@hidden example
+
+There should be only one @code{cbase}, @code{ctbase}, or @code{cbbase}
+function in a given vertical composition. These functions can also
+be written with no arguments: @samp{ctbase()} is a zero-height object
+which means the baseline is the top line of the following item, and
address@hidden()} means the baseline is the bottom line of the preceding
+item.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden crule
+The @code{crule} function builds a ``rule,'' or horizontal line,
+across a vertical composition. By itself @samp{crule()} uses @samp{-}
+characters to build the rule. You can specify any other character,
+e.g., @samp{crule("=")}. The argument must be a character code or
+vector of exactly one character code. It is repeated to match the
+width of the widest item in the stack. For example, a quotient
+with a thick line is @samp{cvert([a + 1, cbase(crule("=")), b^2])}:
+
address@hidden
address@hidden
+a + 1
+=====
+ 2
+ b
address@hidden group
address@hidden example
+
address@hidden
address@hidden
address@hidden ignore
address@hidden clvert
address@hidden
address@hidden
address@hidden ignore
address@hidden crvert
+Finally, the functions @code{clvert} and @code{crvert} act exactly
+like @code{cvert} except that the items are left- or right-justified
+in the stack. Thus @samp{clvert([a, bb, ccc]) + crvert([a, bb, ccc])}
+gives:
+
address@hidden
address@hidden
+a + a
+bb bb
+ccc ccc
address@hidden group
address@hidden example
+
+Like @code{choriz}, the vertical compositions accept a second argument
+which gives the precedence to use when formatting the components.
+Vertical compositions do not support separator strings.
+
address@hidden Other Compositions, Information about Compositions, Vertical
Compositions, Compositions
address@hidden Other Compositions
+
address@hidden
address@hidden
address@hidden
address@hidden ignore
address@hidden csup
+The @code{csup} function builds a superscripted expression. For
+example, @samp{csup(a, b)} looks the same as @samp{a^b} does in Big
+language mode. This is essentially a horizontal composition of
address@hidden and @samp{b}, where @samp{b} is shifted up so that its
+bottom line is one above the baseline.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden csub
+Likewise, the @code{csub} function builds a subscripted expression.
+This shifts @samp{b} down so that its top line is one below the
+bottom line of @samp{a} (note that this is not quite analogous to
address@hidden). Other arrangements can be obtained by using
address@hidden and @code{cvert} directly.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden cflat
+The @code{cflat} function formats its argument in ``flat'' mode,
+as obtained by @samp{d O}, if the current language mode is normal
+or Big. It has no effect in other language modes. For example,
address@hidden(b/c)} is formatted by Big mode like @samp{csup(a, cflat(b/c))}
+to improve its readability.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden cspace
+The @code{cspace} function creates horizontal space. For example,
address@hidden(4)} is effectively the same as @samp{string(" ")}.
+A second string (i.e., vector of characters) argument is repeated
+instead of the space character. For example, @samp{cspace(4, "ab")}
+looks like @samp{abababab}. If the second argument is not a string,
+it is formatted in the normal way and then several copies of that
+are composed together: @samp{cspace(4, a^2)} yields
+
address@hidden
address@hidden
+ 2 2 2 2
+a a a a
address@hidden group
address@hidden example
+
address@hidden
+If the number argument is zero, this is a zero-width object.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden cvspace
+The @code{cvspace} function creates vertical space, or a vertical
+stack of copies of a certain string or formatted object. The
+baseline is the center line of the resulting stack. A numerical
+argument of zero will produce an object which contributes zero
+height if used in a vertical composition.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden ctspace
address@hidden
address@hidden
address@hidden ignore
address@hidden cbspace
+There are also @code{ctspace} and @code{cbspace} functions which
+create vertical space with the baseline the same as the baseline
+of the top or bottom copy, respectively, of the second argument.
+Thus @samp{cvspace(2, a/b) + ctspace(2, a/b) + cbspace(2, a/b)}
+displays as:
+
address@hidden
address@hidden
+ a
+ -
+a b
+- a a
+b + - + -
+a b b
+- a
+b -
+ b
address@hidden group
address@hidden example
+
address@hidden Information about Compositions, User-Defined Compositions, Other
Compositions, Compositions
address@hidden Information about Compositions
+
address@hidden
+The functions in this section are actual functions; they compose their
+arguments according to the current language and other display modes,
+then return a certain measurement of the composition as an integer.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden cwidth
+The @code{cwidth} function measures the width, in characters, of a
+composition. For example, @samp{cwidth(a + b)} is 5, and
address@hidden(a / b)} is 5 in Normal mode, 1 in Big mode, and 11 in
address@hidden mode (for @address@hidden \over address@hidden). The argument
may involve
+the composition functions described in this section.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden cheight
+The @code{cheight} function measures the height of a composition.
+This is the total number of lines in the argument's printed form.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden cascent
address@hidden
address@hidden
address@hidden ignore
address@hidden cdescent
+The functions @code{cascent} and @code{cdescent} measure the amount
+of the height that is above (and including) the baseline, or below
+the baseline, respectively. Thus @samp{cascent(@var{x}) + cdescent(@var{x})}
+always equals @samp{cheight(@var{x})}. For a one-line formula like
address@hidden + b}, @code{cascent} returns 1 and @code{cdescent} returns 0.
+For @samp{a / b} in Big mode, @code{cascent} returns 2 and @code{cdescent}
+returns 1. The only formula for which @code{cascent} will return zero
+is @samp{cvspace(0)} or equivalents.
+
address@hidden User-Defined Compositions, , Information about Compositions,
Compositions
address@hidden User-Defined Compositions
+
address@hidden
address@hidden Z C
address@hidden calc-user-define-composition
+The @kbd{Z C} (@code{calc-user-define-composition}) command lets you
+define the display format for any algebraic function. You provide a
+formula containing a certain number of argument variables on the stack.
+Any time Calc formats a call to the specified function in the current
+language mode and with that number of arguments, Calc effectively
+replaces the function call with that formula with the arguments
+replaced.
+
+Calc builds the default argument list by sorting all the variable names
+that appear in the formula into alphabetical order. You can edit this
+argument list before pressing @key{RET} if you wish. Any variables in
+the formula that do not appear in the argument list will be displayed
+literally; any arguments that do not appear in the formula will not
+affect the display at all.
+
+You can define formats for built-in functions, for functions you have
+defined with @kbd{Z F} (@pxref{Algebraic Definitions}), or for functions
+which have no definitions but are being used as purely syntactic objects.
+You can define different formats for each language mode, and for each
+number of arguments, using a succession of @kbd{Z C} commands. When
+Calc formats a function call, it first searches for a format defined
+for the current language mode (and number of arguments); if there is
+none, it uses the format defined for the Normal language mode. If
+neither format exists, Calc uses its built-in standard format for that
+function (usually just @address@hidden(@var{args})}).
+
+If you execute @kbd{Z C} with the number 0 on the stack instead of a
+formula, any defined formats for the function in the current language
+mode will be removed. The function will revert to its standard format.
+
+For example, the default format for the binomial coefficient function
address@hidden(n, m)} in the Big language mode is
+
address@hidden
address@hidden
+ n
+( )
+ m
address@hidden group
address@hidden example
+
address@hidden
+You might prefer the notation,
+
address@hidden
address@hidden
+ C
+n m
address@hidden group
address@hidden example
+
address@hidden
+To define this notation, first make sure you are in Big mode,
+then put the formula
+
address@hidden
+choriz([cvert([cvspace(1), n]), C, cvert([cvspace(1), m])])
address@hidden smallexample
+
address@hidden
+on the stack and type @kbd{Z C}. Answer the first prompt with
address@hidden The second prompt will be the default argument list
+of @samp{(C m n)}. Edit this list to be @samp{(n m)} and press
address@hidden Now, try it out: For example, turn simplification
+off with @kbd{m O} and enter @samp{choose(a,b) + choose(7,3)}
+as an algebraic entry.
+
address@hidden
address@hidden
+ C + C
+a b 7 3
address@hidden group
address@hidden example
+
+As another example, let's define the usual notation for Stirling
+numbers of the first kind, @samp{stir1(n, m)}. This is just like
+the regular format for binomial coefficients but with square brackets
+instead of parentheses.
+
address@hidden
+choriz([string("["), cvert([n, cbase(cvspace(1)), m]), string("]")])
address@hidden smallexample
+
+Now type @kbd{Z C stir1 @key{RET}}, edit the argument list to
address@hidden(n m)}, and type @key{RET}.
+
+The formula provided to @kbd{Z C} usually will involve composition
+functions, but it doesn't have to. Putting the formula @samp{a + b + c}
+onto the stack and typing @kbd{Z C foo @key{RET} @key{RET}} would define
+the function @samp{foo(x,y,z)} to display like @samp{x + y + z}.
+This ``sum'' will act exactly like a real sum for all formatting
+purposes (it will be parenthesized the same, and so on). However
+it will be computationally unrelated to a sum. For example, the
+formula @samp{2 * foo(1, 2, 3)} will display as @samp{2 (1 + 2 + 3)}.
+Operator precedences have caused the ``sum'' to be written in
+parentheses, but the arguments have not actually been summed.
+(Generally a display format like this would be undesirable, since
+it can easily be confused with a real sum.)
+
+The special function @code{eval} can be used inside a @kbd{Z C}
+composition formula to cause all or part of the formula to be
+evaluated at display time. For example, if the formula is
address@hidden + eval(b + c)}, then @samp{foo(1, 2, 3)} will be displayed
+as @samp{1 + 5}. Evaluation will use the default simplifications,
+regardless of the current simplification mode. There are also
address@hidden and @code{evalextsimp} which simplify as if by
address@hidden s} and @kbd{a e} (respectively). Note that these ``functions''
+operate only in the context of composition formulas (and also in
+rewrite rules, where they serve a similar purpose; @pxref{Rewrite
+Rules}). On the stack, a call to @code{eval} will be left in
+symbolic form.
+
+It is not a good idea to use @code{eval} except as a last resort.
+It can cause the display of formulas to be extremely slow. For
+example, while @samp{eval(a + b)} might seem quite fast and simple,
+there are several situations where it could be slow. For example,
address@hidden and/or @samp{b} could be polar complex numbers, in which
+case doing the sum requires trigonometry. Or, @samp{a} could be
+the factorial @samp{fact(100)} which is unevaluated because you
+have typed @kbd{m O}; @code{eval} will evaluate it anyway to
+produce a large, unwieldy integer.
+
+You can save your display formats permanently using the @kbd{Z P}
+command (@pxref{Creating User Keys}).
+
address@hidden Syntax Tables, , Compositions, Language Modes
address@hidden Syntax Tables
+
address@hidden
address@hidden Syntax tables
address@hidden Parsing formulas, customized
+Syntax tables do for input what compositions do for output: They
+allow you to teach custom notations to Calc's formula parser.
+Calc keeps a separate syntax table for each language mode.
+
+(Note that the Calc ``syntax tables'' discussed here are completely
+unrelated to the syntax tables described in the Emacs manual.)
+
address@hidden Z S
address@hidden calc-edit-user-syntax
+The @kbd{Z S} (@code{calc-edit-user-syntax}) command edits the
+syntax table for the current language mode. If you want your
+syntax to work in any language, define it in the Normal language
+mode. Type @kbd{C-c C-c} to finish editing the syntax table, or
address@hidden k} to cancel the edit. The @kbd{m m} command saves all
+the syntax tables along with the other mode settings;
address@hidden Mode Commands}.
+
address@hidden
+* Syntax Table Basics::
+* Precedence in Syntax Tables::
+* Advanced Syntax Patterns::
+* Conditional Syntax Rules::
address@hidden menu
+
address@hidden Syntax Table Basics, Precedence in Syntax Tables, Syntax Tables,
Syntax Tables
address@hidden Syntax Table Basics
+
address@hidden
address@hidden is the process of converting a raw string of characters,
+such as you would type in during algebraic entry, into a Calc formula.
+Calc's parser works in two stages. First, the input is broken down
+into @dfn{tokens}, such as words, numbers, and punctuation symbols
+like @samp{+}, @samp{:=}, and @samp{+/-}. Space between tokens is
+ignored (except when it serves to separate adjacent words). Next,
+the parser matches this string of tokens against various built-in
+syntactic patterns, such as ``an expression followed by @samp{+}
+followed by another expression'' or ``a name followed by @samp{(},
+zero or more expressions separated by commas, and @samp{)}.''
+
+A @dfn{syntax table} is a list of user-defined @dfn{syntax rules},
+which allow you to specify new patterns to define your own
+favorite input notations. Calc's parser always checks the syntax
+table for the current language mode, then the table for the Normal
+language mode, before it uses its built-in rules to parse an
+algebraic formula you have entered. Each syntax rule should go on
+its own line; it consists of a @dfn{pattern}, a @samp{:=} symbol,
+and a Calc formula with an optional @dfn{condition}. (Syntax rules
+resemble algebraic rewrite rules, but the notation for patterns is
+completely different.)
+
+A syntax pattern is a list of tokens, separated by spaces.
+Except for a few special symbols, tokens in syntax patterns are
+matched literally, from left to right. For example, the rule,
+
address@hidden
+foo ( ) := 2+3
address@hidden example
+
address@hidden
+would cause Calc to parse the formula @samp{4+foo()*5} as if it
+were @samp{4+(2+3)*5}. Notice that the parentheses were written
+as two separate tokens in the rule. As a result, the rule works
+for both @samp{foo()} and @address@hidden ( )}}. If we had written
+the rule as @samp{foo () := 2+3}, then Calc would treat @samp{()}
+as a single, indivisible token, so that @address@hidden( )}} would
+not be recognized by the rule. (It would be parsed as a regular
+zero-argument function call instead.) In fact, this rule would
+also make trouble for the rest of Calc's parser: An unrelated
+formula like @samp{bar()} would now be tokenized into @samp{bar ()}
+instead of @samp{bar ( )}, so that the standard parser for function
+calls would no longer recognize it!
+
+While it is possible to make a token with a mixture of letters
+and punctuation symbols, this is not recommended. It is better to
+break it into several tokens, as we did with @samp{foo()} above.
+
+The symbol @samp{#} in a syntax pattern matches any Calc expression.
+On the righthand side, the things that matched the @samp{#}s can
+be referred to as @samp{#1}, @samp{#2}, and so on (where @samp{#1}
+matches the leftmost @samp{#} in the pattern). For example, these
+rules match a user-defined function, prefix operator, infix operator,
+and postfix operator, respectively:
+
address@hidden
+foo ( # ) := myfunc(#1)
+foo # := myprefix(#1)
+# foo # := myinfix(#1,#2)
+# foo := mypostfix(#1)
address@hidden example
+
+Thus @samp{foo(3)} will parse as @samp{myfunc(3)}, and @samp{2+3 foo}
+will parse as @samp{mypostfix(2+3)}.
+
+It is important to write the first two rules in the order shown,
+because Calc tries rules in order from first to last. If the
+pattern @samp{foo #} came first, it would match anything that could
+match the @samp{foo ( # )} rule, since an expression in parentheses
+is itself a valid expression. Thus the @address@hidden ( # )}} rule would
+never get to match anything. Likewise, the last two rules must be
+written in the order shown or else @samp{3 foo 4} will be parsed as
address@hidden(3) * 4}. (Of course, the best way to avoid these
+ambiguities is not to use the same symbol in more than one way at
+the same time! In case you're not convinced, try the following
+exercise: How will the above rules parse the input @samp{foo(3,4)},
+if at all? Work it out for yourself, then try it in Calc and see.)
+
+Calc is quite flexible about what sorts of patterns are allowed.
+The only rule is that every pattern must begin with a literal
+token (like @samp{foo} in the first two patterns above), or with
+a @samp{#} followed by a literal token (as in the last two
+patterns). After that, any mixture is allowed, although putting
+two @samp{#}s in a row will not be very useful since two
+expressions with nothing between them will be parsed as one
+expression that uses implicit multiplication.
+
+As a more practical example, Maple uses the notation
address@hidden(a(i), i=1..10)} for sums, which Calc's Maple mode doesn't
+recognize at present. To handle this syntax, we simply add the
+rule,
+
address@hidden
+sum ( # , # = # .. # ) := sum(#1,#2,#3,#4)
address@hidden example
+
address@hidden
+to the Maple mode syntax table. As another example, C mode can't
+read assignment operators like @samp{++} and @samp{*=}. We can
+define these operators quite easily:
+
address@hidden
+# *= # := muleq(#1,#2)
+# ++ := postinc(#1)
+++ # := preinc(#1)
address@hidden example
+
address@hidden
+To complete the job, we would use corresponding composition functions
+and @kbd{Z C} to cause these functions to display in their respective
+Maple and C notations. (Note that the C example ignores issues of
+operator precedence, which are discussed in the next section.)
+
+You can enclose any token in quotes to prevent its usual
+interpretation in syntax patterns:
+
address@hidden
+# ":=" # := becomes(#1,#2)
address@hidden example
+
+Quotes also allow you to include spaces in a token, although once
+again it is generally better to use two tokens than one token with
+an embedded space. To include an actual quotation mark in a quoted
+token, precede it with a backslash. (This also works to include
+backslashes in tokens.)
+
address@hidden
+# "bad token" # "/\"\\" # := silly(#1,#2,#3)
address@hidden example
+
address@hidden
+This will parse @samp{3 bad token 4 /"\ 5} to @samp{silly(3,4,5)}.
+
+The token @kbd{#} has a predefined meaning in Calc's formula parser;
+it is not valid to use @samp{"#"} in a syntax rule. However, longer
+tokens that include the @samp{#} character are allowed. Also, while
address@hidden"$"} and @samp{"\""} are allowed as tokens, their presence in
+the syntax table will prevent those characters from working in their
+usual ways (referring to stack entries and quoting strings,
+respectively).
+
+Finally, the notation @samp{%%} anywhere in a syntax table causes
+the rest of the line to be ignored as a comment.
+
address@hidden Precedence in Syntax Tables, Advanced Syntax Patterns, Syntax
Table Basics, Syntax Tables
address@hidden Precedence
+
address@hidden
+Different operators are generally assigned different @dfn{precedences}.
+By default, an operator defined by a rule like
+
address@hidden
+# foo # := foo(#1,#2)
address@hidden example
+
address@hidden
+will have an extremely low precedence, so that @samp{2*3+4 foo 5 == 6}
+will be parsed as @samp{(2*3+4) foo (5 == 6)}. To change the
+precedence of an operator, use the notation @samp{#/@var{p}} in
+place of @samp{#}, where @var{p} is an integer precedence level.
+For example, 185 lies between the precedences for @samp{+} and
address@hidden, so if we change this rule to
+
address@hidden
+#/185 foo #/186 := foo(#1,#2)
address@hidden example
+
address@hidden
+then @samp{2+3 foo 4*5} will be parsed as @samp{2+(3 foo (4*5))}.
+Also, because we've given the righthand expression slightly higher
+precedence, our new operator will be left-associative:
address@hidden foo 2 foo 3} will be parsed as @samp{(1 foo 2) foo 3}.
+By raising the precedence of the lefthand expression instead, we
+can create a right-associative operator.
+
address@hidden Basics}, for a table of precedences of the
+standard Calc operators. For the precedences of operators in other
+language modes, look in the Calc source file @file{calc-lang.el}.
+
address@hidden Advanced Syntax Patterns, Conditional Syntax Rules, Precedence
in Syntax Tables, Syntax Tables
address@hidden Advanced Syntax Patterns
+
address@hidden
+To match a function with a variable number of arguments, you could
+write
+
address@hidden
+foo ( # ) := myfunc(#1)
+foo ( # , # ) := myfunc(#1,#2)
+foo ( # , # , # ) := myfunc(#1,#2,#3)
address@hidden example
+
address@hidden
+but this isn't very elegant. To match variable numbers of items,
+Calc uses some notations inspired regular expressions and the
+``extended BNF'' style used by some language designers.
+
address@hidden
+foo ( @{ # @}*, ) := apply(myfunc,#1)
address@hidden example
+
+The token @address@hidden introduces a repeated or optional portion.
+One of the three tokens @address@hidden, @address@hidden, or @address@hidden
+ends the portion. These will match zero or more, one or more,
+or zero or one copies of the enclosed pattern, respectively.
+In addition, @address@hidden and @address@hidden can be followed by a
+separator token (with no space in between, as shown above).
+Thus @address@hidden # @}*,} matches nothing, or one expression, or
+several expressions separated by commas.
+
+A complete @address@hidden ... @}} item matches as a vector of the
+items that matched inside it. For example, the above rule will
+match @samp{foo(1,2,3)} to get @samp{apply(myfunc,[1,2,3])}.
+The Calc @code{apply} function takes a function name and a vector
+of arguments and builds a call to the function with those
+arguments, so the net result is the formula @samp{myfunc(1,2,3)}.
+
+If the body of a @address@hidden ... @}} contains several @samp{#}s
+(or nested @address@hidden ... @}} constructs), then the items will be
+strung together into the resulting vector. If the body
+does not contain anything but literal tokens, the result will
+always be an empty vector.
+
address@hidden
+foo ( @{ # , # @}+, ) := bar(#1)
+foo ( @{ @{ # @}*, @}*; ) := matrix(#1)
address@hidden example
+
address@hidden
+will parse @samp{foo(1, 2, 3, 4)} as @samp{bar([1, 2, 3, 4])}, and
address@hidden(1, 2; 3, 4)} as @samp{matrix([[1, 2], [3, 4]])}. Also, after
+some thought it's easy to see how this pair of rules will parse
address@hidden(1, 2, 3)} as @samp{matrix([[1, 2, 3]])}, since the first
+rule will only match an even number of arguments. The rule
+
address@hidden
+foo ( # @{ , # , # @}? ) := bar(#1,#2)
address@hidden example
+
address@hidden
+will parse @samp{foo(2,3,4)} as @samp{bar(2,[3,4])}, and
address@hidden(2)} as @samp{bar(2,[])}.
+
+The notation @address@hidden ... @}?.} (note the trailing period) works
+just the same as regular @address@hidden ... @}?}, except that it does not
+count as an argument; the following two rules are equivalent:
+
address@hidden
+foo ( # , @{ also @}? # ) := bar(#1,#3)
+foo ( # , @{ also @}?. # ) := bar(#1,#2)
address@hidden example
+
address@hidden
+Note that in the first case the optional text counts as @samp{#2},
+which will always be an empty vector, but in the second case no
+empty vector is produced.
+
+Another variant is @address@hidden ... @}?$}, which means the body is
+optional only at the end of the input formula. All built-in syntax
+rules in Calc use this for closing delimiters, so that during
+algebraic entry you can type @kbd{[sqrt(2), sqrt(3 @key{RET}}, omitting
+the closing parenthesis and bracket. Calc does this automatically
+for trailing @samp{)}, @samp{]}, and @samp{>} tokens in syntax
+rules, but you can use @address@hidden ... @}?$} explicitly to get
+this effect with any token (such as @samp{"@}"} or @samp{end}).
+Like @address@hidden ... @}?.}, this notation does not count as an
+argument. Conversely, you can use quotes, as in @samp{")"}, to
+prevent a closing-delimiter token from being automatically treated
+as optional.
+
+Calc's parser does not have full backtracking, which means some
+patterns will not work as you might expect:
+
address@hidden
+foo ( @{ # , @}? # , # ) := bar(#1,#2,#3)
address@hidden example
+
address@hidden
+Here we are trying to make the first argument optional, so that
address@hidden(2,3)} parses as @samp{bar([],2,3)}. Unfortunately, Calc
+first tries to match @samp{2,} against the optional part of the
+pattern, finds a match, and so goes ahead to match the rest of the
+pattern. Later on it will fail to match the second comma, but it
+doesn't know how to go back and try the other alternative at that
+point. One way to get around this would be to use two rules:
+
address@hidden
+foo ( # , # , # ) := bar([#1],#2,#3)
+foo ( # , # ) := bar([],#1,#2)
address@hidden example
+
+More precisely, when Calc wants to match an optional or repeated
+part of a pattern, it scans forward attempting to match that part.
+If it reaches the end of the optional part without failing, it
+``finalizes'' its choice and proceeds. If it fails, though, it
+backs up and tries the other alternative. Thus Calc has ``partial''
+backtracking. A fully backtracking parser would go on to make sure
+the rest of the pattern matched before finalizing the choice.
+
address@hidden Conditional Syntax Rules, , Advanced Syntax Patterns, Syntax
Tables
address@hidden Conditional Syntax Rules
+
address@hidden
+It is possible to attach a @dfn{condition} to a syntax rule. For
+example, the rules
+
address@hidden
+foo ( # ) := ifoo(#1) :: integer(#1)
+foo ( # ) := gfoo(#1)
address@hidden example
+
address@hidden
+will parse @samp{foo(3)} as @samp{ifoo(3)}, but will parse
address@hidden(3.5)} and @samp{foo(x)} as calls to @code{gfoo}. Any
+number of conditions may be attached; all must be true for the
+rule to succeed. A condition is ``true'' if it evaluates to a
+nonzero number. @xref{Logical Operations}, for a list of Calc
+functions like @code{integer} that perform logical tests.
+
+The exact sequence of events is as follows: When Calc tries a
+rule, it first matches the pattern as usual. It then substitutes
address@hidden, @samp{#2}, etc., in the conditions, if any. Next, the
+conditions are simplified and evaluated in order from left to right,
+as if by the @address@hidden s}} algebra command (@pxref{Simplifying
Formulas}).
+Each result is true if it is a nonzero number, or an expression
+that can be proven to be nonzero (@pxref{Declarations}). If the
+results of all conditions are true, the expression (such as
address@hidden(#1)}) has its @samp{#}s substituted, and that is the
+result of the parse. If the result of any condition is false, Calc
+goes on to try the next rule in the syntax table.
+
+Syntax rules also support @code{let} conditions, which operate in
+exactly the same way as they do in algebraic rewrite rules.
address@hidden Features of Rewrite Rules}, for details. A @code{let}
+condition is always true, but as a side effect it defines a
+variable which can be used in later conditions, and also in the
+expression after the @samp{:=} sign:
+
address@hidden
+foo ( # ) := hifoo(x) :: let(x := #1 + 0.5) :: dnumint(x)
address@hidden example
+
address@hidden
+The @code{dnumint} function tests if a value is numerically an
+integer, i.e., either a true integer or an integer-valued float.
+This rule will parse @code{foo} with a half-integer argument,
+like @samp{foo(3.5)}, to a call like @samp{hifoo(4.)}.
+
+The lefthand side of a syntax rule @code{let} must be a simple
+variable, not the arbitrary pattern that is allowed in rewrite
+rules.
+
+The @code{matches} function is also treated specially in syntax
+rule conditions (again, in the same way as in rewrite rules).
address@hidden Commands}. If the matching pattern contains
+meta-variables, then those meta-variables may be used in later
+conditions and in the result expression. The arguments to
address@hidden are not evaluated in this situation.
+
address@hidden
+sum ( # , # ) := sum(#1,a,b,c) :: matches(#2, a=[b..c])
address@hidden example
+
address@hidden
+This is another way to implement the Maple mode @code{sum} notation.
+In this approach, we allow @samp{#2} to equal the whole expression
address@hidden Then, we use @code{matches} to break it apart into
+its components. If the expression turns out not to match the pattern,
+the syntax rule will fail. Note that @kbd{Z S} always uses Calc's
+Normal language mode for editing expressions in syntax rules, so we
+must use regular Calc notation for the interval @samp{[b..c]} that
+will correspond to the Maple mode interval @samp{1..10}.
+
address@hidden Modes Variable, Calc Mode Line, Language Modes, Mode Settings
address@hidden The @code{Modes} Variable
+
address@hidden
address@hidden m g
address@hidden calc-get-modes
+The @kbd{m g} (@code{calc-get-modes}) command pushes onto the stack
+a vector of numbers that describes the various mode settings that
+are in effect. With a numeric prefix argument, it pushes only the
address@hidden mode, i.e., the @var{n}th element of this vector. Keyboard
+macros can use the @kbd{m g} command to modify their behavior based
+on the current mode settings.
+
address@hidden @code{Modes} variable
address@hidden Modes
+The modes vector is also available in the special variable
address@hidden In other words, @kbd{m g} is like @kbd{s r Modes @key{RET}}.
+It will not work to store into this variable; in fact, if you do,
address@hidden will cease to track the current modes. (The @kbd{m g}
+command will continue to work, however.)
+
+In general, each number in this vector is suitable as a numeric
+prefix argument to the associated mode-setting command. (Recall
+that the @kbd{~} key takes a number from the stack and gives it as
+a numeric prefix to the next command.)
+
+The elements of the modes vector are as follows:
+
address@hidden
address@hidden
+Current precision. Default is 12; associated command is @kbd{p}.
+
address@hidden
+Binary word size. Default is 32; associated command is @kbd{b w}.
+
address@hidden
+Stack size (not counting the value about to be pushed by @kbd{m g}).
+This is zero if @kbd{m g} is executed with an empty stack.
+
address@hidden
+Number radix. Default is 10; command is @kbd{d r}.
+
address@hidden
+Floating-point format. This is the number of digits, plus the
+constant 0 for normal notation, 10000 for scientific notation,
+20000 for engineering notation, or 30000 for fixed-point notation.
+These codes are acceptable as prefix arguments to the @kbd{d n}
+command, but note that this may lose information: For example,
address@hidden s} and @kbd{C-u 12 d s} have similar (but not quite
+identical) effects if the current precision is 12, but they both
+produce a code of 10012, which will be treated by @kbd{d n} as
address@hidden 12 d s}. If the precision then changes, the float format
+will still be frozen at 12 significant figures.
+
address@hidden
+Angular mode. Default is 1 (degrees). Other values are 2 (radians)
+and 3 (HMS). The @kbd{m d} command accepts these prefixes.
+
address@hidden
+Symbolic mode. Value is 0 or 1; default is 0. Command is @kbd{m s}.
+
address@hidden
+Fraction mode. Value is 0 or 1; default is 0. Command is @kbd{m f}.
+
address@hidden
+Polar mode. Value is 0 (rectangular) or 1 (polar); default is 0.
+Command is @kbd{m p}.
+
address@hidden
+Matrix/Scalar mode. Default value is @mathit{-1}. Value is 0 for Scalar
+mode, @mathit{-2} for Matrix mode, @mathit{-3} for square Matrix mode,
+or @var{N} for
address@hidden @math{N\times N}
address@hidden @address@hidden
+Matrix mode. Command is @kbd{m v}.
+
address@hidden
+Simplification mode. Default is 1. Value is @mathit{-1} for off (@kbd{m O}),
+0 for @kbd{m N}, 2 for @kbd{m B}, 3 for @kbd{m A}, 4 for @kbd{m E},
+or 5 for @address@hidden U}}. The @kbd{m D} command accepts these prefixes.
+
address@hidden
+Infinite mode. Default is @mathit{-1} (off). Value is 1 if the mode is on,
+or 0 if the mode is on with positive zeros. Command is @kbd{m i}.
address@hidden enumerate
+
+For example, the sequence @kbd{M-1 m g @key{RET} 2 + ~ p} increases the
+precision by two, leaving a copy of the old precision on the stack.
+Later, @kbd{~ p} will restore the original precision using that
+stack value. (This sequence might be especially useful inside a
+keyboard macro.)
+
+As another example, @kbd{M-3 m g 1 - ~ @key{DEL}} deletes all but the
+oldest (bottommost) stack entry.
+
+Yet another example: The HP-48 ``round'' command rounds a number
+to the current displayed precision. You could roughly emulate this
+in Calc with the sequence @kbd{M-5 m g 10000 % ~ c c}. (This
+would not work for fixed-point mode, but it wouldn't be hard to
+do a full emulation with the help of the @kbd{Z [} and @kbd{Z ]}
+programming commands. @xref{Conditionals in Macros}.)
+
address@hidden Calc Mode Line, , Modes Variable, Mode Settings
address@hidden The Calc Mode Line
+
address@hidden
address@hidden Mode line indicators
+This section is a summary of all symbols that can appear on the
+Calc mode line, the highlighted bar that appears under the Calc
+stack window (or under an editing window in Embedded mode).
+
+The basic mode line format is:
+
address@hidden
+--%%-Calc: 12 Deg @var{other modes} (Calculator)
address@hidden example
+
+The @samp{%%} is the Emacs symbol for ``read-only''; it shows that
+regular Emacs commands are not allowed to edit the stack buffer
+as if it were text.
+
+The word @samp{Calc:} changes to @samp{CalcEmbed:} if Embedded mode
+is enabled. The words after this describe the various Calc modes
+that are in effect.
+
+The first mode is always the current precision, an integer.
+The second mode is always the angular mode, either @code{Deg},
address@hidden, or @code{Hms}.
+
+Here is a complete list of the remaining symbols that can appear
+on the mode line:
+
address@hidden @code
address@hidden Alg
+Algebraic mode (@kbd{m a}; @pxref{Algebraic Entry}).
+
address@hidden Alg[(
+Incomplete algebraic mode (@kbd{C-u m a}).
+
address@hidden Alg*
+Total algebraic mode (@kbd{m t}).
+
address@hidden Symb
+Symbolic mode (@kbd{m s}; @pxref{Symbolic Mode}).
+
address@hidden Matrix
+Matrix mode (@kbd{m v}; @pxref{Matrix Mode}).
+
address@hidden address@hidden
+Dimensioned Matrix mode (@kbd{C-u @var{n} m v}; @pxref{Matrix Mode}).
+
address@hidden SqMatrix
+Square Matrix mode (@kbd{C-u m v}; @pxref{Matrix Mode}).
+
address@hidden Scalar
+Scalar mode (@kbd{m v}; @pxref{Matrix Mode}).
+
address@hidden Polar
+Polar complex mode (@kbd{m p}; @pxref{Polar Mode}).
+
address@hidden Frac
+Fraction mode (@kbd{m f}; @pxref{Fraction Mode}).
+
address@hidden Inf
+Infinite mode (@kbd{m i}; @pxref{Infinite Mode}).
+
address@hidden +Inf
+Positive Infinite mode (@kbd{C-u 0 m i}).
+
address@hidden NoSimp
+Default simplifications off (@kbd{m O}; @pxref{Simplification Modes}).
+
address@hidden NumSimp
+Default simplifications for numeric arguments only (@kbd{m N}).
+
address@hidden address@hidden
+Binary-integer simplification mode; word size @var{w} (@kbd{m B}, @kbd{b w}).
+
address@hidden AlgSimp
+Algebraic simplification mode (@kbd{m A}).
+
address@hidden ExtSimp
+Extended algebraic simplification mode (@kbd{m E}).
+
address@hidden UnitSimp
+Units simplification mode (@kbd{m U}).
+
address@hidden Bin
+Current radix is 2 (@kbd{d 2}; @pxref{Radix Modes}).
+
address@hidden Oct
+Current radix is 8 (@kbd{d 8}).
+
address@hidden Hex
+Current radix is 16 (@kbd{d 6}).
+
address@hidden address@hidden
+Current radix is @var{n} (@kbd{d r}).
+
address@hidden Zero
+Leading zeros (@kbd{d z}; @pxref{Radix Modes}).
+
address@hidden Big
+Big language mode (@kbd{d B}; @pxref{Normal Language Modes}).
+
address@hidden Flat
+One-line normal language mode (@kbd{d O}).
+
address@hidden Unform
+Unformatted language mode (@kbd{d U}).
+
address@hidden C
+C language mode (@kbd{d C}; @pxref{C FORTRAN Pascal}).
+
address@hidden Pascal
+Pascal language mode (@kbd{d P}).
+
address@hidden Fortran
+FORTRAN language mode (@kbd{d F}).
+
address@hidden TeX
address@hidden language mode (@kbd{d T}; @pxref{TeX and LaTeX Language Modes}).
+
address@hidden LaTeX
address@hidden language mode (@kbd{d L}; @pxref{TeX and LaTeX Language Modes}).
+
address@hidden Eqn
address@hidden language mode (@kbd{d E}; @pxref{Eqn Language Mode}).
+
address@hidden Math
+Mathematica language mode (@kbd{d M}; @pxref{Mathematica Language Mode}).
+
address@hidden Maple
+Maple language mode (@kbd{d W}; @pxref{Maple Language Mode}).
+
address@hidden address@hidden
+Normal float mode with @var{n} digits (@kbd{d n}; @pxref{Float Formats}).
+
address@hidden address@hidden
+Fixed point mode with @var{n} digits after the point (@kbd{d f}).
+
address@hidden Sci
+Scientific notation mode (@kbd{d s}).
+
address@hidden address@hidden
+Scientific notation with @var{n} digits (@kbd{d s}).
+
address@hidden Eng
+Engineering notation mode (@kbd{d e}).
+
address@hidden address@hidden
+Engineering notation with @var{n} digits (@kbd{d e}).
+
address@hidden address@hidden
+Left-justified display indented by @var{n} (@kbd{d <}; @pxref{Justification}).
+
address@hidden Right
+Right-justified display (@kbd{d >}).
+
address@hidden address@hidden
+Right-justified display with width @var{n} (@kbd{d >}).
+
address@hidden Center
+Centered display (@kbd{d =}).
+
address@hidden address@hidden
+Centered display with center column @var{n} (@kbd{d =}).
+
address@hidden address@hidden
+Line breaking with width @var{n} (@kbd{d b}; @pxref{Normal Language Modes}).
+
address@hidden Wide
+No line breaking (@kbd{d b}).
+
address@hidden Break
+Selections show deep structure (@kbd{j b}; @pxref{Making Selections}).
+
address@hidden Save
+Record modes in @file{~/.calc.el} (@kbd{m R}; @pxref{General Mode Commands}).
+
address@hidden Local
+Record modes in Embedded buffer (@kbd{m R}).
+
address@hidden LocEdit
+Record modes as editing-only in Embedded buffer (@kbd{m R}).
+
address@hidden LocPerm
+Record modes as permanent-only in Embedded buffer (@kbd{m R}).
+
address@hidden Global
+Record modes as global in Embedded buffer (@kbd{m R}).
+
address@hidden Manual
+Automatic recomputation turned off (@kbd{m C}; @pxref{Automatic
+Recomputation}).
+
address@hidden Graph
+GNUPLOT process is alive in background (@pxref{Graphics}).
+
address@hidden Sel
+Top-of-stack has a selection (Embedded only; @pxref{Making Selections}).
+
address@hidden Dirty
+The stack display may not be up-to-date (@pxref{Display Modes}).
+
address@hidden Inv
+``Inverse'' prefix was pressed (@kbd{I}; @pxref{Inverse and Hyperbolic}).
+
address@hidden Hyp
+``Hyperbolic'' prefix was pressed (@kbd{H}).
+
address@hidden Keep
+``Keep-arguments'' prefix was pressed (@kbd{K}).
+
address@hidden Narrow
+Stack is truncated (@kbd{d t}; @pxref{Truncating the Stack}).
address@hidden table
+
+In addition, the symbols @code{Active} and @code{~Active} can appear
+as minor modes on an Embedded buffer's mode line. @xref{Embedded Mode}.
+
address@hidden Arithmetic, Scientific Functions, Mode Settings, Top
address@hidden Arithmetic Functions
+
address@hidden
+This chapter describes the Calc commands for doing simple calculations
+on numbers, such as addition, absolute value, and square roots. These
+commands work by removing the top one or two values from the stack,
+performing the desired operation, and pushing the result back onto the
+stack. If the operation cannot be performed, the result pushed is a
+formula instead of a number, such as @samp{2/0} (because division by zero
+is invalid) or @samp{sqrt(x)} (because the argument @samp{x} is a formula).
+
+Most of the commands described here can be invoked by a single keystroke.
+Some of the more obscure ones are two-letter sequences beginning with
+the @kbd{f} (``functions'') prefix key.
+
address@hidden Arguments}, for a discussion of the effect of numeric
+prefix arguments on commands in this chapter which do not otherwise
+interpret a prefix argument.
+
address@hidden
+* Basic Arithmetic::
+* Integer Truncation::
+* Complex Number Functions::
+* Conversions::
+* Date Arithmetic::
+* Financial Functions::
+* Binary Functions::
address@hidden menu
+
address@hidden Basic Arithmetic, Integer Truncation, Arithmetic, Arithmetic
address@hidden Basic Arithmetic
+
address@hidden
address@hidden +
address@hidden calc-plus
address@hidden
address@hidden @null
address@hidden ignore
address@hidden +
+The @kbd{+} (@code{calc-plus}) command adds two numbers. The numbers may
+be any of the standard Calc data types. The resulting sum is pushed back
+onto the stack.
+
+If both arguments of @kbd{+} are vectors or matrices (of matching dimensions),
+the result is a vector or matrix sum. If one argument is a vector and the
+other a scalar (i.e., a non-vector), the scalar is added to each of the
+elements of the vector to form a new vector. If the scalar is not a
+number, the operation is left in symbolic form: Suppose you added @samp{x}
+to the vector @samp{[1,2]}. You may want the result @samp{[1+x,2+x]}, or
+you may plan to substitute a 2-vector for @samp{x} in the future. Since
+the Calculator can't tell which interpretation you want, it makes the
+safest assumption. @xref{Reducing and Mapping}, for a way to add @samp{x}
+to every element of a vector.
+
+If either argument of @kbd{+} is a complex number, the result will in general
+be complex. If one argument is in rectangular form and the other polar,
+the current Polar mode determines the form of the result. If Symbolic
+mode is enabled, the sum may be left as a formula if the necessary
+conversions for polar addition are non-trivial.
+
+If both arguments of @kbd{+} are HMS forms, the forms are added according to
+the usual conventions of hours-minutes-seconds notation. If one argument
+is an HMS form and the other is a number, that number is converted from
+degrees or radians (depending on the current Angular mode) to HMS format
+and then the two HMS forms are added.
+
+If one argument of @kbd{+} is a date form, the other can be either a
+real number, which advances the date by a certain number of days, or
+an HMS form, which advances the date by a certain amount of time.
+Subtracting two date forms yields the number of days between them.
+Adding two date forms is meaningless, but Calc interprets it as the
+subtraction of one date form and the negative of the other. (The
+negative of a date form can be understood by remembering that dates
+are stored as the number of days before or after Jan 1, 1 AD.)
+
+If both arguments of @kbd{+} are error forms, the result is an error form
+with an appropriately computed standard deviation. If one argument is an
+error form and the other is a number, the number is taken to have zero error.
+Error forms may have symbolic formulas as their mean and/or error parts;
+adding these will produce a symbolic error form result. However, adding an
+error form to a plain symbolic formula (as in @samp{(a +/- b) + c}) will not
+work, for the same reasons just mentioned for vectors. Instead you must
+write @samp{(a +/- b) + (c +/- 0)}.
+
+If both arguments of @kbd{+} are modulo forms with equal values of @expr{M},
+or if one argument is a modulo form and the other a plain number, the
+result is a modulo form which represents the sum, modulo @expr{M}, of
+the two values.
+
+If both arguments of @kbd{+} are intervals, the result is an interval
+which describes all possible sums of the possible input values. If
+one argument is a plain number, it is treated as the interval
address@hidden@samp{[x ..@: x]}}.
+
+If one argument of @kbd{+} is an infinity and the other is not, the
+result is that same infinity. If both arguments are infinite and in
+the same direction, the result is the same infinity, but if they are
+infinite in different directions the result is @code{nan}.
+
address@hidden -
address@hidden calc-minus
address@hidden
address@hidden @null
address@hidden ignore
address@hidden -
+The @kbd{-} (@code{calc-minus}) command subtracts two values. The top
+number on the stack is subtracted from the one behind it, so that the
+computation @kbd{5 @key{RET} 2 -} produces 3, not @mathit{-3}. All options
+available for @kbd{+} are available for @kbd{-} as well.
+
address@hidden *
address@hidden calc-times
address@hidden
address@hidden @null
address@hidden ignore
address@hidden *
+The @kbd{*} (@code{calc-times}) command multiplies two numbers. If one
+argument is a vector and the other a scalar, the scalar is multiplied by
+the elements of the vector to produce a new vector. If both arguments
+are vectors, the interpretation depends on the dimensions of the
+vectors: If both arguments are matrices, a matrix multiplication is
+done. If one argument is a matrix and the other a plain vector, the
+vector is interpreted as a row vector or column vector, whichever is
+dimensionally correct. If both arguments are plain vectors, the result
+is a single scalar number which is the dot product of the two vectors.
+
+If one argument of @kbd{*} is an HMS form and the other a number, the
+HMS form is multiplied by that amount. It is an error to multiply two
+HMS forms together, or to attempt any multiplication involving date
+forms. Error forms, modulo forms, and intervals can be multiplied;
+see the comments for addition of those forms. When two error forms
+or intervals are multiplied they are considered to be statistically
+independent; thus, @samp{[-2 ..@: 3] * [-2 ..@: 3]} is @samp{[-6 ..@: 9]},
+whereas @address@hidden ..@: 3] ^ 2}} is @samp{[0 ..@: 9]}.
+
address@hidden /
address@hidden calc-divide
address@hidden
address@hidden @null
address@hidden ignore
address@hidden /
+The @kbd{/} (@code{calc-divide}) command divides two numbers.
+
+When combining multiplication and division in an algebraic formula, it
+is good style to use parentheses to distinguish between possible
+interpretations; the expression @samp{a/b*c} should be written
address@hidden(a/b)*c} or @samp{a/(b*c)}, as appropriate. Without the
+parentheses, Calc will interpret @samp{a/b*c} as @samp{a/(b*c)}, since
+in algebraic entry Calc gives division a lower precedence than
+multiplication. (This is not standard across all computer languages, and
+Calc may change the precedence depending on the language mode being used.
address@hidden Modes}.) This default ordering can be changed by setting
+the customizable variable @code{calc-multiplication-has-precedence} to
address@hidden (@pxref{Customizing Calc}); this will give multiplication and
+division equal precedences. Note that Calc's default choice of
+precedence allows @samp{a b / c d} to be used as a shortcut for
address@hidden
address@hidden
+a b
+---.
+c d
address@hidden group
address@hidden smallexample
+
+When dividing a scalar @expr{B} by a square matrix @expr{A}, the
+computation performed is @expr{B} times the inverse of @expr{A}. This
+also occurs if @expr{B} is itself a vector or matrix, in which case the
+effect is to solve the set of linear equations represented by @expr{B}.
+If @expr{B} is a matrix with the same number of rows as @expr{A}, or a
+plain vector (which is interpreted here as a column vector), then the
+equation @expr{A X = B} is solved for the vector or matrix @expr{X}.
+Otherwise, if @expr{B} is a non-square matrix with the same number of
address@hidden as @expr{A}, the equation @expr{X A = B} is solved. If
+you wish a vector @expr{B} to be interpreted as a row vector to be
+solved as @expr{X A = B}, make it into a one-row matrix with @kbd{C-u 1
+v p} first. To force a left-handed solution with a square matrix
address@hidden, transpose @expr{A} and @expr{B} before dividing, then
+transpose the result.
+
+HMS forms can be divided by real numbers or by other HMS forms. Error
+forms can be divided in any combination of ways. Modulo forms where both
+values and the modulo are integers can be divided to get an integer modulo
+form result. Intervals can be divided; dividing by an interval that
+encompasses zero or has zero as a limit will result in an infinite
+interval.
+
address@hidden ^
address@hidden calc-power
address@hidden
address@hidden @null
address@hidden ignore
address@hidden ^
+The @kbd{^} (@code{calc-power}) command raises a number to a power. If
+the power is an integer, an exact result is computed using repeated
+multiplications. For non-integer powers, Calc uses Newton's method or
+logarithms and exponentials. Square matrices can be raised to integer
+powers. If either argument is an error (or interval or modulo) form,
+the result is also an error (or interval or modulo) form.
+
address@hidden I ^
address@hidden nroot
+If you press the @kbd{I} (inverse) key first, the @kbd{I ^} command
+computes an Nth root: @kbd{125 @key{RET} 3 I ^} computes the number 5.
+(This is entirely equivalent to @kbd{125 @key{RET} 1:3 ^}.)
+
address@hidden \
address@hidden calc-idiv
address@hidden idiv
address@hidden
address@hidden @null
address@hidden ignore
address@hidden \
+The @kbd{\} (@code{calc-idiv}) command divides two numbers on the stack
+to produce an integer result. It is equivalent to dividing with
address@hidden/}, then rounding down with @kbd{F} (@code{calc-floor}), only a
bit
+more convenient and efficient. Also, since it is an all-integer
+operation when the arguments are integers, it avoids problems that
address@hidden/ F} would have with floating-point roundoff.
+
address@hidden %
address@hidden calc-mod
address@hidden
address@hidden @null
address@hidden ignore
address@hidden %
+The @kbd{%} (@code{calc-mod}) command performs a ``modulo'' (or ``remainder'')
+operation. Mathematically, @samp{a%b = a - (a\b)*b}, and is defined
+for all real numbers @expr{a} and @expr{b} (except @expr{b=0}). For
+positive @expr{b}, the result will always be between 0 (inclusive) and
address@hidden (exclusive). Modulo does not work for HMS forms and error forms.
+If @expr{a} is a modulo form, its modulo is changed to @expr{b}, which
+must be positive real number.
+
address@hidden :
address@hidden calc-fdiv
address@hidden fdiv
+The @kbd{:} (@code{calc-fdiv}) address@hidden command
+divides the two integers on the top of the stack to produce a fractional
+result. This is a convenient shorthand for enabling Fraction mode (with
address@hidden f}) temporarily and using @samp{/}. Note that during numeric
entry
+the @kbd{:} key is interpreted as a fraction separator, so to divide 8 by 6
+you would have to type @kbd{8 @key{RET} 6 @key{RET} :}. (Of course, in
+this case, it would be much easier simply to enter the fraction directly
+as @kbd{8:6 @key{RET}}!)
+
address@hidden n
address@hidden calc-change-sign
+The @kbd{n} (@code{calc-change-sign}) command negates the number on the top
+of the stack. It works on numbers, vectors and matrices, HMS forms, date
+forms, error forms, intervals, and modulo forms.
+
address@hidden A
address@hidden calc-abs
address@hidden abs
+The @kbd{A} (@code{calc-abs}) address@hidden command computes the absolute
+value of a number. The result of @code{abs} is always a nonnegative
+real number: With a complex argument, it computes the complex magnitude.
+With a vector or matrix argument, it computes the Frobenius norm, i.e.,
+the square root of the sum of the squares of the absolute values of the
+elements. The absolute value of an error form is defined by replacing
+the mean part with its absolute value and leaving the error part the same.
+The absolute value of a modulo form is undefined. The absolute value of
+an interval is defined in the obvious way.
+
address@hidden f A
address@hidden calc-abssqr
address@hidden abssqr
+The @kbd{f A} (@code{calc-abssqr}) address@hidden command computes the
+absolute value squared of a number, vector or matrix, or error form.
+
address@hidden f s
address@hidden calc-sign
address@hidden sign
+The @kbd{f s} (@code{calc-sign}) address@hidden command returns 1 if its
+argument is positive, @mathit{-1} if its argument is negative, or 0 if its
+argument is zero. In algebraic form, you can also write @samp{sign(a,x)}
+which evaluates to @samp{x * sign(a)}, i.e., either @samp{x}, @samp{-x}, or
+zero depending on the sign of @samp{a}.
+
address@hidden &
address@hidden calc-inv
address@hidden inv
address@hidden Reciprocal
+The @kbd{&} (@code{calc-inv}) address@hidden command computes the
+reciprocal of a number, i.e., @expr{1 / x}. Operating on a square
+matrix, it computes the inverse of that matrix.
+
address@hidden Q
address@hidden calc-sqrt
address@hidden sqrt
+The @kbd{Q} (@code{calc-sqrt}) address@hidden command computes the square
+root of a number. For a negative real argument, the result will be a
+complex number whose form is determined by the current Polar mode.
+
address@hidden f h
address@hidden calc-hypot
address@hidden hypot
+The @kbd{f h} (@code{calc-hypot}) address@hidden command computes the square
+root of the sum of the squares of two numbers. That is, @samp{hypot(a,b)}
+is the length of the hypotenuse of a right triangle with sides @expr{a}
+and @expr{b}. If the arguments are complex numbers, their squared
+magnitudes are used.
+
address@hidden f Q
address@hidden calc-isqrt
address@hidden isqrt
+The @kbd{f Q} (@code{calc-isqrt}) address@hidden command computes the
+integer square root of an integer. This is the true square root of the
+number, rounded down to an integer. For example, @samp{isqrt(10)}
+produces 3. Note that, like @kbd{\} address@hidden, this uses exact
+integer arithmetic throughout to avoid roundoff problems. If the input
+is a floating-point number or other non-integer value, this is exactly
+the same as @samp{floor(sqrt(x))}.
+
address@hidden f n
address@hidden f x
address@hidden calc-min
address@hidden min
address@hidden calc-max
address@hidden max
+The @kbd{f n} (@code{calc-min}) address@hidden and @kbd{f x} (@code{calc-max})
address@hidden commands take the minimum or maximum of two real numbers,
+respectively. These commands also work on HMS forms, date forms,
+intervals, and infinities. (In algebraic expressions, these functions
+take any number of arguments and return the maximum or minimum among
+all the arguments.)
+
address@hidden f M
address@hidden f X
address@hidden calc-mant-part
address@hidden mant
address@hidden calc-xpon-part
address@hidden xpon
+The @kbd{f M} (@code{calc-mant-part}) address@hidden function extracts
+the ``mantissa'' part @expr{m} of its floating-point argument; @kbd{f X}
+(@code{calc-xpon-part}) address@hidden extracts the ``exponent'' part
address@hidden The original number is equal to
address@hidden @math{m \times 10^e},
address@hidden @expr{m * 10^e},
+where @expr{m} is in the interval @samp{[1.0 ..@: 10.0)} except that
address@hidden if the original number is zero. For integers
+and fractions, @code{mant} returns the number unchanged and @code{xpon}
+returns zero. The @kbd{v u} (@code{calc-unpack}) command can also be
+used to ``unpack'' a floating-point number; this produces an integer
+mantissa and exponent, with the constraint that the mantissa is not
+a multiple of ten (again except for the @expr{m=e=0} case).
+
address@hidden f S
address@hidden calc-scale-float
address@hidden scf
+The @kbd{f S} (@code{calc-scale-float}) address@hidden function scales a number
+by a given power of ten. Thus, @samp{scf(mant(x), xpon(x)) = x} for any
+real @samp{x}. The second argument must be an integer, but the first
+may actually be any numeric value. For example, @samp{scf(5,-2) = 0.05}
+or @samp{1:20} depending on the current Fraction mode.
+
address@hidden f [
address@hidden f ]
address@hidden calc-decrement
address@hidden calc-increment
address@hidden decr
address@hidden incr
+The @kbd{f [} (@code{calc-decrement}) address@hidden and @kbd{f ]}
+(@code{calc-increment}) address@hidden functions decrease or increase
+a number by one unit. For integers, the effect is obvious. For
+floating-point numbers, the change is by one unit in the last place.
+For example, incrementing @samp{12.3456} when the current precision
+is 6 digits yields @samp{12.3457}. If the current precision had been
+8 digits, the result would have been @samp{12.345601}. Incrementing
address@hidden produces
address@hidden @math{10^{-p}},
address@hidden @expr{10^-p},
+where @expr{p} is the current
+precision. These operations are defined only on integers and floats.
+With numeric prefix arguments, they change the number by @expr{n} units.
+
+Note that incrementing followed by decrementing, or vice-versa, will
+almost but not quite always cancel out. Suppose the precision is
+6 digits and the number @samp{9.99999} is on the stack. Incrementing
+will produce @samp{10.0000}; decrementing will produce @samp{9.9999}.
+One digit has been dropped. This is an unavoidable consequence of the
+way floating-point numbers work.
+
+Incrementing a date/time form adjusts it by a certain number of seconds.
+Incrementing a pure date form adjusts it by a certain number of days.
+
address@hidden Integer Truncation, Complex Number Functions, Basic Arithmetic,
Arithmetic
address@hidden Integer Truncation
+
address@hidden
+There are four commands for truncating a real number to an integer,
+differing mainly in their treatment of negative numbers. All of these
+commands have the property that if the argument is an integer, the result
+is the same integer. An integer-valued floating-point argument is converted
+to integer form.
+
+If you press @kbd{H} (@code{calc-hyperbolic}) first, the result will be
+expressed as an integer-valued floating-point number.
+
address@hidden Integer part of a number
address@hidden F
address@hidden calc-floor
address@hidden floor
address@hidden ffloor
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H F
+The @kbd{F} (@code{calc-floor}) address@hidden or @code{ffloor}] command
+truncates a real number to the next lower integer, i.e., toward minus
+infinity. Thus @kbd{3.6 F} produces 3, but @kbd{_3.6 F} produces
address@hidden
+
address@hidden I F
address@hidden calc-ceiling
address@hidden ceil
address@hidden fceil
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H I F
+The @kbd{I F} (@code{calc-ceiling}) address@hidden or @code{fceil}]
+command truncates toward positive infinity. Thus @kbd{3.6 I F} produces
+4, and @kbd{_3.6 I F} produces @mathit{-3}.
+
address@hidden R
address@hidden calc-round
address@hidden round
address@hidden fround
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H R
+The @kbd{R} (@code{calc-round}) address@hidden or @code{fround}] command
+rounds to the nearest integer. When the fractional part is .5 exactly,
+this command rounds away from zero. (All other rounding in the
+Calculator uses this convention as well.) Thus @kbd{3.5 R} produces 4
+but @kbd{3.4 R} produces 3; @kbd{_3.5 R} produces @mathit{-4}.
+
address@hidden I R
address@hidden calc-trunc
address@hidden trunc
address@hidden ftrunc
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H I R
+The @kbd{I R} (@code{calc-trunc}) address@hidden or @code{ftrunc}]
+command truncates toward zero. In other words, it ``chops off''
+everything after the decimal point. Thus @kbd{3.6 I R} produces 3 and
address@hidden I R} produces @mathit{-3}.
+
+These functions may not be applied meaningfully to error forms, but they
+do work for intervals. As a convenience, applying @code{floor} to a
+modulo form floors the value part of the form. Applied to a vector,
+these functions operate on all elements of the vector one by one.
+Applied to a date form, they operate on the internal numerical
+representation of dates, converting a date/time form into a pure date.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden rounde
address@hidden
address@hidden
address@hidden ignore
address@hidden roundu
address@hidden
address@hidden
address@hidden ignore
address@hidden frounde
address@hidden
address@hidden
address@hidden ignore
address@hidden froundu
+There are two more rounding functions which can only be entered in
+algebraic notation. The @code{roundu} function is like @code{round}
+except that it rounds up, toward plus infinity, when the fractional
+part is .5. This distinction matters only for negative arguments.
+Also, @code{rounde} rounds to an even number in the case of a tie,
+rounding up or down as necessary. For example, @samp{rounde(3.5)} and
address@hidden(4.5)} both return 4, but @samp{rounde(5.5)} returns 6.
+The advantage of round-to-even is that the net error due to rounding
+after a long calculation tends to cancel out to zero. An important
+subtle point here is that the number being fed to @code{rounde} will
+already have been rounded to the current precision before @code{rounde}
+begins. For example, @samp{rounde(2.500001)} with a current precision
+of 6 will incorrectly, or at least surprisingly, yield 2 because the
+argument will first have been rounded down to @expr{2.5} (which
address@hidden sees as an exact tie between 2 and 3).
+
+Each of these functions, when written in algebraic formulas, allows
+a second argument which specifies the number of digits after the
+decimal point to keep. For example, @samp{round(123.4567, 2)} will
+produce the answer 123.46, and @samp{round(123.4567, -1)} will
+produce 120 (i.e., the cutoff is one digit to the @emph{left} of
+the decimal point). A second argument of zero is equivalent to
+no second argument at all.
+
address@hidden Fractional part of a number
+To compute the fractional part of a number (i.e., the amount which, when
+added to address@hidden(address@hidden@tfn{)}', will produce @var{n}) just
take @var{n}
+modulo 1 using the @code{%} command.
+
+Note also the @kbd{\} (integer quotient), @kbd{f I} (integer logarithm),
+and @kbd{f Q} (integer square root) commands, which are analogous to
address@hidden/}, @kbd{B}, and @kbd{Q}, respectively, except that they take
integer
+arguments and return the result rounded down to an integer.
+
address@hidden Complex Number Functions, Conversions, Integer Truncation,
Arithmetic
address@hidden Complex Number Functions
+
address@hidden
address@hidden J
address@hidden calc-conj
address@hidden conj
+The @kbd{J} (@code{calc-conj}) address@hidden command computes the
+complex conjugate of a number. For complex number @expr{a+bi}, the
+complex conjugate is @expr{a-bi}. If the argument is a real number,
+this command leaves it the same. If the argument is a vector or matrix,
+this command replaces each element by its complex conjugate.
+
address@hidden G
address@hidden calc-argument
address@hidden arg
+The @kbd{G} (@code{calc-argument}) address@hidden command computes the
+``argument'' or polar angle of a complex number. For a number in polar
+notation, this is simply the second component of the pair
address@hidden address@hidden(address@hidden@tfn{;address@hidden@tfn{)}'.
address@hidden address@hidden(address@hidden@tfn{;address@hidden@tfn{)}'.
+The result is expressed according to the current angular mode and will
+be in the range @mathit{-180} degrees (exclusive) to @mathit{+180} degrees
+(inclusive), or the equivalent range in radians.
+
address@hidden calc-imaginary
+The @code{calc-imaginary} command multiplies the number on the
+top of the stack by the imaginary number @expr{i = (0,1)}. This
+command is not normally bound to a key in Calc, but it is available
+on the @key{IMAG} button in Keypad mode.
+
address@hidden f r
address@hidden calc-re
address@hidden re
+The @kbd{f r} (@code{calc-re}) address@hidden command replaces a complex number
+by its real part. This command has no effect on real numbers. (As an
+added convenience, @code{re} applied to a modulo form extracts
+the value part.)
+
address@hidden f i
address@hidden calc-im
address@hidden im
+The @kbd{f i} (@code{calc-im}) address@hidden command replaces a complex number
+by its imaginary part; real numbers are converted to zero. With a vector
+or matrix argument, these functions operate element-wise.
+
address@hidden
address@hidden v p
address@hidden ignore
address@hidden v p (complex)
address@hidden calc-pack
+The @kbd{v p} (@code{calc-pack}) command can pack the top two numbers on
+the stack into a composite object such as a complex number. With
+a prefix argument of @mathit{-1}, it produces a rectangular complex number;
+with an argument of @mathit{-2}, it produces a polar complex number.
+(Also, @pxref{Building Vectors}.)
+
address@hidden
address@hidden v u
address@hidden ignore
address@hidden v u (complex)
address@hidden calc-unpack
+The @kbd{v u} (@code{calc-unpack}) command takes the complex number
+(or other composite object) on the top of the stack and unpacks it
+into its separate components.
+
address@hidden Conversions, Date Arithmetic, Complex Number Functions,
Arithmetic
address@hidden Conversions
+
address@hidden
+The commands described in this section convert numbers from one form
+to another; they are two-key sequences beginning with the letter @kbd{c}.
+
address@hidden c f
address@hidden calc-float
address@hidden pfloat
+The @kbd{c f} (@code{calc-float}) address@hidden command converts the
+number on the top of the stack to floating-point form. For example,
address@hidden is converted to @expr{23.0}, @expr{3:2} is converted to
address@hidden, and @expr{2.3} is left the same. If the value is a composite
+object such as a complex number or vector, each of the components is
+converted to floating-point. If the value is a formula, all numbers
+in the formula are converted to floating-point. Note that depending
+on the current floating-point precision, conversion to floating-point
+format may lose information.
+
+As a special exception, integers which appear as powers or subscripts
+are not floated by @kbd{c f}. If you really want to float a power,
+you can use a @kbd{j s} command to select the power followed by @kbd{c f}.
+Because @kbd{c f} cannot examine the formula outside of the selection,
+it does not notice that the thing being floated is a power.
address@hidden Subformulas}.
+
+The normal @kbd{c f} command is ``pervasive'' in the sense that it
+applies to all numbers throughout the formula. The @code{pfloat}
+algebraic function never stays around in a formula; @samp{pfloat(a + 1)}
+changes to @samp{a + 1.0} as soon as it is evaluated.
+
address@hidden H c f
address@hidden float
+With the Hyperbolic flag, @kbd{H c f} address@hidden operates
+only on the number or vector of numbers at the top level of its
+argument. Thus, @samp{float(1)} is 1.0, but @samp{float(a + 1)}
+is left unevaluated because its argument is not a number.
+
+You should use @kbd{H c f} if you wish to guarantee that the final
+value, once all the variables have been assigned, is a float; you
+would use @kbd{c f} if you wish to do the conversion on the numbers
+that appear right now.
+
address@hidden c F
address@hidden calc-fraction
address@hidden pfrac
+The @kbd{c F} (@code{calc-fraction}) address@hidden command converts a
+floating-point number into a fractional approximation. By default, it
+produces a fraction whose decimal representation is the same as the
+input number, to within the current precision. You can also give a
+numeric prefix argument to specify a tolerance, either directly, or,
+if the prefix argument is zero, by using the number on top of the stack
+as the tolerance. If the tolerance is a positive integer, the fraction
+is correct to within that many significant figures. If the tolerance is
+a non-positive integer, it specifies how many digits fewer than the current
+precision to use. If the tolerance is a floating-point number, the
+fraction is correct to within that absolute amount.
+
address@hidden H c F
address@hidden frac
+The @code{pfrac} function is pervasive, like @code{pfloat}.
+There is also a non-pervasive version, @kbd{H c F} address@hidden,
+which is analogous to @kbd{H c f} discussed above.
+
address@hidden c d
address@hidden calc-to-degrees
address@hidden deg
+The @kbd{c d} (@code{calc-to-degrees}) address@hidden command converts a
+number into degrees form. The value on the top of the stack may be an
+HMS form (interpreted as degrees-minutes-seconds), or a real number which
+will be interpreted in radians regardless of the current angular mode.
+
address@hidden c r
address@hidden calc-to-radians
address@hidden rad
+The @kbd{c r} (@code{calc-to-radians}) address@hidden command converts an
+HMS form or angle in degrees into an angle in radians.
+
address@hidden c h
address@hidden calc-to-hms
address@hidden hms
+The @kbd{c h} (@code{calc-to-hms}) address@hidden command converts a real
+number, interpreted according to the current angular mode, to an HMS
+form describing the same angle. In algebraic notation, the @code{hms}
+function also accepts three arguments: @samp{hms(@var{h}, @var{m}, @var{s})}.
+(The three-argument version is independent of the current angular mode.)
+
address@hidden calc-from-hms
+The @code{calc-from-hms} command converts the HMS form on the top of the
+stack into a real number according to the current angular mode.
+
address@hidden c p
address@hidden I c p
address@hidden calc-polar
address@hidden polar
address@hidden rect
+The @kbd{c p} (@code{calc-polar}) command converts the complex number on
+the top of the stack from polar to rectangular form, or from rectangular
+to polar form, whichever is appropriate. Real numbers are left the same.
+This command is equivalent to the @code{rect} or @code{polar}
+functions in algebraic formulas, depending on the direction of
+conversion. (It uses @code{polar}, except that if the argument is
+already a polar complex number, it uses @code{rect} instead. The
address@hidden c p} command always uses @code{rect}.)
+
address@hidden c c
address@hidden calc-clean
address@hidden pclean
+The @kbd{c c} (@code{calc-clean}) address@hidden command ``cleans'' the
+number on the top of the stack. Floating point numbers are re-rounded
+according to the current precision. Polar numbers whose angular
+components have strayed from the @mathit{-180} to @mathit{+180} degree range
+are normalized. (Note that results will be undesirable if the current
+angular mode is different from the one under which the number was
+produced!) Integers and fractions are generally unaffected by this
+operation. Vectors and formulas are cleaned by cleaning each component
+number (i.e., pervasively).
+
+If the simplification mode is set below the default level, it is raised
+to the default level for the purposes of this command. Thus, @kbd{c c}
+applies the default simplifications even if their automatic application
+is disabled. @xref{Simplification Modes}.
+
address@hidden Roundoff errors, correcting
+A numeric prefix argument to @kbd{c c} sets the floating-point precision
+to that value for the duration of the command. A positive prefix (of at
+least 3) sets the precision to the specified value; a negative or zero
+prefix decreases the precision by the specified amount.
+
address@hidden c 0-9
address@hidden calc-clean-num
+The keystroke sequences @kbd{c 0} through @kbd{c 9} are equivalent
+to @kbd{c c} with the corresponding negative prefix argument. If roundoff
+errors have changed 2.0 into 1.999999, typing @kbd{c 1} to clip off one
+decimal place often conveniently does the trick.
+
+The @kbd{c c} command with a numeric prefix argument, and the @kbd{c 0}
+through @kbd{c 9} commands, also ``clip'' very small floating-point
+numbers to zero. If the exponent is less than or equal to the negative
+of the specified precision, the number is changed to 0.0. For example,
+if the current precision is 12, then @kbd{c 2} changes the vector
address@hidden, 1e-9, 1e-10, 1e-11]} to @samp{[1e-8, 1e-9, 0, 0]}.
+Numbers this small generally arise from roundoff noise.
+
+If the numbers you are using really are legitimately this small,
+you should avoid using the @kbd{c 0} through @kbd{c 9} commands.
+(The plain @kbd{c c} command rounds to the current precision but
+does not clip small numbers.)
+
+One more property of @kbd{c 0} through @kbd{c 9}, and of @kbd{c c} with
+a prefix argument, is that integer-valued floats are converted to
+plain integers, so that @kbd{c 1} on @samp{[1., 1.5, 2., 2.5, 3.]}
+produces @samp{[1, 1.5, 2, 2.5, 3]}. This is not done for huge
+numbers (@samp{1e100} is technically an integer-valued float, but
+you wouldn't want it automatically converted to a 100-digit integer).
+
address@hidden H c 0-9
address@hidden H c c
address@hidden clean
+With the Hyperbolic flag, @kbd{H c c} and @kbd{H c 0} through @kbd{H c 9}
+operate non-pervasively address@hidden
+
address@hidden Date Arithmetic, Financial Functions, Conversions, Arithmetic
address@hidden Date Arithmetic
+
address@hidden
address@hidden Date arithmetic, additional functions
+The commands described in this section perform various conversions
+and calculations involving date forms (@pxref{Date Forms}). They
+use the @kbd{t} (for time/date) prefix key followed by shifted
+letters.
+
+The simplest date arithmetic is done using the regular @kbd{+} and @kbd{-}
+commands. In particular, adding a number to a date form advances the
+date form by a certain number of days; adding an HMS form to a date
+form advances the date by a certain amount of time; and subtracting two
+date forms produces a difference measured in days. The commands
+described here provide additional, more specialized operations on dates.
+
+Many of these commands accept a numeric prefix argument; if you give
+plain @kbd{C-u} as the prefix, these commands will instead take the
+additional argument from the top of the stack.
+
address@hidden
+* Date Conversions::
+* Date Functions::
+* Time Zones::
+* Business Days::
address@hidden menu
+
address@hidden Date Conversions, Date Functions, Date Arithmetic, Date
Arithmetic
address@hidden Date Conversions
+
address@hidden
address@hidden t D
address@hidden calc-date
address@hidden date
+The @kbd{t D} (@code{calc-date}) address@hidden command converts a
+date form into a number, measured in days since Jan 1, 1 AD. The
+result will be an integer if @var{date} is a pure date form, or a
+fraction or float if @var{date} is a date/time form. Or, if its
+argument is a number, it converts this number into a date form.
+
+With a numeric prefix argument, @kbd{t D} takes that many objects
+(up to six) from the top of the stack and interprets them in one
+of the following ways:
+
+The @samp{date(@var{year}, @var{month}, @var{day})} function
+builds a pure date form out of the specified year, month, and
+day, which must all be integers. @var{Year} is a year number,
+such as 1991 (@emph{not} the same as 91!). @var{Month} must be
+an integer in the range 1 to 12; @var{day} must be in the range
+1 to 31. If the specified month has fewer than 31 days and
address@hidden is too large, the equivalent day in the following
+month will be used.
+
+The @samp{date(@var{month}, @var{day})} function builds a
+pure date form using the current year, as determined by the
+real-time clock.
+
+The @samp{date(@var{year}, @var{month}, @var{day}, @var{hms})}
+function builds a date/time form using an @var{hms} form.
+
+The @samp{date(@var{year}, @var{month}, @var{day}, @var{hour},
address@hidden, @var{second})} function builds a date/time form.
address@hidden should be an integer in the range 0 to 23;
address@hidden should be an integer in the range 0 to 59;
address@hidden should be any real number in the range @samp{[0 .. 60)}.
+The last two arguments default to zero if omitted.
+
address@hidden t J
address@hidden calc-julian
address@hidden julian
address@hidden Julian day counts, conversions
+The @kbd{t J} (@code{calc-julian}) address@hidden command converts
+a date form into a Julian day count, which is the number of days
+since noon (GMT) on Jan 1, 4713 BC. A pure date is converted to an
+integer Julian count representing noon of that day. A date/time form
+is converted to an exact floating-point Julian count, adjusted to
+interpret the date form in the current time zone but the Julian
+day count in Greenwich Mean Time. A numeric prefix argument allows
+you to specify the time zone; @pxref{Time Zones}. Use a prefix of
+zero to suppress the time zone adjustment. Note that pure date forms
+are never time-zone adjusted.
+
+This command can also do the opposite conversion, from a Julian day
+count (either an integer day, or a floating-point day and time in
+the GMT zone), into a pure date form or a date/time form in the
+current or specified time zone.
+
address@hidden t U
address@hidden calc-unix-time
address@hidden unixtime
address@hidden Unix time format, conversions
+The @kbd{t U} (@code{calc-unix-time}) address@hidden command
+converts a date form into a Unix time value, which is the number of
+seconds since midnight on Jan 1, 1970, or vice-versa. The numeric result
+will be an integer if the current precision is 12 or less; for higher
+precisions, the result may be a float with (@address@hidden)
+digits after the decimal. Just as for @kbd{t J}, the numeric time
+is interpreted in the GMT time zone and the date form is interpreted
+in the current or specified zone. Some systems use Unix-like
+numbering but with the local time zone; give a prefix of zero to
+suppress the adjustment if so.
+
address@hidden t C
address@hidden calc-convert-time-zones
address@hidden tzconv
address@hidden Time Zones, converting between
+The @kbd{t C} (@code{calc-convert-time-zones}) address@hidden
+command converts a date form from one time zone to another. You
+are prompted for each time zone name in turn; you can answer with
+any suitable Calc time zone expression (@pxref{Time Zones}).
+If you answer either prompt with a blank line, the local time
+zone is used for that prompt. You can also answer the first
+prompt with @kbd{$} to take the two time zone names from the
+stack (and the date to be converted from the third stack level).
+
address@hidden Date Functions, Business Days, Date Conversions, Date Arithmetic
address@hidden Date Functions
+
address@hidden
address@hidden t N
address@hidden calc-now
address@hidden now
+The @kbd{t N} (@code{calc-now}) address@hidden command pushes the
+current date and time on the stack as a date form. The time is
+reported in terms of the specified time zone; with no numeric prefix
+argument, @kbd{t N} reports for the current time zone.
+
address@hidden t P
address@hidden calc-date-part
+The @kbd{t P} (@code{calc-date-part}) command extracts one part
+of a date form. The prefix argument specifies the part; with no
+argument, this command prompts for a part code from 1 to 9.
+The various part codes are described in the following paragraphs.
+
address@hidden year
+The @kbd{M-1 t P} address@hidden function extracts the year number
+from a date form as an integer, e.g., 1991. This and the
+following functions will also accept a real number for an
+argument, which is interpreted as a standard Calc day number.
+Note that this function will never return zero, since the year
+1 BC immediately precedes the year 1 AD.
+
address@hidden month
+The @kbd{M-2 t P} address@hidden function extracts the month number
+from a date form as an integer in the range 1 to 12.
+
address@hidden day
+The @kbd{M-3 t P} address@hidden function extracts the day number
+from a date form as an integer in the range 1 to 31.
+
address@hidden hour
+The @kbd{M-4 t P} address@hidden function extracts the hour from
+a date form as an integer in the range 0 (midnight) to 23. Note
+that 24-hour time is always used. This returns zero for a pure
+date form. This function (and the following two) also accept
+HMS forms as input.
+
address@hidden minute
+The @kbd{M-5 t P} address@hidden function extracts the minute
+from a date form as an integer in the range 0 to 59.
+
address@hidden second
+The @kbd{M-6 t P} address@hidden function extracts the second
+from a date form. If the current precision is 12 or less,
+the result is an integer in the range 0 to 59. For higher
+precisions, the result may instead be a floating-point number.
+
address@hidden weekday
+The @kbd{M-7 t P} address@hidden function extracts the weekday
+number from a date form as an integer in the range 0 (Sunday)
+to 6 (Saturday).
+
address@hidden yearday
+The @kbd{M-8 t P} address@hidden function extracts the day-of-year
+number from a date form as an integer in the range 1 (January 1)
+to 366 (December 31 of a leap year).
+
address@hidden time
+The @kbd{M-9 t P} address@hidden function extracts the time portion
+of a date form as an HMS form. This returns @samp{0@@ 0' 0"}
+for a pure date form.
+
address@hidden t M
address@hidden calc-new-month
address@hidden newmonth
+The @kbd{t M} (@code{calc-new-month}) address@hidden command
+computes a new date form that represents the first day of the month
+specified by the input date. The result is always a pure date
+form; only the year and month numbers of the input are retained.
+With a numeric prefix argument @var{n} in the range from 1 to 31,
address@hidden M} computes the @var{n}th day of the month. (If @var{n}
+is greater than the actual number of days in the month, or if
address@hidden is zero, the last day of the month is used.)
+
address@hidden t Y
address@hidden calc-new-year
address@hidden newyear
+The @kbd{t Y} (@code{calc-new-year}) address@hidden command
+computes a new pure date form that represents the first day of
+the year specified by the input. The month, day, and time
+of the input date form are lost. With a numeric prefix argument
address@hidden in the range from 1 to 366, @kbd{t Y} computes the
address@hidden day of the year (366 is treated as 365 in non-leap
+years). A prefix argument of 0 computes the last day of the
+year (December 31). A negative prefix argument from @mathit{-1} to
address@hidden computes the first day of the @var{n}th month of the year.
+
address@hidden t W
address@hidden calc-new-week
address@hidden newweek
+The @kbd{t W} (@code{calc-new-week}) address@hidden command
+computes a new pure date form that represents the Sunday on or before
+the input date. With a numeric prefix argument, it can be made to
+use any day of the week as the starting day; the argument must be in
+the range from 0 (Sunday) to 6 (Saturday). This function always
+subtracts between 0 and 6 days from the input date.
+
+Here's an example use of @code{newweek}: Find the date of the next
+Wednesday after a given date. Using @kbd{M-3 t W} or @samp{newweek(d, 3)}
+will give you the @emph{preceding} Wednesday, so @samp{newweek(d+7, 3)}
+will give you the following Wednesday. A further look at the definition
+of @code{newweek} shows that if the input date is itself a Wednesday,
+this formula will return the Wednesday one week in the future. An
+exercise for the reader is to modify this formula to yield the same day
+if the input is already a Wednesday. Another interesting exercise is
+to preserve the time-of-day portion of the input (@code{newweek} resets
+the time to midnight; hint:@: how can @code{newweek} be defined in terms
+of the @code{weekday} function?).
+
address@hidden
address@hidden
address@hidden ignore
address@hidden pwday
+The @samp{pwday(@var{date})} function (not on any key) computes the
+day-of-month number of the Sunday on or before @var{date}. With
+two arguments, @samp{pwday(@var{date}, @var{day})} computes the day
+number of the Sunday on or before day number @var{day} of the month
+specified by @var{date}. The @var{day} must be in the range from
+7 to 31; if the day number is greater than the actual number of days
+in the month, the true number of days is used instead. Thus
address@hidden(@var{date}, 7)} finds the first Sunday of the month, and
address@hidden(@var{date}, 31)} finds the last Sunday of the month.
+With a third @var{weekday} argument, @code{pwday} can be made to look
+for any day of the week instead of Sunday.
+
address@hidden t I
address@hidden calc-inc-month
address@hidden incmonth
+The @kbd{t I} (@code{calc-inc-month}) address@hidden command
+increases a date form by one month, or by an arbitrary number of
+months specified by a numeric prefix argument. The time portion,
+if any, of the date form stays the same. The day also stays the
+same, except that if the new month has fewer days the day
+number may be reduced to lie in the valid range. For example,
address@hidden(<Jan 31, 1991>)} produces @samp{<Feb 28, 1991>}.
+Because of this, @kbd{t I t I} and @kbd{M-2 t I} do not always give
+the same results (@samp{<Mar 28, 1991>} versus @samp{<Mar 31, 1991>}
+in this case).
+
address@hidden
address@hidden
address@hidden ignore
address@hidden incyear
+The @samp{incyear(@var{date}, @var{step})} function increases
+a date form by the specified number of years, which may be
+any positive or negative integer. Note that @samp{incyear(d, n)}
+is equivalent to @address@hidden(d, 12*n)}}, but these do not have
+simple equivalents in terms of day arithmetic because
+months and years have varying lengths. If the @var{step}
+argument is omitted, 1 year is assumed. There is no keyboard
+command for this function; use @kbd{C-u 12 t I} instead.
+
+There is no @code{newday} function at all because @kbd{F} address@hidden
+serves this purpose. Similarly, instead of @code{incday} and
address@hidden simply use @expr{d + n} or @expr{d + 7 n}.
+
address@hidden Arithmetic}, for the @kbd{f ]} address@hidden command
+which can adjust a date/time form by a certain number of seconds.
+
address@hidden Business Days, Time Zones, Date Functions, Date Arithmetic
address@hidden Business Days
+
address@hidden
+Often time is measured in ``business days'' or ``working days,''
+where weekends and holidays are skipped. Calc's normal date
+arithmetic functions use calendar days, so that subtracting two
+consecutive Mondays will yield a difference of 7 days. By contrast,
+subtracting two consecutive Mondays would yield 5 business days
+(assuming two-day weekends and the absence of holidays).
+
address@hidden t +
address@hidden t -
address@hidden badd
address@hidden bsub
address@hidden calc-business-days-plus
address@hidden calc-business-days-minus
+The @kbd{t +} (@code{calc-business-days-plus}) address@hidden
+and @kbd{t -} (@code{calc-business-days-minus}) address@hidden
+commands perform arithmetic using business days. For @kbd{t +},
+one argument must be a date form and the other must be a real
+number (positive or negative). If the number is not an integer,
+then a certain amount of time is added as well as a number of
+days; for example, adding 0.5 business days to a time in Friday
+evening will produce a time in Monday morning. It is also
+possible to add an HMS form; adding @samp{12@@ 0' 0"} also adds
+half a business day. For @kbd{t -}, the arguments are either a
+date form and a number or HMS form, or two date forms, in which
+case the result is the number of business days between the two
+dates.
+
address@hidden @code{Holidays} variable
address@hidden Holidays
+By default, Calc considers any day that is not a Saturday or
+Sunday to be a business day. You can define any number of
+additional holidays by editing the variable @code{Holidays}.
+(There is an @address@hidden H}} convenience command for editing this
+variable.) Initially, @code{Holidays} contains the vector
address@hidden, sun]}. Entries in the @code{Holidays} vector may
+be any of the following kinds of objects:
+
address@hidden @bullet
address@hidden
+Date forms (pure dates, not date/time forms). These specify
+particular days which are to be treated as holidays.
+
address@hidden
+Intervals of date forms. These specify a range of days, all of
+which are holidays (e.g., Christmas week). @xref{Interval Forms}.
+
address@hidden
+Nested vectors of date forms. Each date form in the vector is
+considered to be a holiday.
+
address@hidden
+Any Calc formula which evaluates to one of the above three things.
+If the formula involves the variable @expr{y}, it stands for a
+yearly repeating holiday; @expr{y} will take on various year
+numbers like 1992. For example, @samp{date(y, 12, 25)} specifies
+Christmas day, and @samp{newweek(date(y, 11, 7), 4) + 21} specifies
+Thanksgiving (which is held on the fourth Thursday of November).
+If the formula involves the variable @expr{m}, that variable
+takes on month numbers from 1 to 12: @samp{date(y, m, 15)} is
+a holiday that takes place on the 15th of every month.
+
address@hidden
+A weekday name, such as @code{sat} or @code{sun}. This is really
+a variable whose name is a three-letter, lower-case day name.
+
address@hidden
+An interval of year numbers (integers). This specifies the span of
+years over which this holiday list is to be considered valid. Any
+business-day arithmetic that goes outside this range will result
+in an error message. Use this if you are including an explicit
+list of holidays, rather than a formula to generate them, and you
+want to make sure you don't accidentally go beyond the last point
+where the holidays you entered are complete. If there is no
+limiting interval in the @code{Holidays} vector, the default
address@hidden .. 2737]} is used. (This is the absolute range of years
+for which Calc's business-day algorithms will operate.)
+
address@hidden
+An interval of HMS forms. This specifies the span of hours that
+are to be considered one business day. For example, if this
+range is @samp{[9@@ 0' 0" .. 17@@ 0' 0"]} (i.e., 9am to 5pm), then
+the business day is only eight hours long, so that @kbd{1.5 t +}
+on @samp{<4:00pm Fri Dec 13, 1991>} will add one business day and
+four business hours to produce @samp{<12:00pm Tue Dec 17, 1991>}.
+Likewise, @kbd{t -} will now express differences in time as
+fractions of an eight-hour day. Times before 9am will be treated
+as 9am by business date arithmetic, and times at or after 5pm will
+be treated as 4:59:59pm. If there is no HMS interval in @code{Holidays},
+the full 24-hour day @samp{[0@ 0' 0" .. 24@ 0' 0"]} is assumed.
+(Regardless of the type of bounds you specify, the interval is
+treated as inclusive on the low end and exclusive on the high end,
+so that the work day goes from 9am up to, but not including, 5pm.)
address@hidden itemize
+
+If the @code{Holidays} vector is empty, then @kbd{t +} and
address@hidden -} will act just like @kbd{+} and @kbd{-} because there will
+then be no difference between business days and calendar days.
+
+Calc expands the intervals and formulas you give into a complete
+list of holidays for internal use. This is done mainly to make
+sure it can detect multiple holidays. (For example,
address@hidden<Jan 1, 1989>} is both New Year's Day and a Sunday, but
+Calc's algorithms take care to count it only once when figuring
+the number of holidays between two dates.)
+
+Since the complete list of holidays for all the years from 1 to
+2737 would be huge, Calc actually computes only the part of the
+list between the smallest and largest years that have been involved
+in business-day calculations so far. Normally, you won't have to
+worry about this. Keep in mind, however, that if you do one
+calculation for 1992, and another for 1792, even if both involve
+only a small range of years, Calc will still work out all the
+holidays that fall in that 200-year span.
+
+If you add a (positive) number of days to a date form that falls on a
+weekend or holiday, the date form is treated as if it were the most
+recent business day. (Thus adding one business day to a Friday,
+Saturday, or Sunday will all yield the following Monday.) If you
+subtract a number of days from a weekend or holiday, the date is
+effectively on the following business day. (So subtracting one business
+day from Saturday, Sunday, or Monday yields the preceding Friday.) The
+difference between two dates one or both of which fall on holidays
+equals the number of actual business days between them. These
+conventions are consistent in the sense that, if you add @var{n}
+business days to any date, the difference between the result and the
+original date will come out to @var{n} business days. (It can't be
+completely consistent though; a subtraction followed by an addition
+might come out a bit differently, since @kbd{t +} is incapable of
+producing a date that falls on a weekend or holiday.)
+
address@hidden
address@hidden
address@hidden ignore
address@hidden holiday
+There is a @code{holiday} function, not on any keys, that takes
+any date form and returns 1 if that date falls on a weekend or
+holiday, as defined in @code{Holidays}, or 0 if the date is a
+business day.
+
address@hidden Time Zones, , Business Days, Date Arithmetic
address@hidden Time Zones
+
address@hidden
address@hidden Time zones
address@hidden Daylight saving time
+Time zones and daylight saving time are a complicated business.
+The conversions to and from Julian and Unix-style dates automatically
+compute the correct time zone and daylight saving adjustment to use,
+provided they can figure out this information. This section describes
+Calc's time zone adjustment algorithm in detail, in case you want to
+do conversions in different time zones or in case Calc's algorithms
+can't determine the right correction to use.
+
+Adjustments for time zones and daylight saving time are done by
address@hidden U}, @kbd{t J}, @kbd{t N}, and @kbd{t C}, but not by any other
+commands. In particular, @samp{<may 1 1991> - <apr 1 1991>} evaluates
+to exactly 30 days even though there is a daylight-saving
+transition in between. This is also true for Julian pure dates:
address@hidden(<may 1 1991>) - julian(<apr 1 1991>)}. But Julian
+and Unix date/times will adjust for daylight saving time: using Calc's
+default daylight saving time rule (see the explanation below),
address@hidden(<12am may 1 1991>) - julian(<12am apr 1 1991>)}
+evaluates to @samp{29.95833} (that's 29 days and 23 hours)
+because one hour was lost when daylight saving commenced on
+April 7, 1991.
+
+In brief, the idiom @samp{julian(@var{date1}) - julian(@var{date2})}
+computes the actual number of 24-hour periods between two dates, whereas
address@hidden@var{date1} - @var{date2}} computes the number of calendar
+days between two dates without taking daylight saving into account.
+
address@hidden calc-time-zone
address@hidden
address@hidden
address@hidden ignore
address@hidden tzone
+The @code{calc-time-zone} address@hidden command converts the time
+zone specified by its numeric prefix argument into a number of
+seconds difference from Greenwich mean time (GMT). If the argument
+is a number, the result is simply that value multiplied by 3600.
+Typical arguments for North America are 5 (Eastern) or 8 (Pacific). If
+Daylight Saving time is in effect, one hour should be subtracted from
+the normal difference.
+
+If you give a prefix of plain @kbd{C-u}, @code{calc-time-zone} (like other
+date arithmetic commands that include a time zone argument) takes the
+zone argument from the top of the stack. (In the case of @kbd{t J}
+and @kbd{t U}, the normal argument is then taken from the second-to-top
+stack position.) This allows you to give a non-integer time zone
+adjustment. The time-zone argument can also be an HMS form, or
+it can be a variable which is a time zone name in upper- or lower-case.
+For example @samp{tzone(PST) = tzone(8)} and @samp{tzone(pdt) = tzone(7)}
+(for Pacific standard and daylight saving times, respectively).
+
+North American and European time zone names are defined as follows;
+note that for each time zone there is one name for standard time,
+another for daylight saving time, and a third for ``generalized'' time
+in which the daylight saving adjustment is computed from context.
+
address@hidden
address@hidden
+YST PST MST CST EST AST NST GMT WET MET MEZ
+ 9 8 7 6 5 4 3.5 0 -1 -2 -2
+
+YDT PDT MDT CDT EDT ADT NDT BST WETDST METDST MESZ
+ 8 7 6 5 4 3 2.5 -1 -2 -3 -3
+
+YGT PGT MGT CGT EGT AGT NGT BGT WEGT MEGT MEGZ
+9/8 8/7 7/6 6/5 5/4 4/3 3.5/2.5 0/-1 -1/-2 -2/-3 -2/-3
address@hidden group
address@hidden smallexample
+
address@hidden math-tzone-names
+To define time zone names that do not appear in the above table,
+you must modify the Lisp variable @code{math-tzone-names}. This
+is a list of lists describing the different time zone names; its
+structure is best explained by an example. The three entries for
+Pacific Time look like this:
+
address@hidden
address@hidden
+( ( "PST" 8 0 ) ; Name as an upper-case string, then standard
+ ( "PDT" 8 -1 ) ; adjustment, then daylight saving adjustment.
+ ( "PGT" 8 "PST" "PDT" ) ) ; Generalized time zone.
address@hidden group
address@hidden smallexample
+
address@hidden @code{TimeZone} variable
address@hidden TimeZone
+With no arguments, @code{calc-time-zone} or @samp{tzone()} will by
+default get the time zone and daylight saving information from the
+calendar (@pxref{Daylight Saving,Calendar/Diary,The Calendar and the Diary,
+emacs,The GNU Emacs Manual}). To use a different time zone, or if the
+calendar does not give the desired result, you can set the Calc variable
address@hidden (which is by default @code{nil}) to an appropriate
+time zone name. (The easiest way to do this is to edit the
address@hidden variable using Calc's @kbd{s T} command, then use the
address@hidden p} (@code{calc-permanent-variable}) command to save the value of
address@hidden permanently.)
+If the time zone given by @code{TimeZone} is a generalized time zone,
+e.g., @code{EGT}, Calc examines the date being converted to tell whether
+to use standard or daylight saving time. But if the current time zone
+is explicit, e.g., @code{EST} or @code{EDT}, then that adjustment is
+used exactly and Calc's daylight saving algorithm is not consulted.
+The special time zone name @code{local}
+is equivalent to no argument; i.e., it uses the information obtained
+from the calendar.
+
+The @kbd{t J} and @code{t U} commands with no numeric prefix
+arguments do the same thing as @samp{tzone()}; namely, use the
+information from the calendar if @code{TimeZone} is @code{nil},
+otherwise use the time zone given by @code{TimeZone}.
+
address@hidden math-daylight-savings-hook
address@hidden math-std-daylight-savings
+When Calc computes the daylight saving information itself (i.e., when
+the @code{TimeZone} variable is set), it will by default consider
+daylight saving time to begin at 2 a.m.@: on the second Sunday of March
+(for years from 2007 on) or on the last Sunday in April (for years
+before 2007), and to end at 2 a.m.@: on the first Sunday of
+November. (for years from 2007 on) or the last Sunday in October (for
+years before 2007). These are the rules that have been in effect in
+much of North America since 1966 and take into account the rule change
+that began in 2007. If you are in a country that uses different rules
+for computing daylight saving time, you have two choices: Write your own
+daylight saving hook, or control time zones explicitly by setting the
address@hidden variable and/or always giving a time-zone argument for
+the conversion functions.
+
+The Lisp variable @code{math-daylight-savings-hook} holds the
+name of a function that is used to compute the daylight saving
+adjustment for a given date. The default is
address@hidden, which computes an adjustment
+(either 0 or @mathit{-1}) using the North American rules given above.
+
+The daylight saving hook function is called with four arguments:
+The date, as a floating-point number in standard Calc format;
+a six-element list of the date decomposed into year, month, day,
+hour, minute, and second, respectively; a string which contains
+the generalized time zone name in upper-case, e.g., @code{"WEGT"};
+and a special adjustment to be applied to the hour value when
+converting into a generalized time zone (see below).
+
address@hidden math-prev-weekday-in-month
+The Lisp function @code{math-prev-weekday-in-month} is useful for
+daylight saving computations. This is an internal version of
+the user-level @code{pwday} function described in the previous
+section. It takes four arguments: The floating-point date value,
+the corresponding six-element date list, the day-of-month number,
+and the weekday number (0-6).
+
+The default daylight saving hook ignores the time zone name, but a
+more sophisticated hook could use different algorithms for different
+time zones. It would also be possible to use different algorithms
+depending on the year number, but the default hook always uses the
+algorithm for 1987 and later. Here is a listing of the default
+daylight saving hook:
+
address@hidden
+(defun math-std-daylight-savings (date dt zone bump)
+ (cond ((< (nth 1 dt) 4) 0)
+ ((= (nth 1 dt) 4)
+ (let ((sunday (math-prev-weekday-in-month date dt 7 0)))
+ (cond ((< (nth 2 dt) sunday) 0)
+ ((= (nth 2 dt) sunday)
+ (if (>= (nth 3 dt) (+ 3 bump)) -1 0))
+ (t -1))))
+ ((< (nth 1 dt) 10) -1)
+ ((= (nth 1 dt) 10)
+ (let ((sunday (math-prev-weekday-in-month date dt 31 0)))
+ (cond ((< (nth 2 dt) sunday) -1)
+ ((= (nth 2 dt) sunday)
+ (if (>= (nth 3 dt) (+ 2 bump)) 0 -1))
+ (t 0))))
+ (t 0))
+)
address@hidden smallexample
+
address@hidden
+The @code{bump} parameter is equal to zero when Calc is converting
+from a date form in a generalized time zone into a GMT date value.
+It is @mathit{-1} when Calc is converting in the other direction. The
+adjustments shown above ensure that the conversion behaves correctly
+and reasonably around the 2 a.m.@: transition in each direction.
+
+There is a ``missing'' hour between 2 a.m.@: and 3 a.m.@: at the
+beginning of daylight saving time; converting a date/time form that
+falls in this hour results in a time value for the following hour,
+from 3 a.m.@: to 4 a.m. At the end of daylight saving time, the
+hour from 1 a.m.@: to 2 a.m.@: repeats itself; converting a date/time
+form that falls in this hour results in a time value for the first
+manifestation of that time (@emph{not} the one that occurs one hour
+later).
+
+If @code{math-daylight-savings-hook} is @code{nil}, then the
+daylight saving adjustment is always taken to be zero.
+
+In algebraic formulas, @samp{tzone(@var{zone}, @var{date})}
+computes the time zone adjustment for a given zone name at a
+given date. The @var{date} is ignored unless @var{zone} is a
+generalized time zone. If @var{date} is a date form, the
+daylight saving computation is applied to it as it appears.
+If @var{date} is a numeric date value, it is adjusted for the
+daylight-saving version of @var{zone} before being given to
+the daylight saving hook. This odd-sounding rule ensures
+that the daylight-saving computation is always done in
+local time, not in the GMT time that a numeric @var{date}
+is typically represented in.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden dsadj
+The @samp{dsadj(@var{date}, @var{zone})} function computes the
+daylight saving adjustment that is appropriate for @var{date} in
+time zone @var{zone}. If @var{zone} is explicitly in or not in
+daylight saving time (e.g., @code{PDT} or @code{PST}) the
address@hidden is ignored. If @var{zone} is a generalized time zone,
+the algorithms described above are used. If @var{zone} is omitted,
+the computation is done for the current time zone.
+
address@hidden Bugs}, for the address of Calc's author, if you
+should wish to contribute your improved versions of
address@hidden and @code{math-daylight-savings-hook}
+to the Calc distribution.
+
address@hidden Financial Functions, Binary Functions, Date Arithmetic,
Arithmetic
address@hidden Financial Functions
+
address@hidden
+Calc's financial or business functions use the @kbd{b} prefix
+key followed by a shifted letter. (The @kbd{b} prefix followed by
+a lower-case letter is used for operations on binary numbers.)
+
+Note that the rate and the number of intervals given to these
+functions must be on the same time scale, e.g., both months or
+both years. Mixing an annual interest rate with a time expressed
+in months will give you very wrong answers!
+
+It is wise to compute these functions to a higher precision than
+you really need, just to make sure your answer is correct to the
+last penny; also, you may wish to check the definitions at the end
+of this section to make sure the functions have the meaning you expect.
+
address@hidden
+* Percentages::
+* Future Value::
+* Present Value::
+* Related Financial Functions::
+* Depreciation Functions::
+* Definitions of Financial Functions::
address@hidden menu
+
address@hidden Percentages, Future Value, Financial Functions, Financial
Functions
address@hidden Percentages
+
address@hidden M-%
address@hidden calc-percent
address@hidden %
address@hidden percent
+The @kbd{M-%} (@code{calc-percent}) command takes a percentage value,
+say 5.4, and converts it to an equivalent actual number. For example,
address@hidden M-%} enters 0.054 on the stack. (That's the @key{META} or
address@hidden key combined with @kbd{%}.)
+
+Actually, @kbd{M-%} creates a formula of the form @samp{5.4%}.
+You can enter @samp{5.4%} yourself during algebraic entry. The
address@hidden operator simply means, ``the preceding value divided by
+100.'' The @samp{%} operator has very high precedence, so that
address@hidden is interpreted as @samp{1+(8%)}, not as @samp{(1+8)%}.
+(The @samp{%} operator is just a postfix notation for the
address@hidden function, just like @samp{20!} is the notation for
address@hidden(20)}, or twenty-factorial.)
+
+The formula @samp{5.4%} would normally evaluate immediately to
+0.054, but the @kbd{M-%} command suppresses evaluation as it puts
+the formula onto the stack. However, the next Calc command that
+uses the formula @samp{5.4%} will evaluate it as its first step.
+The net effect is that you get to look at @samp{5.4%} on the stack,
+but Calc commands see it as @samp{0.054}, which is what they expect.
+
+In particular, @samp{5.4%} and @samp{0.054} are suitable values
+for the @var{rate} arguments of the various financial functions,
+but the number @samp{5.4} is probably @emph{not} suitable---it
+represents a rate of 540 percent!
+
+The key sequence @kbd{M-% *} effectively means ``percent-of.''
+For example, @kbd{68 @key{RET} 25 M-% *} computes 17, which is 25% of
+68 (and also 68% of 25, which comes out to the same thing).
+
address@hidden c %
address@hidden calc-convert-percent
+The @kbd{c %} (@code{calc-convert-percent}) command converts the
+value on the top of the stack from numeric to percentage form.
+For example, if 0.08 is on the stack, @kbd{c %} converts it to
address@hidden The quantity is the same, it's just represented
+differently. (Contrast this with @kbd{M-%}, which would convert
+this number to @samp{0.08%}.) The @kbd{=} key is a convenient way
+to convert a formula like @samp{8%} back to numeric form, 0.08.
+
+To compute what percentage one quantity is of another quantity,
+use @kbd{/ c %}. For example, @address@hidden @key{RET} 68 / c %}} displays
address@hidden
+
address@hidden b %
address@hidden calc-percent-change
address@hidden relch
+The @kbd{b %} (@code{calc-percent-change}) address@hidden command
+calculates the percentage change from one number to another.
+For example, @kbd{40 @key{RET} 50 b %} produces the answer @samp{25%},
+since 50 is 25% larger than 40. A negative result represents a
+decrease: @kbd{50 @key{RET} 40 b %} produces @samp{-20%}, since 40 is
+20% smaller than 50. (The answers are different in magnitude
+because, in the first case, we're increasing by 25% of 40, but
+in the second case, we're decreasing by 20% of 50.) The effect
+of @kbd{40 @key{RET} 50 b %} is to compute @expr{(50-40)/40}, converting
+the answer to percentage form as if by @kbd{c %}.
+
address@hidden Future Value, Present Value, Percentages, Financial Functions
address@hidden Future Value
+
address@hidden
address@hidden b F
address@hidden calc-fin-fv
address@hidden fv
+The @kbd{b F} (@code{calc-fin-fv}) address@hidden command computes
+the future value of an investment. It takes three arguments
+from the stack: @samp{fv(@var{rate}, @var{n}, @var{payment})}.
+If you give payments of @var{payment} every year for @var{n}
+years, and the money you have paid earns interest at @var{rate} per
+year, then this function tells you what your investment would be
+worth at the end of the period. (The actual interval doesn't
+have to be years, as long as @var{n} and @var{rate} are expressed
+in terms of the same intervals.) This function assumes payments
+occur at the @emph{end} of each interval.
+
address@hidden I b F
address@hidden fvb
+The @kbd{I b F} address@hidden command does the same computation,
+but assuming your payments are at the beginning of each interval.
+Suppose you plan to deposit $1000 per year in a savings account
+earning 5.4% interest, starting right now. How much will be
+in the account after five years? @code{fvb(5.4%, 5, 1000) = 5870.73}.
+Thus you will have earned $870 worth of interest over the years.
+Using the stack, this calculation would have been
address@hidden M-% 5 @key{RET} 1000 I b F}. Note that the rate is expressed
+as a number between 0 and 1, @emph{not} as a percentage.
+
address@hidden H b F
address@hidden fvl
+The @kbd{H b F} address@hidden command computes the future value
+of an initial lump sum investment. Suppose you could deposit
+those five thousand dollars in the bank right now; how much would
+they be worth in five years? @code{fvl(5.4%, 5, 5000) = 6503.89}.
+
+The algebraic functions @code{fv} and @code{fvb} accept an optional
+fourth argument, which is used as an initial lump sum in the sense
+of @code{fvl}. In other words, @code{fv(@var{rate}, @var{n},
address@hidden, @var{initial}) = fv(@var{rate}, @var{n}, @var{payment})
++ fvl(@var{rate}, @var{n}, @var{initial})}.
+
+To illustrate the relationships between these functions, we could
+do the @code{fvb} calculation ``by hand'' using @code{fvl}. The
+final balance will be the sum of the contributions of our five
+deposits at various times. The first deposit earns interest for
+five years: @code{fvl(5.4%, 5, 1000) = 1300.78}. The second
+deposit only earns interest for four years: @code{fvl(5.4%, 4, 1000) =
+1234.13}. And so on down to the last deposit, which earns one
+year's interest: @code{fvl(5.4%, 1, 1000) = 1054.00}. The sum of
+these five values is, sure enough, $5870.73, just as was computed
+by @code{fvb} directly.
+
+What does @code{fv(5.4%, 5, 1000) = 5569.96} mean? The payments
+are now at the ends of the periods. The end of one year is the same
+as the beginning of the next, so what this really means is that we've
+lost the payment at year zero (which contributed $1300.78), but we're
+now counting the payment at year five (which, since it didn't have
+a chance to earn interest, counts as $1000). Indeed, @expr{5569.96 =
+5870.73 - 1300.78 + 1000} (give or take a bit of roundoff error).
+
address@hidden Present Value, Related Financial Functions, Future Value,
Financial Functions
address@hidden Present Value
+
address@hidden
address@hidden b P
address@hidden calc-fin-pv
address@hidden pv
+The @kbd{b P} (@code{calc-fin-pv}) address@hidden command computes
+the present value of an investment. Like @code{fv}, it takes
+three arguments: @code{pv(@var{rate}, @var{n}, @var{payment})}.
+It computes the present value of a series of regular payments.
+Suppose you have the chance to make an investment that will
+pay $2000 per year over the next four years; as you receive
+these payments you can put them in the bank at 9% interest.
+You want to know whether it is better to make the investment, or
+to keep the money in the bank where it earns 9% interest right
+from the start. The calculation @code{pv(9%, 4, 2000)} gives the
+result 6479.44. If your initial investment must be less than this,
+say, $6000, then the investment is worthwhile. But if you had to
+put up $7000, then it would be better just to leave it in the bank.
+
+Here is the interpretation of the result of @code{pv}: You are
+trying to compare the return from the investment you are
+considering, which is @code{fv(9%, 4, 2000) = 9146.26}, with
+the return from leaving the money in the bank, which is
address@hidden(9%, 4, @var{x})} where @var{x} is the amount of money
+you would have to put up in advance. The @code{pv} function
+finds the break-even point, @expr{x = 6479.44}, at which
address@hidden(9%, 4, 6479.44)} is also equal to 9146.26. This is
+the largest amount you should be willing to invest.
+
address@hidden I b P
address@hidden pvb
+The @kbd{I b P} address@hidden command solves the same problem,
+but with payments occurring at the beginning of each interval.
+It has the same relationship to @code{fvb} as @code{pv} has
+to @code{fv}. For example @code{pvb(9%, 4, 2000) = 7062.59},
+a larger number than @code{pv} produced because we get to start
+earning interest on the return from our investment sooner.
+
address@hidden H b P
address@hidden pvl
+The @kbd{H b P} address@hidden command computes the present value of
+an investment that will pay off in one lump sum at the end of the
+period. For example, if we get our $8000 all at the end of the
+four years, @code{pvl(9%, 4, 8000) = 5667.40}. This is much
+less than @code{pv} reported, because we don't earn any interest
+on the return from this investment. Note that @code{pvl} and
address@hidden are simple inverses: @code{fvl(9%, 4, 5667.40) = 8000}.
+
+You can give an optional fourth lump-sum argument to @code{pv}
+and @code{pvb}; this is handled in exactly the same way as the
+fourth argument for @code{fv} and @code{fvb}.
+
address@hidden b N
address@hidden calc-fin-npv
address@hidden npv
+The @kbd{b N} (@code{calc-fin-npv}) address@hidden command computes
+the net present value of a series of irregular investments.
+The first argument is the interest rate. The second argument is
+a vector which represents the expected return from the investment
+at the end of each interval. For example, if the rate represents
+a yearly interest rate, then the vector elements are the return
+from the first year, second year, and so on.
+
+Thus, @code{npv(9%, [2000,2000,2000,2000]) = pv(9%, 4, 2000) = 6479.44}.
+Obviously this function is more interesting when the payments are
+not all the same!
+
+The @code{npv} function can actually have two or more arguments.
+Multiple arguments are interpreted in the same way as for the
+vector statistical functions like @code{vsum}.
address@hidden Statistics}. Basically, if there are several
+payment arguments, each either a vector or a plain number, all these
+values are collected left-to-right into the complete list of payments.
+A numeric prefix argument on the @kbd{b N} command says how many
+payment values or vectors to take from the stack.
+
address@hidden I b N
address@hidden npvb
+The @kbd{I b N} address@hidden command computes the net present
+value where payments occur at the beginning of each interval
+rather than at the end.
+
address@hidden Related Financial Functions, Depreciation Functions, Present
Value, Financial Functions
address@hidden Related Financial Functions
+
address@hidden
+The functions in this section are basically inverses of the
+present value functions with respect to the various arguments.
+
address@hidden b M
address@hidden calc-fin-pmt
address@hidden pmt
+The @kbd{b M} (@code{calc-fin-pmt}) address@hidden command computes
+the amount of periodic payment necessary to amortize a loan.
+Thus @code{pmt(@var{rate}, @var{n}, @var{amount})} equals the
+value of @var{payment} such that @code{pv(@var{rate}, @var{n},
address@hidden) = @var{amount}}.
+
address@hidden I b M
address@hidden pmtb
+The @kbd{I b M} address@hidden command does the same computation
+but using @code{pvb} instead of @code{pv}. Like @code{pv} and
address@hidden, these functions can also take a fourth argument which
+represents an initial lump-sum investment.
+
address@hidden H b M
+The @kbd{H b M} key just invokes the @code{fvl} function, which is
+the inverse of @code{pvl}. There is no explicit @code{pmtl} function.
+
address@hidden b #
address@hidden calc-fin-nper
address@hidden nper
+The @kbd{b #} (@code{calc-fin-nper}) address@hidden command computes
+the number of regular payments necessary to amortize a loan.
+Thus @code{nper(@var{rate}, @var{payment}, @var{amount})} equals
+the value of @var{n} such that @code{pv(@var{rate}, @var{n},
address@hidden) = @var{amount}}. If @var{payment} is too small
+ever to amortize a loan for @var{amount} at interest rate @var{rate},
+the @code{nper} function is left in symbolic form.
+
address@hidden I b #
address@hidden nperb
+The @kbd{I b #} address@hidden command does the same computation
+but using @code{pvb} instead of @code{pv}. You can give a fourth
+lump-sum argument to these functions, but the computation will be
+rather slow in the four-argument case.
+
address@hidden H b #
address@hidden nperl
+The @kbd{H b #} address@hidden command does the same computation
+using @code{pvl}. By exchanging @var{payment} and @var{amount} you
+can also get the solution for @code{fvl}. For example,
address@hidden(8%, 2000, 1000) = 9.006}, so if you place $1000 in a
+bank account earning 8%, it will take nine years to grow to $2000.
+
address@hidden b T
address@hidden calc-fin-rate
address@hidden rate
+The @kbd{b T} (@code{calc-fin-rate}) address@hidden command computes
+the rate of return on an investment. This is also an inverse of @code{pv}:
address@hidden(@var{n}, @var{payment}, @var{amount})} computes the value of
address@hidden such that @code{pv(@var{rate}, @var{n}, @var{payment}) =
address@hidden The result is expressed as a formula like @samp{6.3%}.
+
address@hidden I b T
address@hidden H b T
address@hidden rateb
address@hidden ratel
+The @kbd{I b T} address@hidden and @kbd{H b T} address@hidden
+commands solve the analogous equations with @code{pvb} or @code{pvl}
+in place of @code{pv}. Also, @code{rate} and @code{rateb} can
+accept an optional fourth argument just like @code{pv} and @code{pvb}.
+To redo the above example from a different perspective,
address@hidden(9, 2000, 1000) = 8.00597%}, which says you will need an
+interest rate of 8% in order to double your account in nine years.
+
address@hidden b I
address@hidden calc-fin-irr
address@hidden irr
+The @kbd{b I} (@code{calc-fin-irr}) address@hidden command is the
+analogous function to @code{rate} but for net present value.
+Its argument is a vector of payments. Thus @code{irr(@var{payments})}
+computes the @var{rate} such that @code{npv(@var{rate}, @var{payments}) = 0};
+this rate is known as the @dfn{internal rate of return}.
+
address@hidden I b I
address@hidden irrb
+The @kbd{I b I} address@hidden command computes the internal rate of
+return assuming payments occur at the beginning of each period.
+
address@hidden Depreciation Functions, Definitions of Financial Functions,
Related Financial Functions, Financial Functions
address@hidden Depreciation Functions
+
address@hidden
+The functions in this section calculate @dfn{depreciation}, which is
+the amount of value that a possession loses over time. These functions
+are characterized by three parameters: @var{cost}, the original cost
+of the asset; @var{salvage}, the value the asset will have at the end
+of its expected ``useful life''; and @var{life}, the number of years
+(or other periods) of the expected useful life.
+
+There are several methods for calculating depreciation that differ in
+the way they spread the depreciation over the lifetime of the asset.
+
address@hidden b S
address@hidden calc-fin-sln
address@hidden sln
+The @kbd{b S} (@code{calc-fin-sln}) address@hidden command computes the
+``straight-line'' depreciation. In this method, the asset depreciates
+by the same amount every year (or period). For example,
address@hidden(12000, 2000, 5)} returns 2000. The asset costs $12000
+initially and will be worth $2000 after five years; it loses $2000
+per year.
+
address@hidden b Y
address@hidden calc-fin-syd
address@hidden syd
+The @kbd{b Y} (@code{calc-fin-syd}) address@hidden command computes the
+accelerated ``sum-of-years'-digits'' depreciation. Here the depreciation
+is higher during the early years of the asset's life. Since the
+depreciation is different each year, @kbd{b Y} takes a fourth @var{period}
+parameter which specifies which year is requested, from 1 to @var{life}.
+If @var{period} is outside this range, the @code{syd} function will
+return zero.
+
address@hidden b D
address@hidden calc-fin-ddb
address@hidden ddb
+The @kbd{b D} (@code{calc-fin-ddb}) address@hidden command computes an
+accelerated depreciation using the double-declining balance method.
+It also takes a fourth @var{period} parameter.
+
+For symmetry, the @code{sln} function will accept a @var{period}
+parameter as well, although it will ignore its value except that the
+return value will as usual be zero if @var{period} is out of range.
+
+For example, pushing the vector @expr{[1,2,3,4,5]} (perhaps with @kbd{v x 5})
+and then mapping @kbd{V M ' [sln(12000,2000,5,$), syd(12000,2000,5,$),
+ddb(12000,2000,5,$)] @key{RET}} produces a matrix that allows us to compare
+the three depreciation methods:
+
address@hidden
address@hidden
+[ [ 2000, 3333, 4800 ]
+ [ 2000, 2667, 2880 ]
+ [ 2000, 2000, 1728 ]
+ [ 2000, 1333, 592 ]
+ [ 2000, 667, 0 ] ]
address@hidden group
address@hidden example
+
address@hidden
+(Values have been rounded to nearest integers in this figure.)
+We see that @code{sln} depreciates by the same amount each year,
address@hidden depreciates more at the beginning and less at the end,
+and @kbd{ddb} weights the depreciation even more toward the beginning.
+
+Summing columns with @kbd{V R : +} yields @expr{[10000, 10000, 10000]};
+the total depreciation in any method is (by definition) the
+difference between the cost and the salvage value.
+
address@hidden Definitions of Financial Functions, , Depreciation Functions,
Financial Functions
address@hidden Definitions
+
address@hidden
+For your reference, here are the actual formulas used to compute
+Calc's financial functions.
+
+Calc will not evaluate a financial function unless the @var{rate} or
address@hidden argument is known. However, @var{payment} or @var{amount} can
+be a variable. Calc expands these functions according to the
+formulas below for symbolic arguments only when you use the @kbd{a "}
+(@code{calc-expand-formula}) command, or when taking derivatives or
+integrals or solving equations involving the functions.
+
address@hidden
+These formulas are shown using the conventions of Big display
+mode (@kbd{d B}); for example, the formula for @code{fv} written
+linearly is @samp{pmt * ((1 + rate)^n) - 1) / rate}.
+
address@hidden
+ n
+ (1 + rate) - 1
+fv(rate, n, pmt) = pmt * ---------------
+ rate
+
+ n
+ ((1 + rate) - 1) (1 + rate)
+fvb(rate, n, pmt) = pmt * ----------------------------
+ rate
+
+ n
+fvl(rate, n, pmt) = pmt * (1 + rate)
+
+ -n
+ 1 - (1 + rate)
+pv(rate, n, pmt) = pmt * ----------------
+ rate
+
+ -n
+ (1 - (1 + rate) ) (1 + rate)
+pvb(rate, n, pmt) = pmt * -----------------------------
+ rate
+
+ -n
+pvl(rate, n, pmt) = pmt * (1 + rate)
+
+ -1 -2 -3
+npv(rate, [a, b, c]) = a*(1 + rate) + b*(1 + rate) + c*(1 + rate)
+
+ -1 -2
+npvb(rate, [a, b, c]) = a + b*(1 + rate) + c*(1 + rate)
+
+ -n
+ (amt - x * (1 + rate) ) * rate
+pmt(rate, n, amt, x) = -------------------------------
+ -n
+ 1 - (1 + rate)
+
+ -n
+ (amt - x * (1 + rate) ) * rate
+pmtb(rate, n, amt, x) = -------------------------------
+ -n
+ (1 - (1 + rate) ) (1 + rate)
+
+ amt * rate
+nper(rate, pmt, amt) = - log(1 - ------------, 1 + rate)
+ pmt
+
+ amt * rate
+nperb(rate, pmt, amt) = - log(1 - ---------------, 1 + rate)
+ pmt * (1 + rate)
+
+ amt
+nperl(rate, pmt, amt) = - log(---, 1 + rate)
+ pmt
+
+ 1/n
+ pmt
+ratel(n, pmt, amt) = ------ - 1
+ 1/n
+ amt
+
+ cost - salv
+sln(cost, salv, life) = -----------
+ life
+
+ (cost - salv) * (life - per + 1)
+syd(cost, salv, life, per) = --------------------------------
+ life * (life + 1) / 2
+
+ book * 2
+ddb(cost, salv, life, per) = --------, book = cost - depreciation so far
+ life
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+$$ \code{fv}(r, n, p) = p { (1 + r)^n - 1 \over r } $$
+$$ \code{fvb}(r, n, p) = p { ((1 + r)^n - 1) (1 + r) \over r } $$
+$$ \code{fvl}(r, n, p) = p (1 + r)^n $$
+$$ \code{pv}(r, n, p) = p { 1 - (1 + r)^{-n} \over r } $$
+$$ \code{pvb}(r, n, p) = p { (1 - (1 + r)^{-n}) (1 + r) \over r } $$
+$$ \code{pvl}(r, n, p) = p (1 + r)^{-n} $$
+$$ \code{npv}(r, [a,b,c]) = a (1 + r)^{-1} + b (1 + r)^{-2} + c (1 + r)^{-3} $$
+$$ \code{npvb}(r, [a,b,c]) = a + b (1 + r)^{-1} + c (1 + r)^{-2} $$
+$$ \code{pmt}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over 1 - (1 + r)^{-n} }$$
+$$ \code{pmtb}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over
+ (1 - (1 + r)^{-n}) (1 + r) } $$
+$$ \code{nper}(r, p, a) = -\code{log}(1 - { a r \over p }, 1 + r) $$
+$$ \code{nperb}(r, p, a) = -\code{log}(1 - { a r \over p (1 + r) }, 1 + r) $$
+$$ \code{nperl}(r, p, a) = -\code{log}({a \over p}, 1 + r) $$
+$$ \code{ratel}(n, p, a) = { p^{1/n} \over a^{1/n} } - 1 $$
+$$ \code{sln}(c, s, l) = { c - s \over l } $$
+$$ \code{syd}(c, s, l, p) = { (c - s) (l - p + 1) \over l (l+1) / 2 } $$
+$$ \code{ddb}(c, s, l, p) = { 2 (c - \hbox{depreciation so far}) \over l } $$
address@hidden tex
+
address@hidden
+In @code{pmt} and @code{pmtb}, @expr{x=0} if omitted.
+
+These functions accept any numeric objects, including error forms,
+intervals, and even (though not very usefully) complex numbers. The
+above formulas specify exactly the behavior of these functions with
+all sorts of inputs.
+
+Note that if the first argument to the @code{log} in @code{nper} is
+negative, @code{nper} leaves itself in symbolic form rather than
+returning a (financially meaningless) complex number.
+
address@hidden(num, pmt, amt)} solves the equation
address@hidden(rate, num, pmt) = amt} for @samp{rate} using @kbd{H a R}
+(@code{calc-find-root}), with the interval @samp{[.01% .. 100%]}
+for an initial guess. The @code{rateb} function is the same except
+that it uses @code{pvb}. Note that @code{ratel} can be solved
+directly; its formula is shown in the above list.
+
+Similarly, @samp{irr(pmts)} solves the equation @samp{npv(rate, pmts) = 0}
+for @samp{rate}.
+
+If you give a fourth argument to @code{nper} or @code{nperb}, Calc
+will also use @kbd{H a R} to solve the equation using an initial
+guess interval of @samp{[0 .. 100]}.
+
+A fourth argument to @code{fv} simply sums the two components
+calculated from the above formulas for @code{fv} and @code{fvl}.
+The same is true of @code{fvb}, @code{pv}, and @code{pvb}.
+
+The @kbd{ddb} function is computed iteratively; the ``book'' value
+starts out equal to @var{cost}, and decreases according to the above
+formula for the specified number of periods. If the book value
+would decrease below @var{salvage}, it only decreases to @var{salvage}
+and the depreciation is zero for all subsequent periods. The @code{ddb}
+function returns the amount the book value decreased in the specified
+period.
+
address@hidden Binary Functions, , Financial Functions, Arithmetic
address@hidden Binary Number Functions
+
address@hidden
+The commands in this chapter all use two-letter sequences beginning with
+the @kbd{b} prefix.
+
address@hidden Binary numbers
+The ``binary'' operations actually work regardless of the currently
+displayed radix, although their results make the most sense in a radix
+like 2, 8, or 16 (as obtained by the @kbd{d 2}, @kbd{d 8}, or @address@hidden
6}}
+commands, respectively). You may also wish to enable display of leading
+zeros with @kbd{d z}. @xref{Radix Modes}.
+
address@hidden Word size for binary operations
+The Calculator maintains a current @dfn{word size} @expr{w}, an
+arbitrary positive or negative integer. For a positive word size, all
+of the binary operations described here operate modulo @expr{2^w}. In
+particular, negative arguments are converted to positive integers modulo
address@hidden by all binary functions.
+
+If the word size is negative, binary operations produce 2's complement
+integers from
address@hidden @math{-2^{-w-1}}
address@hidden @expr{-(2^(-w-1))}
+to
address@hidden @math{2^{-w-1}-1}
address@hidden @expr{2^(-w-1)-1}
+inclusive. Either mode accepts inputs in any range; the sign of
address@hidden affects only the results produced.
+
address@hidden b c
address@hidden calc-clip
address@hidden clip
+The @kbd{b c} (@code{calc-clip})
address@hidden command can be used to clip a number by reducing it modulo
address@hidden The commands described in this chapter automatically clip
+their results to the current word size. Note that other operations like
+addition do not use the current word size, since integer addition
+generally is not ``binary.'' (However, @pxref{Simplification Modes},
address@hidden) For example, with a word size of 8
+bits @kbd{b c} converts a number to the range 0 to 255; with a word
+size of @mathit{-8} @kbd{b c} converts to the range @mathit{-128} to 127.
+
address@hidden b w
address@hidden calc-word-size
+The default word size is 32 bits. All operations except the shifts and
+rotates allow you to specify a different word size for that one
+operation by giving a numeric prefix argument: @kbd{C-u 8 b c} clips the
+top of stack to the range 0 to 255 regardless of the current word size.
+To set the word size permanently, use @kbd{b w} (@code{calc-word-size}).
+This command displays a prompt with the current word size; press @key{RET}
+immediately to keep this word size, or type a new word size at the prompt.
+
+When the binary operations are written in symbolic form, they take an
+optional second (or third) word-size parameter. When a formula like
address@hidden(a,b)} is finally evaluated, the word size current at that time
+will be used, but when @samp{and(a,b,-8)} is evaluated, a word size of
address@hidden will always be used. A symbolic binary function will be left
+in symbolic form unless the all of its argument(s) are integers or
+integer-valued floats.
+
+If either or both arguments are modulo forms for which @expr{M} is a
+power of two, that power of two is taken as the word size unless a
+numeric prefix argument overrides it. The current word size is never
+consulted when modulo-power-of-two forms are involved.
+
address@hidden b a
address@hidden calc-and
address@hidden and
+The @kbd{b a} (@code{calc-and}) address@hidden command computes the bitwise
+AND of the two numbers on the top of the stack. In other words, for each
+of the @expr{w} binary digits of the two numbers (pairwise), the corresponding
+bit of the result is 1 if and only if both input bits are 1:
address@hidden(2#1100, 2#1010) = 2#1000}.
+
address@hidden b o
address@hidden calc-or
address@hidden or
+The @kbd{b o} (@code{calc-or}) address@hidden command computes the bitwise
+inclusive OR of two numbers. A bit is 1 if either of the input bits, or
+both, are 1: @samp{or(2#1100, 2#1010) = 2#1110}.
+
address@hidden b x
address@hidden calc-xor
address@hidden xor
+The @kbd{b x} (@code{calc-xor}) address@hidden command computes the bitwise
+exclusive OR of two numbers. A bit is 1 if exactly one of the input bits
+is 1: @samp{xor(2#1100, 2#1010) = 2#0110}.
+
address@hidden b d
address@hidden calc-diff
address@hidden diff
+The @kbd{b d} (@code{calc-diff}) address@hidden command computes the bitwise
+difference of two numbers; this is defined by @samp{diff(a,b) = and(a,not(b))},
+so that @samp{diff(2#1100, 2#1010) = 2#0100}.
+
address@hidden b n
address@hidden calc-not
address@hidden not
+The @kbd{b n} (@code{calc-not}) address@hidden command computes the bitwise
+NOT of a number. A bit is 1 if the input bit is 0 and vice-versa.
+
address@hidden b l
address@hidden calc-lshift-binary
address@hidden lsh
+The @kbd{b l} (@code{calc-lshift-binary}) address@hidden command shifts a
+number left by one bit, or by the number of bits specified in the numeric
+prefix argument. A negative prefix argument performs a logical right shift,
+in which zeros are shifted in on the left. In symbolic form, @samp{lsh(a)}
+is short for @samp{lsh(a,1)}, which in turn is short for @samp{lsh(a,n,w)}.
+Bits shifted ``off the end,'' according to the current word size, are lost.
+
address@hidden H b l
address@hidden H b r
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden H b L
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H b R
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H b t
+The @kbd{H b l} command also does a left shift, but it takes two arguments
+from the stack (the value to shift, and, at top-of-stack, the number of
+bits to shift). This version interprets the prefix argument just like
+the regular binary operations, i.e., as a word size. The Hyperbolic flag
+has a similar effect on the rest of the binary shift and rotate commands.
+
address@hidden b r
address@hidden calc-rshift-binary
address@hidden rsh
+The @kbd{b r} (@code{calc-rshift-binary}) address@hidden command shifts a
+number right by one bit, or by the number of bits specified in the numeric
+prefix argument: @samp{rsh(a,n) = lsh(a,-n)}.
+
address@hidden b L
address@hidden calc-lshift-arith
address@hidden ash
+The @kbd{b L} (@code{calc-lshift-arith}) address@hidden command shifts a
+number left. It is analogous to @code{lsh}, except that if the shift
+is rightward (the prefix argument is negative), an arithmetic shift
+is performed as described below.
+
address@hidden b R
address@hidden calc-rshift-arith
address@hidden rash
+The @kbd{b R} (@code{calc-rshift-arith}) address@hidden command performs
+an ``arithmetic'' shift to the right, in which the leftmost bit (according
+to the current word size) is duplicated rather than shifting in zeros.
+This corresponds to dividing by a power of two where the input is interpreted
+as a signed, twos-complement number. (The distinction between the @samp{rsh}
+and @samp{rash} operations is totally independent from whether the word
+size is positive or negative.) With a negative prefix argument, this
+performs a standard left shift.
+
address@hidden b t
address@hidden calc-rotate-binary
address@hidden rot
+The @kbd{b t} (@code{calc-rotate-binary}) address@hidden command rotates a
+number one bit to the left. The leftmost bit (according to the current
+word size) is dropped off the left and shifted in on the right. With a
+numeric prefix argument, the number is rotated that many bits to the left
+or right.
+
address@hidden Operations}, for the @kbd{b p} and @kbd{b u} commands that
+pack and unpack binary integers into sets. (For example, @kbd{b u}
+unpacks the number @samp{2#11001} to the set of bit-numbers
address@hidden, 3, 4]}.) Type @kbd{b u V #} to count the number of ``1''
+bits in a binary integer.
+
+Another interesting use of the set representation of binary integers
+is to reverse the bits in, say, a 32-bit integer. Type @kbd{b u} to
+unpack; type @kbd{31 @key{TAB} -} to replace each bit-number in the set
+with 31 minus that bit-number; type @kbd{b p} to pack the set back
+into a binary integer.
+
address@hidden Scientific Functions, Matrix Functions, Arithmetic, Top
address@hidden Scientific Functions
+
address@hidden
+The functions described here perform trigonometric and other transcendental
+calculations. They generally produce floating-point answers correct to the
+full current precision. The @kbd{H} (Hyperbolic) and @kbd{I} (Inverse)
+flag keys must be used to get some of these functions from the keyboard.
+
address@hidden P
address@hidden calc-pi
address@hidden @code{pi} variable
address@hidden pi
address@hidden H P
address@hidden @code{e} variable
address@hidden e
address@hidden I P
address@hidden @code{gamma} variable
address@hidden gamma
address@hidden Gamma constant, Euler's
address@hidden Euler's gamma constant
address@hidden H I P
address@hidden @code{phi} variable
address@hidden Phi, golden ratio
address@hidden Golden ratio
+One miscellaneous command is address@hidden (@code{calc-pi}), which pushes
+the value of @cpi{} (at the current precision) onto the stack. With the
+Hyperbolic flag, it pushes the value @expr{e}, the base of natural logarithms.
+With the Inverse flag, it pushes Euler's constant
address@hidden @math{\gamma}
address@hidden @expr{gamma}
+(about 0.5772). With both Inverse and Hyperbolic, it
+pushes the ``golden ratio''
address@hidden @math{\phi}
address@hidden @expr{phi}
+(about 1.618). (At present, Euler's constant is not available
+to unlimited precision; Calc knows only the first 100 digits.)
+In Symbolic mode, these commands push the
+actual variables @samp{pi}, @samp{e}, @samp{gamma}, and @samp{phi},
+respectively, instead of their values; @pxref{Symbolic Mode}.
+
address@hidden
address@hidden Q
address@hidden ignore
address@hidden
address@hidden I Q
address@hidden ignore
address@hidden I Q
address@hidden sqr
+The @kbd{Q} (@code{calc-sqrt}) address@hidden function is described elsewhere;
address@hidden Arithmetic}. With the Inverse flag address@hidden, this command
+computes the square of the argument.
+
address@hidden Arguments}, for a discussion of the effect of numeric
+prefix arguments on commands in this chapter which do not otherwise
+interpret a prefix argument.
+
address@hidden
+* Logarithmic Functions::
+* Trigonometric and Hyperbolic Functions::
+* Advanced Math Functions::
+* Branch Cuts::
+* Random Numbers::
+* Combinatorial Functions::
+* Probability Distribution Functions::
address@hidden menu
+
address@hidden Logarithmic Functions, Trigonometric and Hyperbolic Functions,
Scientific Functions, Scientific Functions
address@hidden Logarithmic Functions
+
address@hidden
address@hidden L
address@hidden calc-ln
address@hidden ln
address@hidden
address@hidden @null
address@hidden ignore
address@hidden I E
+The address@hidden (@code{calc-ln}) address@hidden command computes the natural
+logarithm of the real or complex number on the top of the stack. With
+the Inverse flag it computes the exponential function instead, although
+this is redundant with the @kbd{E} command.
+
address@hidden E
address@hidden calc-exp
address@hidden exp
address@hidden
address@hidden @null
address@hidden ignore
address@hidden I L
+The address@hidden (@code{calc-exp}) address@hidden command computes the
+exponential, i.e., @expr{e} raised to the power of the number on the stack.
+The meanings of the Inverse and Hyperbolic flags follow from those for
+the @code{calc-ln} command.
+
address@hidden H L
address@hidden H E
address@hidden calc-log10
address@hidden log10
address@hidden exp10
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H I L
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H I E
+The @kbd{H L} (@code{calc-log10}) address@hidden command computes the common
+(base-10) logarithm of a number. (With the Inverse flag address@hidden,
+it raises ten to a given power.) Note that the common logarithm of a
+complex number is computed by taking the natural logarithm and dividing
+by
address@hidden @math{\ln10}.
address@hidden @expr{ln(10)}.
+
address@hidden B
address@hidden I B
address@hidden calc-log
address@hidden log
address@hidden alog
+The @kbd{B} (@code{calc-log}) address@hidden command computes a logarithm
+to any base. For example, @kbd{1024 @key{RET} 2 B} produces 10, since
address@hidden @math{2^{10} = 1024}.
address@hidden @expr{2^10 = 1024}.
+In certain cases like @samp{log(3,9)}, the result
+will be either @expr{1:2} or @expr{0.5} depending on the current Fraction
+mode setting. With the Inverse flag address@hidden, this command is
+similar to @kbd{^} except that the order of the arguments is reversed.
+
address@hidden f I
address@hidden calc-ilog
address@hidden ilog
+The @kbd{f I} (@code{calc-ilog}) address@hidden command computes the
+integer logarithm of a number to any base. The number and the base must
+themselves be positive integers. This is the true logarithm, rounded
+down to an integer. Thus @kbd{ilog(x,10)} is 3 for all @expr{x} in the
+range from 1000 to 9999. If both arguments are positive integers, exact
+integer arithmetic is used; otherwise, this is equivalent to
address@hidden(log(x,b))}.
+
address@hidden f E
address@hidden calc-expm1
address@hidden expm1
+The @kbd{f E} (@code{calc-expm1}) address@hidden command computes
address@hidden @math{e^x - 1},
address@hidden @expr{exp(x)-1},
+but using an algorithm that produces a more accurate
+answer when the result is close to zero, i.e., when
address@hidden @math{e^x}
address@hidden @expr{exp(x)}
+is close to one.
+
address@hidden f L
address@hidden calc-lnp1
address@hidden lnp1
+The @kbd{f L} (@code{calc-lnp1}) address@hidden command computes
address@hidden @math{\ln(x+1)},
address@hidden @expr{ln(x+1)},
+producing a more accurate answer when @expr{x} is close to zero.
+
address@hidden Trigonometric and Hyperbolic Functions, Advanced Math Functions,
Logarithmic Functions, Scientific Functions
address@hidden Trigonometric/Hyperbolic Functions
+
address@hidden
address@hidden S
address@hidden calc-sin
address@hidden sin
+The address@hidden (@code{calc-sin}) address@hidden command computes the sine
+of an angle or complex number. If the input is an HMS form, it is interpreted
+as degrees-minutes-seconds; otherwise, the input is interpreted according
+to the current angular mode. It is best to use Radians mode when operating
+on complex numbers.
+
+Calc's ``units'' mechanism includes angular units like @code{deg},
address@hidden, and @code{grad}. While @samp{sin(45 deg)} is not evaluated
+all the time, the @kbd{u s} (@code{calc-simplify-units}) command will
+simplify @samp{sin(45 deg)} by taking the sine of 45 degrees, regardless
+of the current angular mode. @xref{Basic Operations on Units}.
+
+Also, the symbolic variable @code{pi} is not ordinarily recognized in
+arguments to trigonometric functions, as in @samp{sin(3 pi / 4)}, but
+the @kbd{a s} (@code{calc-simplify}) command recognizes many such
+formulas when the current angular mode is Radians @emph{and} Symbolic
+mode is enabled; this example would be replaced by @samp{sqrt(2) / 2}.
address@hidden Mode}. Beware, this simplification occurs even if you
+have stored a different value in the variable @samp{pi}; this is one
+reason why changing built-in variables is a bad idea. Arguments of
+the form @expr{x} plus a multiple of @cpiover{2} are also simplified.
+Calc includes similar formulas for @code{cos} and @code{tan}.
+
+The @kbd{a s} command knows all angles which are integer multiples of
address@hidden, @cpiover{10}, or @cpiover{8} radians. In Degrees mode,
+analogous simplifications occur for integer multiples of 15 or 18
+degrees, and for arguments plus multiples of 90 degrees.
+
address@hidden I S
address@hidden calc-arcsin
address@hidden arcsin
+With the Inverse flag, @code{calc-sin} computes an arcsine. This is also
+available as the @code{calc-arcsin} command or @code{arcsin} algebraic
+function. The returned argument is converted to degrees, radians, or HMS
+notation depending on the current angular mode.
+
address@hidden H S
address@hidden calc-sinh
address@hidden sinh
address@hidden H I S
address@hidden calc-arcsinh
address@hidden arcsinh
+With the Hyperbolic flag, @code{calc-sin} computes the hyperbolic
+sine, also available as @code{calc-sinh} address@hidden With the
+Hyperbolic and Inverse flags, it computes the hyperbolic arcsine
+(@code{calc-arcsinh}) address@hidden
+
address@hidden C
address@hidden calc-cos
address@hidden cos
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden I C
address@hidden calc-arccos
address@hidden
address@hidden @null
address@hidden ignore
address@hidden arccos
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H C
address@hidden calc-cosh
address@hidden
address@hidden @null
address@hidden ignore
address@hidden cosh
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H I C
address@hidden calc-arccosh
address@hidden
address@hidden @null
address@hidden ignore
address@hidden arccosh
address@hidden
address@hidden @null
address@hidden ignore
address@hidden T
address@hidden calc-tan
address@hidden
address@hidden @null
address@hidden ignore
address@hidden tan
address@hidden
address@hidden @null
address@hidden ignore
address@hidden I T
address@hidden calc-arctan
address@hidden
address@hidden @null
address@hidden ignore
address@hidden arctan
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H T
address@hidden calc-tanh
address@hidden
address@hidden @null
address@hidden ignore
address@hidden tanh
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H I T
address@hidden calc-arctanh
address@hidden
address@hidden @null
address@hidden ignore
address@hidden arctanh
+The address@hidden (@code{calc-cos}) address@hidden command computes the cosine
+of an angle or complex number, and address@hidden (@code{calc-tan})
address@hidden
+computes the tangent, along with all the various inverse and hyperbolic
+variants of these functions.
+
address@hidden f T
address@hidden calc-arctan2
address@hidden arctan2
+The @kbd{f T} (@code{calc-arctan2}) address@hidden command takes two
+numbers from the stack and computes the arc tangent of their ratio. The
+result is in the full range from @mathit{-180} (exclusive) to @mathit{+180}
+(inclusive) degrees, or the analogous range in radians. A similar
+result would be obtained with @kbd{/} followed by @kbd{I T}, but the
+value would only be in the range from @mathit{-90} to @mathit{+90} degrees
+since the division loses information about the signs of the two
+components, and an error might result from an explicit division by zero
+which @code{arctan2} would avoid. By (arbitrary) definition,
address@hidden(0,0)=0}.
+
address@hidden calc-sincos
address@hidden
address@hidden
address@hidden ignore
address@hidden sincos
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden arcsincos
+The @code{calc-sincos} address@hidden command computes the sine and
+cosine of a number, returning them as a vector of the form
address@hidden@var{cos}, @var{sin}]}.
+With the Inverse flag address@hidden, this command takes a two-element
+vector as an argument and computes @code{arctan2} of the elements.
+(This command does not accept the Hyperbolic flag.)
+
address@hidden calc-sec
address@hidden sec
address@hidden calc-csc
address@hidden csc
address@hidden calc-cot
address@hidden cot
address@hidden calc-sech
address@hidden sech
address@hidden calc-csch
address@hidden csch
address@hidden calc-coth
address@hidden coth
+The remaining trigonometric functions, @code{calc-sec} address@hidden,
address@hidden address@hidden and @code{calc-sec} address@hidden, are also
+available. With the Hyperbolic flag, these compute their hyperbolic
+counterparts, which are also available separately as @code{calc-sech}
address@hidden, @code{calc-csch} address@hidden and @code{calc-sech}
address@hidden (These commmands do not accept the Inverse flag.)
+
address@hidden Advanced Math Functions, Branch Cuts, Trigonometric and
Hyperbolic Functions, Scientific Functions
address@hidden Advanced Mathematical Functions
+
address@hidden
+Calc can compute a variety of less common functions that arise in
+various branches of mathematics. All of the functions described in
+this section allow arbitrary complex arguments and, except as noted,
+will work to arbitrarily large precisions. They can not at present
+handle error forms or intervals as arguments.
+
+NOTE: These functions are still experimental. In particular, their
+accuracy is not guaranteed in all domains. It is advisable to set the
+current precision comfortably higher than you actually need when
+using these functions. Also, these functions may be impractically
+slow for some values of the arguments.
+
address@hidden f g
address@hidden calc-gamma
address@hidden gamma
+The @kbd{f g} (@code{calc-gamma}) address@hidden command computes the Euler
+gamma function. For positive integer arguments, this is related to the
+factorial function: @samp{gamma(n+1) = fact(n)}. For general complex
+arguments the gamma function can be defined by the following definite
+integral:
address@hidden @math{\Gamma(a) = \int_0^\infty t^{a-1} e^t dt}.
address@hidden @expr{gamma(a) = integ(t^(a-1) exp(t), t, 0, inf)}.
+(The actual implementation uses far more efficient computational methods.)
+
address@hidden f G
address@hidden gammaP
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden I f G
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H f G
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H I f G
address@hidden calc-inc-gamma
address@hidden
address@hidden @null
address@hidden ignore
address@hidden gammaQ
address@hidden
address@hidden @null
address@hidden ignore
address@hidden gammag
address@hidden
address@hidden @null
address@hidden ignore
address@hidden gammaG
+The @kbd{f G} (@code{calc-inc-gamma}) address@hidden command computes
+the incomplete gamma function, denoted @samp{P(a,x)}. This is defined by
+the integral,
address@hidden @math{P(a,x) = \left( \int_0^x t^{a-1} e^t dt \right) /
\Gamma(a)}.
address@hidden @expr{gammaP(a,x) = integ(t^(a-1) exp(t), t, 0, x) / gamma(a)}.
+This implies that @samp{gammaP(a,inf) = 1} for any @expr{a} (see the
+definition of the normal gamma function).
+
+Several other varieties of incomplete gamma function are defined.
+The complement of @expr{P(a,x)}, called @expr{Q(a,x) = 1-P(a,x)} by
+some authors, is computed by the @kbd{I f G} address@hidden command.
+You can think of this as taking the other half of the integral, from
address@hidden to infinity.
+
address@hidden
+The functions corresponding to the integrals that define @expr{P(a,x)}
+and @expr{Q(a,x)} but without the normalizing @expr{1/gamma(a)}
+factor are called @expr{g(a,x)} and @expr{G(a,x)}, respectively
+(where @expr{g} and @expr{G} represent the lower- and upper-case Greek
+letter gamma). You can obtain these using the @kbd{H f G} address@hidden
+and @kbd{H I f G} address@hidden commands.
address@hidden ifnottex
address@hidden
+\turnoffactive
+The functions corresponding to the integrals that define $P(a,x)$
+and $Q(a,x)$ but without the normalizing $1/\Gamma(a)$
+factor are called $\gamma(a,x)$ and $\Gamma(a,x)$, respectively.
+You can obtain these using the \kbd{H f G} [\code{gammag}] and
+\kbd{I H f G} [\code{gammaG}] commands.
address@hidden tex
+
address@hidden f b
address@hidden calc-beta
address@hidden beta
+The @kbd{f b} (@code{calc-beta}) address@hidden command computes the
+Euler beta function, which is defined in terms of the gamma function as
address@hidden @math{B(a,b) = \Gamma(a) \Gamma(b) / \Gamma(a+b)},
address@hidden @expr{beta(a,b) = gamma(a) gamma(b) / gamma(a+b)},
+or by
address@hidden @math{B(a,b) = \int_0^1 t^{a-1} (1-t)^{b-1} dt}.
address@hidden @expr{beta(a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, 1)}.
+
address@hidden f B
address@hidden H f B
address@hidden calc-inc-beta
address@hidden betaI
address@hidden betaB
+The @kbd{f B} (@code{calc-inc-beta}) address@hidden command computes
+the incomplete beta function @expr{I(x,a,b)}. It is defined by
address@hidden @math{I(x,a,b) = \left( \int_0^x t^{a-1} (1-t)^{b-1} dt \right)
/ B(a,b)}.
address@hidden @expr{betaI(x,a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, x) /
beta(a,b)}.
+Once again, the @kbd{H} (hyperbolic) prefix gives the corresponding
+un-normalized version address@hidden
+
address@hidden f e
address@hidden I f e
address@hidden calc-erf
address@hidden erf
address@hidden erfc
+The @kbd{f e} (@code{calc-erf}) address@hidden command computes the
+error function
address@hidden @math{\hbox{erf}(x) = {2 \over \sqrt{\pi}} \int_0^x e^{-t^2} dt}.
address@hidden @expr{erf(x) = 2 integ(exp(-(t^2)), t, 0, x) / sqrt(pi)}.
+The complementary error function @kbd{I f e} (@code{calc-erfc}) address@hidden
+is the corresponding integral from @samp{x} to infinity; the sum
address@hidden @math{\hbox{erf}(x) + \hbox{erfc}(x) = 1}.
address@hidden @expr{erf(x) + erfc(x) = 1}.
+
address@hidden f j
address@hidden f y
address@hidden calc-bessel-J
address@hidden calc-bessel-Y
address@hidden besJ
address@hidden besY
+The @kbd{f j} (@code{calc-bessel-J}) address@hidden and @kbd{f y}
+(@code{calc-bessel-Y}) address@hidden commands compute the Bessel
+functions of the first and second kinds, respectively.
+In @samp{besJ(n,x)} and @samp{besY(n,x)} the ``order'' parameter
address@hidden is often an integer, but is not required to be one.
+Calc's implementation of the Bessel functions currently limits the
+precision to 8 digits, and may not be exact even to that precision.
+Use with care!
+
address@hidden Branch Cuts, Random Numbers, Advanced Math Functions, Scientific
Functions
address@hidden Branch Cuts and Principal Values
+
address@hidden
address@hidden Branch cuts
address@hidden Principal values
+All of the logarithmic, trigonometric, and other scientific functions are
+defined for complex numbers as well as for reals.
+This section describes the values
+returned in cases where the general result is a family of possible values.
+Calc follows section 12.5.3 of Steele's @dfn{Common Lisp, the Language},
+second edition, in these matters. This section will describe each
+function briefly; for a more detailed discussion (including some nifty
+diagrams), consult Steele's book.
+
+Note that the branch cuts for @code{arctan} and @code{arctanh} were
+changed between the first and second editions of Steele. Versions of
+Calc starting with 2.00 follow the second edition.
+
+The new branch cuts exactly match those of the HP-28/48 calculators.
+They also match those of Mathematica 1.2, except that Mathematica's
address@hidden cut is always in the right half of the complex plane,
+and its @code{arctanh} cut is always in the top half of the plane.
+Calc's cuts are continuous with quadrants I and III for @code{arctan},
+or II and IV for @code{arctanh}.
+
+Note: The current implementations of these functions with complex arguments
+are designed with proper behavior around the branch cuts in mind, @emph{not}
+efficiency or accuracy. You may need to increase the floating precision
+and wait a while to get suitable answers from them.
+
+For @samp{sqrt(a+bi)}: When @expr{a<0} and @expr{b} is small but positive
+or zero, the result is close to the @expr{+i} axis. For @expr{b} small and
+negative, the result is close to the @expr{-i} axis. The result always lies
+in the right half of the complex plane.
+
+For @samp{ln(a+bi)}: The real part is defined as @samp{ln(abs(a+bi))}.
+The imaginary part is defined as @samp{arg(a+bi) = arctan2(b,a)}.
+Thus the branch cuts for @code{sqrt} and @code{ln} both lie on the
+negative real axis.
+
+The following table describes these branch cuts in another way.
+If the real and imaginary parts of @expr{z} are as shown, then
+the real and imaginary parts of @expr{f(z)} will be as shown.
+Here @code{eps} stands for a small positive value; each
+occurrence of @code{eps} may stand for a different small value.
+
address@hidden
+ z sqrt(z) ln(z)
+----------------------------------------
+ +, 0 +, 0 any, 0
+ -, 0 0, + any, pi
+ -, +eps +eps, + +eps, +
+ -, -eps +eps, - +eps, -
address@hidden smallexample
+
+For @samp{z1^z2}: This is defined by @samp{exp(ln(z1)*z2)}.
+One interesting consequence of this is that @samp{(-8)^1:3} does
+not evaluate to @mathit{-2} as you might expect, but to the complex
+number @expr{(1., 1.732)}. Both of these are valid cube roots
+of @mathit{-8} (as is @expr{(1., -1.732)}); Calc chooses a perhaps
+less-obvious root for the sake of mathematical consistency.
+
+For @samp{arcsin(z)}: This is defined by @samp{-i*ln(i*z + sqrt(1-z^2))}.
+The branch cuts are on the real axis, less than @mathit{-1} and greater than 1.
+
+For @samp{arccos(z)}: This is defined by @samp{-i*ln(z + i*sqrt(1-z^2))},
+or equivalently by @samp{pi/2 - arcsin(z)}. The branch cuts are on
+the real axis, less than @mathit{-1} and greater than 1.
+
+For @samp{arctan(z)}: This is defined by
address@hidden(ln(1+i*z) - ln(1-i*z)) / (2*i)}. The branch cuts are on the
+imaginary axis, below @expr{-i} and above @expr{i}.
+
+For @samp{arcsinh(z)}: This is defined by @samp{ln(z + sqrt(1+z^2))}.
+The branch cuts are on the imaginary axis, below @expr{-i} and
+above @expr{i}.
+
+For @samp{arccosh(z)}: This is defined by
address@hidden(z + (z+1)*sqrt((z-1)/(z+1)))}. The branch cut is on the
+real axis less than 1.
+
+For @samp{arctanh(z)}: This is defined by @samp{(ln(1+z) - ln(1-z)) / 2}.
+The branch cuts are on the real axis, less than @mathit{-1} and greater than 1.
+
+The following tables for @code{arcsin}, @code{arccos}, and
address@hidden assume the current angular mode is Radians. The
+hyperbolic functions operate independently of the angular mode.
+
address@hidden
+ z arcsin(z) arccos(z)
+-------------------------------------------------------
+ (-1..1), 0 (-pi/2..pi/2), 0 (0..pi), 0
+ (-1..1), +eps (-pi/2..pi/2), +eps (0..pi), -eps
+ (-1..1), -eps (-pi/2..pi/2), -eps (0..pi), +eps
+ <-1, 0 -pi/2, + pi, -
+ <-1, +eps -pi/2 + eps, + pi - eps, -
+ <-1, -eps -pi/2 + eps, - pi - eps, +
+ >1, 0 pi/2, - 0, +
+ >1, +eps pi/2 - eps, + +eps, -
+ >1, -eps pi/2 - eps, - +eps, +
address@hidden smallexample
+
address@hidden
+ z arccosh(z) arctanh(z)
+-----------------------------------------------------
+ (-1..1), 0 0, (0..pi) any, 0
+ (-1..1), +eps +eps, (0..pi) any, +eps
+ (-1..1), -eps +eps, (-pi..0) any, -eps
+ <-1, 0 +, pi -, pi/2
+ <-1, +eps +, pi - eps -, pi/2 - eps
+ <-1, -eps +, -pi + eps -, -pi/2 + eps
+ >1, 0 +, 0 +, -pi/2
+ >1, +eps +, +eps +, pi/2 - eps
+ >1, -eps +, -eps +, -pi/2 + eps
address@hidden smallexample
+
address@hidden
+ z arcsinh(z) arctan(z)
+-----------------------------------------------------
+ 0, (-1..1) 0, (-pi/2..pi/2) 0, any
+ 0, <-1 -, -pi/2 -pi/2, -
+ +eps, <-1 +, -pi/2 + eps pi/2 - eps, -
+ -eps, <-1 -, -pi/2 + eps -pi/2 + eps, -
+ 0, >1 +, pi/2 pi/2, +
+ +eps, >1 +, pi/2 - eps pi/2 - eps, +
+ -eps, >1 -, pi/2 - eps -pi/2 + eps, +
address@hidden smallexample
+
+Finally, the following identities help to illustrate the relationship
+between the complex trigonometric and hyperbolic functions. They
+are valid everywhere, including on the branch cuts.
+
address@hidden
+sin(i*z) = i*sinh(z) arcsin(i*z) = i*arcsinh(z)
+cos(i*z) = cosh(z) arcsinh(i*z) = i*arcsin(z)
+tan(i*z) = i*tanh(z) arctan(i*z) = i*arctanh(z)
+sinh(i*z) = i*sin(z) cosh(i*z) = cos(z)
address@hidden smallexample
+
+The ``advanced math'' functions (gamma, Bessel, etc.@:) are also defined
+for general complex arguments, but their branch cuts and principal values
+are not rigorously specified at present.
+
address@hidden Random Numbers, Combinatorial Functions, Branch Cuts, Scientific
Functions
address@hidden Random Numbers
+
address@hidden
address@hidden k r
address@hidden calc-random
address@hidden random
+The @kbd{k r} (@code{calc-random}) address@hidden command produces
+random numbers of various sorts.
+
+Given a positive numeric prefix argument @expr{M}, it produces a random
+integer @expr{N} in the range
address@hidden @math{0 \le N < M}.
address@hidden @expr{0 <= N < M}.
+Each of the @expr{M} values appears with equal probability.
+
+With no numeric prefix argument, the @kbd{k r} command takes its argument
+from the stack instead. Once again, if this is a positive integer @expr{M}
+the result is a random integer less than @expr{M}. However, note that
+while numeric prefix arguments are limited to six digits or so, an @expr{M}
+taken from the stack can be arbitrarily large. If @expr{M} is negative,
+the result is a random integer in the range
address@hidden @math{M < N \le 0}.
address@hidden @expr{M < N <= 0}.
+
+If the value on the stack is a floating-point number @expr{M}, the result
+is a random floating-point number @expr{N} in the range
address@hidden @math{0 \le N < M}
address@hidden @expr{0 <= N < M}
+or
address@hidden @math{M < N \le 0},
address@hidden @expr{M < N <= 0},
+according to the sign of @expr{M}.
+
+If @expr{M} is zero, the result is a Gaussian-distributed random real
+number; the distribution has a mean of zero and a standard deviation
+of one. The algorithm used generates random numbers in pairs; thus,
+every other call to this function will be especially fast.
+
+If @expr{M} is an error form
address@hidden @math{m} @code{+/-} @math{\sigma}
address@hidden @samp{m +/- s}
+where @var{m} and
address@hidden @math{\sigma}
address@hidden @var{s}
+are both real numbers, the result uses a Gaussian distribution with mean
address@hidden and standard deviation
address@hidden @math{\sigma}.
address@hidden @var{s}.
+
+If @expr{M} is an interval form, the lower and upper bounds specify the
+acceptable limits of the random numbers. If both bounds are integers,
+the result is a random integer in the specified range. If either bound
+is floating-point, the result is a random real number in the specified
+range. If the interval is open at either end, the result will be sure
+not to equal that end value. (This makes a big difference for integer
+intervals, but for floating-point intervals it's relatively minor:
+with a precision of 6, @samp{random([1.0..2.0))} will return any of one
+million numbers from 1.00000 to 1.99999; @samp{random([1.0..2.0])} may
+additionally return 2.00000, but the probability of this happening is
+extremely small.)
+
+If @expr{M} is a vector, the result is one element taken at random from
+the vector. All elements of the vector are given equal probabilities.
+
address@hidden RandSeed
+The sequence of numbers produced by @kbd{k r} is completely random by
+default, i.e., the sequence is seeded each time you start Calc using
+the current time and other information. You can get a reproducible
+sequence by storing a particular ``seed value'' in the Calc variable
address@hidden Any integer will do for a seed; integers of from 1
+to 12 digits are good. If you later store a different integer into
address@hidden, Calc will switch to a different pseudo-random
+sequence. If you ``unstore'' @code{RandSeed}, Calc will re-seed itself
+from the current time. If you store the same integer that you used
+before back into @code{RandSeed}, you will get the exact same sequence
+of random numbers as before.
+
address@hidden calc-rrandom
+The @code{calc-rrandom} command (not on any key) produces a random real
+number between zero and one. It is equivalent to @samp{random(1.0)}.
+
address@hidden k a
address@hidden calc-random-again
+The @kbd{k a} (@code{calc-random-again}) command produces another random
+number, re-using the most recent value of @expr{M}. With a numeric
+prefix argument @var{n}, it produces @var{n} more random numbers using
+that value of @expr{M}.
+
address@hidden k h
address@hidden calc-shuffle
address@hidden shuffle
+The @kbd{k h} (@code{calc-shuffle}) command produces a vector of several
+random values with no duplicates. The value on the top of the stack
+specifies the set from which the random values are drawn, and may be any
+of the @expr{M} formats described above. The numeric prefix argument
+gives the length of the desired list. (If you do not provide a numeric
+prefix argument, the length of the list is taken from the top of the
+stack, and @expr{M} from second-to-top.)
+
+If @expr{M} is a floating-point number, zero, or an error form (so
+that the random values are being drawn from the set of real numbers)
+there is little practical difference between using @kbd{k h} and using
address@hidden r} several times. But if the set of possible values consists
+of just a few integers, or the elements of a vector, then there is
+a very real chance that multiple @kbd{k r}'s will produce the same
+number more than once. The @kbd{k h} command produces a vector whose
+elements are always distinct. (Actually, there is a slight exception:
+If @expr{M} is a vector, no given vector element will be drawn more
+than once, but if several elements of @expr{M} are equal, they may
+each make it into the result vector.)
+
+One use of @kbd{k h} is to rearrange a list at random. This happens
+if the prefix argument is equal to the number of values in the list:
address@hidden, 1.5, 2, 2.5, 3] 5 k h} might produce the permuted list
address@hidden, 1, 1.5, 3, 2]}. As a convenient feature, if the argument
address@hidden is negative it is replaced by the size of the set represented
+by @expr{M}. Naturally, this is allowed only when @expr{M} specifies
+a small discrete set of possibilities.
+
+To do the equivalent of @kbd{k h} but with duplications allowed,
+given @expr{M} on the stack and with @var{n} just entered as a numeric
+prefix, use @kbd{v b} to build a vector of copies of @expr{M}, then use
address@hidden M k r} to ``map'' the normal @kbd{k r} function over the
+elements of this vector. @xref{Matrix Functions}.
+
address@hidden
+* Random Number Generator:: (Complete description of Calc's algorithm)
address@hidden menu
+
address@hidden Random Number Generator, , Random Numbers, Random Numbers
address@hidden Random Number Generator
+
+Calc's random number generator uses several methods to ensure that
+the numbers it produces are highly random. Knuth's @emph{Art of
+Computer Programming}, Volume II, contains a thorough description
+of the theory of random number generators and their measurement and
+characterization.
+
+If @code{RandSeed} has no stored value, Calc calls Emacs' built-in
address@hidden function to get a stream of random numbers, which it
+then treats in various ways to avoid problems inherent in the simple
+random number generators that many systems use to implement @code{random}.
+
+When Calc's random number generator is first invoked, it ``seeds''
+the low-level random sequence using the time of day, so that the
+random number sequence will be different every time you use Calc.
+
+Since Emacs Lisp doesn't specify the range of values that will be
+returned by its @code{random} function, Calc exercises the function
+several times to estimate the range. When Calc subsequently uses
+the @code{random} function, it takes only 10 bits of the result
+near the most-significant end. (It avoids at least the bottom
+four bits, preferably more, and also tries to avoid the top two
+bits.) This strategy works well with the linear congruential
+generators that are typically used to implement @code{random}.
+
+If @code{RandSeed} contains an integer, Calc uses this integer to
+seed an ``additive congruential'' method (Knuth's algorithm 3.2.2A,
+computing
address@hidden @math{X_{n-55} - X_{n-24}}.
address@hidden @expr{X_n-55 - X_n-24}).
+This method expands the seed
+value into a large table which is maintained internally; the variable
address@hidden is changed from, e.g., 42 to the vector @expr{[42]}
+to indicate that the seed has been absorbed into this table. When
address@hidden contains a vector, @kbd{k r} and related commands
+continue to use the same internal table as last time. There is no
+way to extract the complete state of the random number generator
+so that you can restart it from any point; you can only restart it
+from the same initial seed value. A simple way to restart from the
+same seed is to type @kbd{s r RandSeed} to get the seed vector,
address@hidden u} to unpack it back into a number, then @kbd{s t RandSeed}
+to reseed the generator with that number.
+
+Calc uses a ``shuffling'' method as described in algorithm 3.2.2B
+of Knuth. It fills a table with 13 random 10-bit numbers. Then,
+to generate a new random number, it uses the previous number to
+index into the table, picks the value it finds there as the new
+random number, then replaces that table entry with a new value
+obtained from a call to the base random number generator (either
+the additive congruential generator or the @code{random} function
+supplied by the system). If there are any flaws in the base
+generator, shuffling will tend to even them out. But if the system
+provides an excellent @code{random} function, shuffling will not
+damage its randomness.
+
+To create a random integer of a certain number of digits, Calc
+builds the integer three decimal digits at a time. For each group
+of three digits, Calc calls its 10-bit shuffling random number generator
+(which returns a value from 0 to 1023); if the random value is 1000
+or more, Calc throws it out and tries again until it gets a suitable
+value.
+
+To create a random floating-point number with precision @var{p}, Calc
+simply creates a random @var{p}-digit integer and multiplies by
address@hidden @math{10^{-p}}.
address@hidden @expr{10^-p}.
+The resulting random numbers should be very clean, but note
+that relatively small numbers will have few significant random digits.
+In other words, with a precision of 12, you will occasionally get
+numbers on the order of
address@hidden @math{10^{-9}}
address@hidden @expr{10^-9}
+or
address@hidden @math{10^{-10}},
address@hidden @expr{10^-10},
+but those numbers will only have two or three random digits since they
+correspond to small integers times
address@hidden @math{10^{-12}}.
address@hidden @expr{10^-12}.
+
+To create a random integer in the interval @samp{[0 .. @var{m})}, Calc
+counts the digits in @var{m}, creates a random integer with three
+additional digits, then reduces modulo @var{m}. Unless @var{m} is a
+power of ten the resulting values will be very slightly biased toward
+the lower numbers, but this bias will be less than 0.1%. (For example,
+if @var{m} is 42, Calc will reduce a random integer less than 100000
+modulo 42 to get a result less than 42. It is easy to show that the
+numbers 40 and 41 will be only 2380/2381 as likely to result from this
+modulo operation as numbers 39 and below.) If @var{m} is a power of
+ten, however, the numbers should be completely unbiased.
+
+The Gaussian random numbers generated by @samp{random(0.0)} use the
+``polar'' method described in Knuth section 3.4.1C. This method
+generates a pair of Gaussian random numbers at a time, so only every
+other call to @samp{random(0.0)} will require significant calculations.
+
address@hidden Combinatorial Functions, Probability Distribution Functions,
Random Numbers, Scientific Functions
address@hidden Combinatorial Functions
+
address@hidden
+Commands relating to combinatorics and number theory begin with the
address@hidden key prefix.
+
address@hidden k g
address@hidden calc-gcd
address@hidden gcd
+The @kbd{k g} (@code{calc-gcd}) address@hidden command computes the
+Greatest Common Divisor of two integers. It also accepts fractions;
+the GCD of two fractions is defined by taking the GCD of the
+numerators, and the LCM of the denominators. This definition is
+consistent with the idea that @samp{a / gcd(a,x)} should yield an
+integer for any @samp{a} and @samp{x}. For other types of arguments,
+the operation is left in symbolic form.
+
address@hidden k l
address@hidden calc-lcm
address@hidden lcm
+The @kbd{k l} (@code{calc-lcm}) address@hidden command computes the
+Least Common Multiple of two integers or fractions. The product of
+the LCM and GCD of two numbers is equal to the product of the
+numbers.
+
address@hidden k E
address@hidden calc-extended-gcd
address@hidden egcd
+The @kbd{k E} (@code{calc-extended-gcd}) address@hidden command computes
+the GCD of two integers @expr{x} and @expr{y} and returns a vector
address@hidden, a, b]} where
address@hidden @math{g = \gcd(x,y) = a x + b y}.
address@hidden @expr{g = gcd(x,y) = a x + b y}.
+
address@hidden !
address@hidden calc-factorial
address@hidden fact
address@hidden
address@hidden @null
address@hidden ignore
address@hidden !
+The @kbd{!} (@code{calc-factorial}) address@hidden command computes the
+factorial of the number at the top of the stack. If the number is an
+integer, the result is an exact integer. If the number is an
+integer-valued float, the result is a floating-point approximation. If
+the number is a non-integral real number, the generalized factorial is used,
+as defined by the Euler Gamma function. Please note that computation of
+large factorials can be slow; using floating-point format will help
+since fewer digits must be maintained. The same is true of many of
+the commands in this section.
+
address@hidden k d
address@hidden calc-double-factorial
address@hidden dfact
address@hidden
address@hidden @null
address@hidden ignore
address@hidden !!
+The @kbd{k d} (@code{calc-double-factorial}) address@hidden command
+computes the ``double factorial'' of an integer. For an even integer,
+this is the product of even integers from 2 to @expr{N}. For an odd
+integer, this is the product of odd integers from 3 to @expr{N}. If
+the argument is an integer-valued float, the result is a floating-point
+approximation. This function is undefined for negative even integers.
+The notation @expr{N!!} is also recognized for double factorials.
+
address@hidden k c
address@hidden calc-choose
address@hidden choose
+The @kbd{k c} (@code{calc-choose}) address@hidden command computes the
+binomial coefficient @address@hidden, where @expr{M} is the number
+on the top of the stack and @expr{N} is second-to-top. If both arguments
+are integers, the result is an exact integer. Otherwise, the result is a
+floating-point approximation. The binomial coefficient is defined for all
+real numbers by
address@hidden @math{N! \over M! (N-M)!\,}.
address@hidden @expr{N! / M! (N-M)!}.
+
address@hidden H k c
address@hidden calc-perm
address@hidden perm
address@hidden
+The @kbd{H k c} (@code{calc-perm}) address@hidden command computes the
+number-of-permutations function @expr{N! / (N-M)!}.
address@hidden ifnottex
address@hidden
+The \kbd{H k c} (\code{calc-perm}) [\code{perm}] command computes the
+number-of-perm\-utations function $N! \over (N-M)!\,$.
address@hidden tex
+
address@hidden k b
address@hidden H k b
address@hidden calc-bernoulli-number
address@hidden bern
+The @kbd{k b} (@code{calc-bernoulli-number}) address@hidden command
+computes a given Bernoulli number. The value at the top of the stack
+is a nonnegative integer @expr{n} that specifies which Bernoulli number
+is desired. The @kbd{H k b} command computes a Bernoulli polynomial,
+taking @expr{n} from the second-to-top position and @expr{x} from the
+top of the stack. If @expr{x} is a variable or formula the result is
+a polynomial in @expr{x}; if @expr{x} is a number the result is a number.
+
address@hidden k e
address@hidden H k e
address@hidden calc-euler-number
address@hidden euler
+The @kbd{k e} (@code{calc-euler-number}) address@hidden command similarly
+computes an Euler number, and @address@hidden k e}} computes an Euler
polynomial.
+Bernoulli and Euler numbers occur in the Taylor expansions of several
+functions.
+
address@hidden k s
address@hidden H k s
address@hidden calc-stirling-number
address@hidden stir1
address@hidden stir2
+The @kbd{k s} (@code{calc-stirling-number}) address@hidden command
+computes a Stirling number of the first
address@hidden address@hidden@math{n \brack m},
address@hidden kind,
+given two integers @expr{n} and @expr{m} on the stack. The @kbd{H k s}
address@hidden command computes a Stirling number of the second
address@hidden address@hidden@math{n \brace m}.
address@hidden kind.
+These are the number of @expr{m}-cycle permutations of @expr{n} objects,
+and the number of ways to partition @expr{n} objects into @expr{m}
+non-empty sets, respectively.
+
address@hidden k p
address@hidden calc-prime-test
address@hidden Primes
+The @kbd{k p} (@code{calc-prime-test}) command checks if the integer on
+the top of the stack is prime. For integers less than eight million, the
+answer is always exact and reasonably fast. For larger integers, a
+probabilistic method is used (see Knuth vol. II, section 4.5.4, algorithm P).
+The number is first checked against small prime factors (up to 13). Then,
+any number of iterations of the algorithm are performed. Each step either
+discovers that the number is non-prime, or substantially increases the
+certainty that the number is prime. After a few steps, the chance that
+a number was mistakenly described as prime will be less than one percent.
+(Indeed, this is a worst-case estimate of the probability; in practice
+even a single iteration is quite reliable.) After the @kbd{k p} command,
+the number will be reported as definitely prime or non-prime if possible,
+or otherwise ``probably'' prime with a certain probability of error.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden prime
+The normal @kbd{k p} command performs one iteration of the primality
+test. Pressing @kbd{k p} repeatedly for the same integer will perform
+additional iterations. Also, @kbd{k p} with a numeric prefix performs
+the specified number of iterations. There is also an algebraic function
address@hidden(n)} or @samp{prime(n,iters)} which returns 1 if @expr{n}
+is (probably) prime and 0 if not.
+
address@hidden k f
address@hidden calc-prime-factors
address@hidden prfac
+The @kbd{k f} (@code{calc-prime-factors}) address@hidden command
+attempts to decompose an integer into its prime factors. For numbers up
+to 25 million, the answer is exact although it may take some time. The
+result is a vector of the prime factors in increasing order. For larger
+inputs, prime factors above 5000 may not be found, in which case the
+last number in the vector will be an unfactored integer greater than 25
+million (with a warning message). For negative integers, the first
+element of the list will be @mathit{-1}. For inputs @mathit{-1}, @mathit{0},
and
address@hidden, the result is a list of the same number.
+
address@hidden k n
address@hidden calc-next-prime
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden nextprime
+The @kbd{k n} (@code{calc-next-prime}) address@hidden command finds
+the next prime above a given number. Essentially, it searches by calling
address@hidden on successive integers until it finds one that
+passes the test. This is quite fast for integers less than eight million,
+but once the probabilistic test comes into play the search may be rather
+slow. Ordinarily this command stops for any prime that passes one iteration
+of the primality test. With a numeric prefix argument, a number must pass
+the specified number of iterations before the search stops. (This only
+matters when searching above eight million.) You can always use additional
address@hidden p} commands to increase your certainty that the number is indeed
+prime.
+
address@hidden I k n
address@hidden calc-prev-prime
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden prevprime
+The @kbd{I k n} (@code{calc-prev-prime}) address@hidden command
+analogously finds the next prime less than a given number.
+
address@hidden k t
address@hidden calc-totient
address@hidden totient
+The @kbd{k t} (@code{calc-totient}) address@hidden command computes the
+Euler ``totient''
address@hidden address@hidden@math{\phi(n)},
address@hidden function,
+the number of integers less than @expr{n} which
+are relatively prime to @expr{n}.
+
address@hidden k m
address@hidden calc-moebius
address@hidden moebius
+The @kbd{k m} (@code{calc-moebius}) address@hidden command computes the
address@hidden M@"obius @math{\mu}
address@hidden Moebius ``mu''
+function. If the input number is a product of @expr{k}
+distinct factors, this is @expr{(-1)^k}. If the input number has any
+duplicate factors (i.e., can be divided by the same prime more than once),
+the result is zero.
+
address@hidden Probability Distribution Functions, , Combinatorial Functions,
Scientific Functions
address@hidden Probability Distribution Functions
+
address@hidden
+The functions in this section compute various probability distributions.
+For continuous distributions, this is the integral of the probability
+density function from @expr{x} to infinity. (These are the ``upper
+tail'' distribution functions; there are also corresponding ``lower
+tail'' functions which integrate from minus infinity to @expr{x}.)
+For discrete distributions, the upper tail function gives the sum
+from @expr{x} to infinity; the lower tail function gives the sum
+from minus infinity up to, but not including,@w{ address@hidden
+
+To integrate from @expr{x} to @expr{y}, just use the distribution
+function twice and subtract. For example, the probability that a
+Gaussian random variable with mean 2 and standard deviation 1 will
+lie in the range from 2.5 to 2.8 is @samp{utpn(2.5,2,1) - utpn(2.8,2,1)}
+(``the probability that it is greater than 2.5, but not greater than 2.8''),
+or equivalently @samp{ltpn(2.8,2,1) - ltpn(2.5,2,1)}.
+
address@hidden k B
address@hidden I k B
address@hidden calc-utpb
address@hidden utpb
address@hidden ltpb
+The @kbd{k B} (@code{calc-utpb}) address@hidden function uses the
+binomial distribution. Push the parameters @var{n}, @var{p}, and
+then @var{x} onto the stack; the result (@samp{utpb(x,n,p)}) is the
+probability that an event will occur @var{x} or more times out
+of @var{n} trials, if its probability of occurring in any given
+trial is @var{p}. The @kbd{I k B} address@hidden function is
+the probability that the event will occur fewer than @var{x} times.
+
+The other probability distribution functions similarly take the
+form @kbd{k @var{X}} (@address@hidden) address@hidden@var{x}}]
+and @kbd{I k @var{X}} address@hidden@var{x}}], for various letters
address@hidden The arguments to the algebraic functions are the value of
+the random variable first, then whatever other parameters define the
+distribution. Note these are among the few Calc functions where the
+order of the arguments in algebraic form differs from the order of
+arguments as found on the stack. (The random variable comes last on
+the stack, so that you can type, e.g., @kbd{2 @key{RET} 1 @key{RET} 2.5
+k N address@hidden @key{DEL} 2.8 k N -}, using @address@hidden @key{DEL}} to
+recover the original arguments but substitute a new value for @expr{x}.)
+
address@hidden k C
address@hidden calc-utpc
address@hidden utpc
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden I k C
address@hidden
address@hidden @null
address@hidden ignore
address@hidden ltpc
+The @samp{utpc(x,v)} function uses the chi-square distribution with
address@hidden @math{\nu}
address@hidden @expr{v}
+degrees of freedom. It is the probability that a model is
+correct if its chi-square statistic is @expr{x}.
+
address@hidden k F
address@hidden calc-utpf
address@hidden utpf
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden I k F
address@hidden
address@hidden @null
address@hidden ignore
address@hidden ltpf
+The @samp{utpf(F,v1,v2)} function uses the F distribution, used in
+various statistical tests. The parameters
address@hidden @math{\nu_1}
address@hidden @expr{v1}
+and
address@hidden @math{\nu_2}
address@hidden @expr{v2}
+are the degrees of freedom in the numerator and denominator,
+respectively, used in computing the statistic @expr{F}.
+
address@hidden k N
address@hidden calc-utpn
address@hidden utpn
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden I k N
address@hidden
address@hidden @null
address@hidden ignore
address@hidden ltpn
+The @samp{utpn(x,m,s)} function uses a normal (Gaussian) distribution
+with mean @expr{m} and standard deviation
address@hidden @math{\sigma}.
address@hidden @expr{s}.
+It is the probability that such a normal-distributed random variable
+would exceed @expr{x}.
+
address@hidden k P
address@hidden calc-utpp
address@hidden utpp
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden I k P
address@hidden
address@hidden @null
address@hidden ignore
address@hidden ltpp
+The @samp{utpp(n,x)} function uses a Poisson distribution with
+mean @expr{x}. It is the probability that @expr{n} or more such
+Poisson random events will occur.
+
address@hidden k T
address@hidden calc-ltpt
address@hidden utpt
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden I k T
address@hidden
address@hidden @null
address@hidden ignore
address@hidden ltpt
+The @samp{utpt(t,v)} function uses the Student's ``t'' distribution
+with
address@hidden @math{\nu}
address@hidden @expr{v}
+degrees of freedom. It is the probability that a
+t-distributed random variable will be greater than @expr{t}.
+(Note: This computes the distribution function
address@hidden @math{A(t|\nu)}
address@hidden @expr{A(t|v)}
+where
address@hidden @math{A(0|\nu) = 1}
address@hidden @expr{A(0|v) = 1}
+and
address@hidden @math{A(\infty|\nu) \to 0}.
address@hidden @expr{A(inf|v) -> 0}.
+The @code{UTPT} operation on the HP-48 uses a different definition which
+returns half of Calc's value: @samp{UTPT(t,v) = .5*utpt(t,v)}.)
+
+While Calc does not provide inverses of the probability distribution
+functions, the @kbd{a R} command can be used to solve for the inverse.
+Since the distribution functions are monotonic, @kbd{a R} is guaranteed
+to be able to find a solution given any initial guess.
address@hidden Solutions}.
+
address@hidden Matrix Functions, Algebra, Scientific Functions, Top
address@hidden Vector/Matrix Functions
+
address@hidden
+Many of the commands described here begin with the @kbd{v} prefix.
+(For convenience, the address@hidden prefix is equivalent to @kbd{v}.)
+The commands usually apply to both plain vectors and matrices; some
+apply only to matrices or only to square matrices. If the argument
+has the wrong dimensions the operation is left in symbolic form.
+
+Vectors are entered and displayed using @samp{[a,b,c]} notation.
+Matrices are vectors of which all elements are vectors of equal length.
+(Though none of the standard Calc commands use this concept, a
+three-dimensional matrix or rank-3 tensor could be defined as a
+vector of matrices, and so on.)
+
address@hidden
+* Packing and Unpacking::
+* Building Vectors::
+* Extracting Elements::
+* Manipulating Vectors::
+* Vector and Matrix Arithmetic::
+* Set Operations::
+* Statistical Operations::
+* Reducing and Mapping::
+* Vector and Matrix Formats::
address@hidden menu
+
address@hidden Packing and Unpacking, Building Vectors, Matrix Functions,
Matrix Functions
address@hidden Packing and Unpacking
+
address@hidden
+Calc's ``pack'' and ``unpack'' commands collect stack entries to build
+composite objects such as vectors and complex numbers. They are
+described in this chapter because they are most often used to build
+vectors.
+
address@hidden v p
address@hidden calc-pack
+The @kbd{v p} (@code{calc-pack}) address@hidden command collects several
+elements from the stack into a matrix, complex number, HMS form, error
+form, etc. It uses a numeric prefix argument to specify the kind of
+object to be built; this argument is referred to as the ``packing mode.''
+If the packing mode is a nonnegative integer, a vector of that
+length is created. For example, @kbd{C-u 5 v p} will pop the top
+five stack elements and push back a single vector of those five
+elements. (@kbd{C-u 0 v p} simply creates an empty vector.)
+
+The same effect can be had by pressing @kbd{[} to push an incomplete
+vector on the stack, using @key{TAB} (@code{calc-roll-down}) to sneak
+the incomplete object up past a certain number of elements, and
+then pressing @kbd{]} to complete the vector.
+
+Negative packing modes create other kinds of composite objects:
+
address@hidden @cite
address@hidden -1
+Two values are collected to build a complex number. For example,
address@hidden @key{RET} 7 C-u -1 v p} creates the complex number
address@hidden(5, 7)}. The result is always a rectangular complex
+number. The two input values must both be real numbers,
+i.e., integers, fractions, or floats. If they are not, Calc
+will instead build a formula like @samp{a + (0, 1) b}. (The
+other packing modes also create a symbolic answer if the
+components are not suitable.)
+
address@hidden -2
+Two values are collected to build a polar complex number.
+The first is the magnitude; the second is the phase expressed
+in either degrees or radians according to the current angular
+mode.
+
address@hidden -3
+Three values are collected into an HMS form. The first
+two values (hours and minutes) must be integers or
+integer-valued floats. The third value may be any real
+number.
+
address@hidden -4
+Two values are collected into an error form. The inputs
+may be real numbers or formulas.
+
address@hidden -5
+Two values are collected into a modulo form. The inputs
+must be real numbers.
+
address@hidden -6
+Two values are collected into the interval @samp{[a .. b]}.
+The inputs may be real numbers, HMS or date forms, or formulas.
+
address@hidden -7
+Two values are collected into the interval @samp{[a .. b)}.
+
address@hidden -8
+Two values are collected into the interval @samp{(a .. b]}.
+
address@hidden -9
+Two values are collected into the interval @samp{(a .. b)}.
+
address@hidden -10
+Two integer values are collected into a fraction.
+
address@hidden -11
+Two values are collected into a floating-point number.
+The first is the mantissa; the second, which must be an
+integer, is the exponent. The result is the mantissa
+times ten to the power of the exponent.
+
address@hidden -12
+This is treated the same as @mathit{-11} by the @kbd{v p} command.
+When unpacking, @mathit{-12} specifies that a floating-point mantissa
+is desired.
+
address@hidden -13
+A real number is converted into a date form.
+
address@hidden -14
+Three numbers (year, month, day) are packed into a pure date form.
+
address@hidden -15
+Six numbers are packed into a date/time form.
address@hidden table
+
+With any of the two-input negative packing modes, either or both
+of the inputs may be vectors. If both are vectors of the same
+length, the result is another vector made by packing corresponding
+elements of the input vectors. If one input is a vector and the
+other is a plain number, the number is packed along with each vector
+element to produce a new vector. For example, @kbd{C-u -4 v p}
+could be used to convert a vector of numbers and a vector of errors
+into a single vector of error forms; @kbd{C-u -5 v p} could convert
+a vector of numbers and a single number @var{M} into a vector of
+numbers modulo @var{M}.
+
+If you don't give a prefix argument to @kbd{v p}, it takes
+the packing mode from the top of the stack. The elements to
+be packed then begin at stack level 2. Thus
address@hidden @key{RET} 2 @key{RET} 4 n v p} is another way to
+enter the error form @samp{1 +/- 2}.
+
+If the packing mode taken from the stack is a vector, the result is a
+matrix with the dimensions specified by the elements of the vector,
+which must each be integers. For example, if the packing mode is
address@hidden, 3]}, then six numbers will be taken from the stack and
+returned in the form @address@hidden, b, c]}, [d, e, f]]}.
+
+If any elements of the vector are negative, other kinds of
+packing are done at that level as described above. For
+example, @samp{[2, 3, -4]} takes 12 objects and creates a
address@hidden @math{2\times3}
address@hidden 2x3
+matrix of error forms: @samp{[[a +/- b, c +/- d ... ]]}.
+Also, @samp{[-4, -10]} will convert four integers into an
+error form consisting of two fractions: @samp{a:b +/- c:d}.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden pack
+There is an equivalent algebraic function,
address@hidden(@var{mode}, @var{items})} where @var{mode} is a
+packing mode (an integer or a vector of integers) and @var{items}
+is a vector of objects to be packed (re-packed, really) according
+to that mode. For example, @samp{pack([3, -4], [a,b,c,d,e,f])}
+yields @samp{[a +/- b, @w{c +/- d}, e +/- f]}. The function is
+left in symbolic form if the packing mode is invalid, or if the
+number of data items does not match the number of items required
+by the mode.
+
address@hidden v u
address@hidden calc-unpack
+The @kbd{v u} (@code{calc-unpack}) command takes the vector, complex
+number, HMS form, or other composite object on the top of the stack and
+``unpacks'' it, pushing each of its elements onto the stack as separate
+objects. Thus, it is the ``inverse'' of @kbd{v p}. If the value
+at the top of the stack is a formula, @kbd{v u} unpacks it by pushing
+each of the arguments of the top-level operator onto the stack.
+
+You can optionally give a numeric prefix argument to @kbd{v u}
+to specify an explicit (un)packing mode. If the packing mode is
+negative and the input is actually a vector or matrix, the result
+will be two or more similar vectors or matrices of the elements.
+For example, given the vector @address@hidden +/- b}, c^2, d +/- 7]},
+the result of @kbd{C-u -4 v u} will be the two vectors
address@hidden, c^2, d]} and @address@hidden, 0, 7]}}.
+
+Note that the prefix argument can have an effect even when the input is
+not a vector. For example, if the input is the number @mathit{-5}, then
address@hidden -1 v u} yields @mathit{-5} and 0 (the components of @mathit{-5}
+when viewed as a rectangular complex number); @kbd{C-u -2 v u} yields 5
+and 180 (assuming Degrees mode); and @kbd{C-u -10 v u} yields @mathit{-5}
+and 1 (the numerator and denominator of @mathit{-5}, viewed as a rational
+number). Plain @kbd{v u} with this input would complain that the input
+is not a composite object.
+
+Unpacking mode @mathit{-11} converts a float into an integer mantissa and
+an integer exponent, where the mantissa is not divisible by 10
+(except that 0.0 is represented by a mantissa and exponent of 0).
+Unpacking mode @mathit{-12} converts a float into a floating-point mantissa
+and integer exponent, where the mantissa (for non-zero numbers)
+is guaranteed to lie in the range [1 .. 10). In both cases,
+the mantissa is shifted left or right (and the exponent adjusted
+to compensate) in order to satisfy these constraints.
+
+Positive unpacking modes are treated differently than for @kbd{v p}.
+A mode of 1 is much like plain @kbd{v u} with no prefix argument,
+except that in addition to the components of the input object,
+a suitable packing mode to re-pack the object is also pushed.
+Thus, @kbd{C-u 1 v u} followed by @kbd{v p} will re-build the
+original object.
+
+A mode of 2 unpacks two levels of the object; the resulting
+re-packing mode will be a vector of length 2. This might be used
+to unpack a matrix, say, or a vector of error forms. Higher
+unpacking modes unpack the input even more deeply.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden unpack
+There are two algebraic functions analogous to @kbd{v u}.
+The @samp{unpack(@var{mode}, @var{item})} function unpacks the
address@hidden using the given @var{mode}, returning the result as
+a vector of components. Here the @var{mode} must be an
+integer, not a vector. For example, @samp{unpack(-4, a +/- b)}
+returns @samp{[a, b]}, as does @samp{unpack(1, a +/- b)}.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden unpackt
+The @code{unpackt} function is like @code{unpack} but instead
+of returning a simple vector of items, it returns a vector of
+two things: The mode, and the vector of items. For example,
address@hidden(1, 2:3 +/- 1:4)} returns @samp{[-4, [2:3, 1:4]]},
+and @samp{unpackt(2, 2:3 +/- 1:4)} returns @samp{[[-4, -10], [2, 3, 1, 4]]}.
+The identity for re-building the original object is
address@hidden(pack, unpackt(@var{n}, @var{x})) = @var{x}}. (The
address@hidden function builds a function call given the function
+name and a vector of arguments.)
+
address@hidden Numerator of a fraction, extracting
+Subscript notation is a useful way to extract a particular part
+of an object. For example, to get the numerator of a rational
+number, you can use @samp{unpack(-10, @var{x})_1}.
+
address@hidden Building Vectors, Extracting Elements, Packing and Unpacking,
Matrix Functions
address@hidden Building Vectors
+
address@hidden
+Vectors and matrices can be added,
+subtracted, multiplied, and divided; @pxref{Basic Arithmetic}.
+
address@hidden |
address@hidden calc-concat
address@hidden
address@hidden @null
address@hidden ignore
address@hidden |
+The @kbd{|} (@code{calc-concat}) address@hidden command ``concatenates'' two
vectors
+into one. For example, after @address@hidden 1 , 2 ]} [ 3 , 4 ] |}, the stack
+will contain the single vector @samp{[1, 2, 3, 4]}. If the arguments
+are matrices, the rows of the first matrix are concatenated with the
+rows of the second. (In other words, two matrices are just two vectors
+of row-vectors as far as @kbd{|} is concerned.)
+
+If either argument to @kbd{|} is a scalar (a non-vector), it is treated
+like a one-element vector for purposes of concatenation: @kbd{1 [ 2 , 3 ] |}
+produces the vector @samp{[1, 2, 3]}. Likewise, if one argument is a
+matrix and the other is a plain vector, the vector is treated as a
+one-row matrix.
+
address@hidden H |
address@hidden append
+The @kbd{H |} (@code{calc-append}) address@hidden command concatenates
+two vectors without any special cases. Both inputs must be vectors.
+Whether or not they are matrices is not taken into account. If either
+argument is a scalar, the @code{append} function is left in symbolic form.
+See also @code{cons} and @code{rcons} below.
+
address@hidden I |
address@hidden H I |
+The @kbd{I |} and @kbd{H I |} commands are similar, but they use their
+two stack arguments in the opposite order. Thus @kbd{I |} is equivalent
+to @address@hidden |}, but possibly more convenient and also a bit faster.
+
address@hidden v d
address@hidden calc-diag
address@hidden diag
+The @kbd{v d} (@code{calc-diag}) address@hidden function builds a diagonal
+square matrix. The optional numeric prefix gives the number of rows
+and columns in the matrix. If the value at the top of the stack is a
+vector, the elements of the vector are used as the diagonal elements; the
+prefix, if specified, must match the size of the vector. If the value on
+the stack is a scalar, it is used for each element on the diagonal, and
+the prefix argument is required.
+
+To build a constant square matrix, e.g., a
address@hidden @math{3\times3}
address@hidden 3x3
+matrix filled with ones, use @kbd{0 M-3 v d 1 +}, i.e., build a zero
+matrix first and then add a constant value to that matrix. (Another
+alternative would be to use @kbd{v b} and @kbd{v a}; see below.)
+
address@hidden v i
address@hidden calc-ident
address@hidden idn
+The @kbd{v i} (@code{calc-ident}) address@hidden function builds an identity
+matrix of the specified size. It is a convenient form of @kbd{v d}
+where the diagonal element is always one. If no prefix argument is given,
+this command prompts for one.
+
+In algebraic notation, @samp{idn(a,n)} acts much like @samp{diag(a,n)},
+except that @expr{a} is required to be a scalar (non-vector) quantity.
+If @expr{n} is omitted, @samp{idn(a)} represents @expr{a} times an
+identity matrix of unknown size. Calc can operate algebraically on
+such generic identity matrices, and if one is combined with a matrix
+whose size is known, it is converted automatically to an identity
+matrix of a suitable matching size. The @kbd{v i} command with an
+argument of zero creates a generic identity matrix, @samp{idn(1)}.
+Note that in dimensioned Matrix mode (@pxref{Matrix Mode}), generic
+identity matrices are immediately expanded to the current default
+dimensions.
+
address@hidden v x
address@hidden calc-index
address@hidden index
+The @kbd{v x} (@code{calc-index}) address@hidden function builds a vector
+of consecutive integers from 1 to @var{n}, where @var{n} is the numeric
+prefix argument. If you do not provide a prefix argument, you will be
+prompted to enter a suitable number. If @var{n} is negative, the result
+is a vector of negative integers from @var{n} to @mathit{-1}.
+
+With a prefix argument of just @kbd{C-u}, the @kbd{v x} command takes
+three values from the stack: @var{n}, @var{start}, and @var{incr} (with
address@hidden at top-of-stack). Counting starts at @var{start} and increases
+by @var{incr} for successive vector elements. If @var{start} or @var{n}
+is in floating-point format, the resulting vector elements will also be
+floats. Note that @var{start} and @var{incr} may in fact be any kind
+of numbers or formulas.
+
+When @var{start} and @var{incr} are specified, a negative @var{n} has a
+different interpretation: It causes a geometric instead of arithmetic
+sequence to be generated. For example, @samp{index(-3, a, b)} produces
address@hidden, a b, a b^2]}. If you omit @var{incr} in the algebraic form,
address@hidden(@var{n}, @var{start})}, the default value for @var{incr}
+is one for positive @var{n} or two for negative @var{n}.
+
address@hidden v b
address@hidden calc-build-vector
address@hidden cvec
+The @kbd{v b} (@code{calc-build-vector}) address@hidden function builds a
+vector of @var{n} copies of the value on the top of the stack, where @var{n}
+is the numeric prefix argument. In algebraic formulas, @samp{cvec(x,n,m)}
+can also be used to build an @address@hidden matrix of copies of @var{x}.
+(Interactively, just use @kbd{v b} twice: once to build a row, then again
+to build a matrix of copies of that row.)
+
address@hidden v h
address@hidden I v h
address@hidden calc-head
address@hidden calc-tail
address@hidden head
address@hidden tail
+The @kbd{v h} (@code{calc-head}) address@hidden function returns the first
+element of a vector. The @kbd{I v h} (@code{calc-tail}) address@hidden
+function returns the vector with its first element removed. In both
+cases, the argument must be a non-empty vector.
+
address@hidden v k
address@hidden calc-cons
address@hidden cons
+The @kbd{v k} (@code{calc-cons}) address@hidden function takes a value @var{h}
+and a vector @var{t} from the stack, and produces the vector whose head is
address@hidden and whose tail is @var{t}. This is similar to @kbd{|}, except
+if @var{h} is itself a vector, @kbd{|} will concatenate the two vectors
+whereas @code{cons} will insert @var{h} at the front of the vector @var{t}.
+
address@hidden H v h
address@hidden rhead
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden H I v h
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H v k
address@hidden
address@hidden @null
address@hidden ignore
address@hidden rtail
address@hidden
address@hidden @null
address@hidden ignore
address@hidden rcons
+Each of these three functions also accepts the Hyperbolic flag address@hidden,
address@hidden, @code{rcons}] in which case @var{t} instead represents
+the @emph{last} single element of the vector, with @var{h}
+representing the remainder of the vector. Thus the vector
address@hidden, b, c, d] = cons(a, [b, c, d]) = rcons([a, b, c], d)}.
+Also, @samp{head([a, b, c, d]) = a}, @samp{tail([a, b, c, d]) = [b, c, d]},
address@hidden([a, b, c, d]) = [a, b, c]}, and @samp{rtail([a, b, c, d]) = d}.
+
address@hidden Extracting Elements, Manipulating Vectors, Building Vectors,
Matrix Functions
address@hidden Extracting Vector Elements
+
address@hidden
address@hidden v r
address@hidden calc-mrow
address@hidden mrow
+The @kbd{v r} (@code{calc-mrow}) address@hidden command extracts one row of
+the matrix on the top of the stack, or one element of the plain vector on
+the top of the stack. The row or element is specified by the numeric
+prefix argument; the default is to prompt for the row or element number.
+The matrix or vector is replaced by the specified row or element in the
+form of a vector or scalar, respectively.
+
address@hidden Permutations, applying
+With a prefix argument of @kbd{C-u} only, @kbd{v r} takes the index of
+the element or row from the top of the stack, and the vector or matrix
+from the second-to-top position. If the index is itself a vector of
+integers, the result is a vector of the corresponding elements of the
+input vector, or a matrix of the corresponding rows of the input matrix.
+This command can be used to obtain any permutation of a vector.
+
+With @kbd{C-u}, if the index is an interval form with integer components,
+it is interpreted as a range of indices and the corresponding subvector or
+submatrix is returned.
+
address@hidden Subscript notation
address@hidden a _
address@hidden calc-subscript
address@hidden subscr
address@hidden _
+Subscript notation in algebraic formulas (@samp{a_b}) stands for the
+Calc function @code{subscr}, which is synonymous with @code{mrow}.
+Thus, @samp{[x, y, z]_k} produces @expr{x}, @expr{y}, or @expr{z} if
address@hidden is one, two, or three, respectively. A double subscript
+(@samp{M_i_j}, equivalent to @samp{subscr(subscr(M, i), j)}) will
+access the element at row @expr{i}, column @expr{j} of a matrix.
+The @kbd{a _} (@code{calc-subscript}) command creates a subscript
+formula @samp{a_b} out of two stack entries. (It is on the @kbd{a}
+``algebra'' prefix because subscripted variables are often used
+purely as an algebraic notation.)
+
address@hidden mrrow
+Given a negative prefix argument, @kbd{v r} instead deletes one row or
+element from the matrix or vector on the top of the stack. Thus
address@hidden 2 v r} replaces a matrix with its second row, but @kbd{C-u -2 v
r}
+replaces the matrix with the same matrix with its second row removed.
+In algebraic form this function is called @code{mrrow}.
+
address@hidden getdiag
+Given a prefix argument of zero, @kbd{v r} extracts the diagonal elements
+of a square matrix in the form of a vector. In algebraic form this
+function is called @code{getdiag}.
+
address@hidden v c
address@hidden calc-mcol
address@hidden mcol
address@hidden mrcol
+The @kbd{v c} (@code{calc-mcol}) address@hidden or @code{mrcol}] command is
+the analogous operation on columns of a matrix. Given a plain vector
+it extracts (or removes) one element, just like @kbd{v r}. If the
+index in @kbd{C-u v c} is an interval or vector and the argument is a
+matrix, the result is a submatrix with only the specified columns
+retained (and possibly permuted in the case of a vector index).
+
+To extract a matrix element at a given row and column, use @kbd{v r} to
+extract the row as a vector, then @kbd{v c} to extract the column element
+from that vector. In algebraic formulas, it is often more convenient to
+use subscript notation: @samp{m_i_j} gives row @expr{i}, column @expr{j}
+of matrix @expr{m}.
+
address@hidden v s
address@hidden calc-subvector
address@hidden subvec
+The @kbd{v s} (@code{calc-subvector}) address@hidden command extracts
+a subvector of a vector. The arguments are the vector, the starting
+index, and the ending index, with the ending index in the top-of-stack
+position. The starting index indicates the first element of the vector
+to take. The ending index indicates the first element @emph{past} the
+range to be taken. Thus, @samp{subvec([a, b, c, d, e], 2, 4)} produces
+the subvector @samp{[b, c]}. You could get the same result using
address@hidden([a, b, c, d, e], @w{[2 .. 4)})}.
+
+If either the start or the end index is zero or negative, it is
+interpreted as relative to the end of the vector. Thus
address@hidden([a, b, c, d, e], 2, -2)} also produces @samp{[b, c]}. In
+the algebraic form, the end index can be omitted in which case it
+is taken as zero, i.e., elements from the starting element to the
+end of the vector are used. The infinity symbol, @code{inf}, also
+has this effect when used as the ending index.
+
address@hidden I v s
address@hidden rsubvec
+With the Inverse flag, @kbd{I v s} address@hidden removes a subvector
+from a vector. The arguments are interpreted the same as for the
+normal @kbd{v s} command. Thus, @samp{rsubvec([a, b, c, d, e], 2, 4)}
+produces @samp{[a, d, e]}. It is always true that @code{subvec} and
address@hidden return complementary parts of the input vector.
+
address@hidden Subformulas}, for an alternative way to operate on
+vectors one element at a time.
+
address@hidden Manipulating Vectors, Vector and Matrix Arithmetic, Extracting
Elements, Matrix Functions
address@hidden Manipulating Vectors
+
address@hidden
address@hidden v l
address@hidden calc-vlength
address@hidden vlen
+The @kbd{v l} (@code{calc-vlength}) address@hidden command computes the
+length of a vector. The length of a non-vector is considered to be zero.
+Note that matrices are just vectors of vectors for the purposes of this
+command.
+
address@hidden H v l
address@hidden mdims
+With the Hyperbolic flag, @kbd{H v l} address@hidden computes a vector
+of the dimensions of a vector, matrix, or higher-order object. For
+example, @samp{mdims([[a,b,c],[d,e,f]])} returns @samp{[2, 3]} since
+its argument is a
address@hidden @math{2\times3}
address@hidden 2x3
+matrix.
+
address@hidden v f
address@hidden calc-vector-find
address@hidden find
+The @kbd{v f} (@code{calc-vector-find}) address@hidden command searches
+along a vector for the first element equal to a given target. The target
+is on the top of the stack; the vector is in the second-to-top position.
+If a match is found, the result is the index of the matching element.
+Otherwise, the result is zero. The numeric prefix argument, if given,
+allows you to select any starting index for the search.
+
address@hidden v a
address@hidden calc-arrange-vector
address@hidden arrange
address@hidden Arranging a matrix
address@hidden Reshaping a matrix
address@hidden Flattening a matrix
+The @kbd{v a} (@code{calc-arrange-vector}) address@hidden command
+rearranges a vector to have a certain number of columns and rows. The
+numeric prefix argument specifies the number of columns; if you do not
+provide an argument, you will be prompted for the number of columns.
+The vector or matrix on the top of the stack is @dfn{flattened} into a
+plain vector. If the number of columns is nonzero, this vector is
+then formed into a matrix by taking successive groups of @var{n} elements.
+If the number of columns does not evenly divide the number of elements
+in the vector, the last row will be short and the result will not be
+suitable for use as a matrix. For example, with the matrix
address@hidden, 2], @w{[3, 4]}]} on the stack, @kbd{v a 4} produces
address@hidden, 2, 3, 4]]} (a
address@hidden @math{1\times4}
address@hidden 1x4
+matrix), @kbd{v a 1} produces @samp{[[1], [2], [3], [4]]} (a
address@hidden @math{4\times1}
address@hidden 4x1
+matrix), @kbd{v a 2} produces @samp{[[1, 2], [3, 4]]} (the original
address@hidden @math{2\times2}
address@hidden 2x2
+matrix), @address@hidden a 3}} produces @samp{[[1, 2, 3], [4]]} (not a
+matrix), and @kbd{v a 0} produces the flattened list
address@hidden, 2, @w{3, 4}]}.
+
address@hidden Sorting data
address@hidden V S
address@hidden I V S
address@hidden calc-sort
address@hidden sort
address@hidden rsort
+The @kbd{V S} (@code{calc-sort}) address@hidden command sorts the elements of
+a vector into increasing order. Real numbers, real infinities, and
+constant interval forms come first in this ordering; next come other
+kinds of numbers, then variables (in alphabetical order), then finally
+come formulas and other kinds of objects; these are sorted according
+to a kind of lexicographic ordering with the useful property that
+one vector is less or greater than another if the first corresponding
+unequal elements are less or greater, respectively. Since quoted strings
+are stored by Calc internally as vectors of ASCII character codes
+(@pxref{Strings}), this means vectors of strings are also sorted into
+alphabetical order by this command.
+
+The @kbd{I V S} address@hidden command sorts a vector into decreasing order.
+
address@hidden Permutation, inverse of
address@hidden Inverse of permutation
address@hidden Index tables
address@hidden Rank tables
address@hidden V G
address@hidden I V G
address@hidden calc-grade
address@hidden grade
address@hidden rgrade
+The @kbd{V G} (@code{calc-grade}) address@hidden, @code{rgrade}] command
+produces an index table or permutation vector which, if applied to the
+input vector (as the index of @kbd{C-u v r}, say), would sort the vector.
+A permutation vector is just a vector of integers from 1 to @var{n}, where
+each integer occurs exactly once. One application of this is to sort a
+matrix of data rows using one column as the sort key; extract that column,
+grade it with @kbd{V G}, then use the result to reorder the original matrix
+with @kbd{C-u v r}. Another interesting property of the @code{V G} command
+is that, if the input is itself a permutation vector, the result will
+be the inverse of the permutation. The inverse of an index table is
+a rank table, whose @var{k}th element says where the @var{k}th original
+vector element will rest when the vector is sorted. To get a rank
+table, just use @kbd{V G V G}.
+
+With the Inverse flag, @kbd{I V G} produces an index table that would
+sort the input into decreasing order. Note that @kbd{V S} and @kbd{V G}
+use a ``stable'' sorting algorithm, i.e., any two elements which are equal
+will not be moved out of their original order. Generally there is no way
+to tell with @kbd{V S}, since two elements which are equal look the same,
+but with @kbd{V G} this can be an important issue. In the matrix-of-rows
+example, suppose you have names and telephone numbers as two columns and
+you wish to sort by phone number primarily, and by name when the numbers
+are equal. You can sort the data matrix by names first, and then again
+by phone numbers. Because the sort is stable, any two rows with equal
+phone numbers will remain sorted by name even after the second sort.
+
address@hidden Histograms
address@hidden V H
address@hidden calc-histogram
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden histogram
+The @kbd{V H} (@code{calc-histogram}) address@hidden command builds a
+histogram of a vector of numbers. Vector elements are assumed to be
+integers or real numbers in the range address@hidden) for some ``number of
+bins'' @var{n}, which is the numeric prefix argument given to the
+command. The result is a vector of @var{n} counts of how many times
+each value appeared in the original vector. Non-integers in the input
+are rounded down to integers. Any vector elements outside the specified
+range are ignored. (You can tell if elements have been ignored by noting
+that the counts in the result vector don't add up to the length of the
+input vector.)
+
address@hidden H V H
+With the Hyperbolic flag, @kbd{H V H} pulls two vectors from the stack.
+The second-to-top vector is the list of numbers as before. The top
+vector is an equal-sized list of ``weights'' to attach to the elements
+of the data vector. For example, if the first data element is 4.2 and
+the first weight is 10, then 10 will be added to bin 4 of the result
+vector. Without the hyperbolic flag, every element has a weight of one.
+
address@hidden v t
address@hidden calc-transpose
address@hidden trn
+The @kbd{v t} (@code{calc-transpose}) address@hidden command computes
+the transpose of the matrix at the top of the stack. If the argument
+is a plain vector, it is treated as a row vector and transposed into
+a one-column matrix.
+
address@hidden v v
address@hidden calc-reverse-vector
address@hidden rev
+The @kbd{v v} (@code{calc-reverse-vector}) address@hidden command reverses
+a vector end-for-end. Given a matrix, it reverses the order of the rows.
+(To reverse the columns instead, just use @kbd{v t v v v t}. The same
+principle can be used to apply other vector commands to the columns of
+a matrix.)
+
address@hidden v m
address@hidden calc-mask-vector
address@hidden vmask
+The @kbd{v m} (@code{calc-mask-vector}) address@hidden command uses
+one vector as a mask to extract elements of another vector. The mask
+is in the second-to-top position; the target vector is on the top of
+the stack. These vectors must have the same length. The result is
+the same as the target vector, but with all elements which correspond
+to zeros in the mask vector deleted. Thus, for example,
address@hidden([1, 0, 1, 0, 1], [a, b, c, d, e])} produces @samp{[a, c, e]}.
address@hidden Operations}.
+
address@hidden v e
address@hidden calc-expand-vector
address@hidden vexp
+The @kbd{v e} (@code{calc-expand-vector}) address@hidden command
+expands a vector according to another mask vector. The result is a
+vector the same length as the mask, but with nonzero elements replaced
+by successive elements from the target vector. The length of the target
+vector is normally the number of nonzero elements in the mask. If the
+target vector is longer, its last few elements are lost. If the target
+vector is shorter, the last few nonzero mask elements are left
+unreplaced in the result. Thus @samp{vexp([2, 0, 3, 0, 7], [a, b])}
+produces @samp{[a, 0, b, 0, 7]}.
+
address@hidden H v e
+With the Hyperbolic flag, @kbd{H v e} takes a filler value from the
+top of the stack; the mask and target vectors come from the third and
+second elements of the stack. This filler is used where the mask is
+zero: @samp{vexp([2, 0, 3, 0, 7], [a, b], z)} produces
address@hidden, z, c, z, 7]}. If the filler value is itself a vector,
+then successive values are taken from it, so that the effect is to
+interleave two vectors according to the mask:
address@hidden([2, 0, 3, 7, 0, 0], [a, b], [x, y])} produces
address@hidden, x, b, 7, y, 0]}.
+
+Another variation on the masking idea is to combine @samp{[a, b, c, d, e]}
+with the mask @samp{[1, 0, 1, 0, 1]} to produce @samp{[a, 0, c, 0, e]}.
+You can accomplish this with @kbd{V M a &}, mapping the logical ``and''
+operation across the two vectors. @xref{Logical Operations}. Note that
+the @code{? :} operation also discussed there allows other types of
+masking using vectors.
+
address@hidden Vector and Matrix Arithmetic, Set Operations, Manipulating
Vectors, Matrix Functions
address@hidden Vector and Matrix Arithmetic
+
address@hidden
+Basic arithmetic operations like addition and multiplication are defined
+for vectors and matrices as well as for numbers. Division of matrices, in
+the sense of multiplying by the inverse, is supported. (Division by a
+matrix actually uses LU-decomposition for greater accuracy and speed.)
address@hidden Arithmetic}.
+
+The following functions are applied element-wise if their arguments are
+vectors or matrices: @code{change-sign}, @code{conj}, @code{arg},
address@hidden, @code{im}, @code{polar}, @code{rect}, @code{clean},
address@hidden, @code{frac}. @xref{Function Index}.
+
address@hidden V J
address@hidden calc-conj-transpose
address@hidden ctrn
+The @kbd{V J} (@code{calc-conj-transpose}) address@hidden command computes
+the conjugate transpose of its argument, i.e., @samp{conj(trn(x))}.
+
address@hidden
address@hidden A
address@hidden ignore
address@hidden A (vectors)
address@hidden calc-abs (vectors)
address@hidden
address@hidden abs
address@hidden ignore
address@hidden abs (vectors)
+The @kbd{A} (@code{calc-abs}) address@hidden command computes the
+Frobenius norm of a vector or matrix argument. This is the square
+root of the sum of the squares of the absolute values of the
+elements of the vector or matrix. If the vector is interpreted as
+a point in two- or three-dimensional space, this is the distance
+from that point to the origin.
+
address@hidden v n
address@hidden calc-rnorm
address@hidden rnorm
+The @kbd{v n} (@code{calc-rnorm}) address@hidden command computes
+the row norm, or infinity-norm, of a vector or matrix. For a plain
+vector, this is the maximum of the absolute values of the elements.
+For a matrix, this is the maximum of the row-absolute-value-sums,
+i.e., of the sums of the absolute values of the elements along the
+various rows.
+
address@hidden V N
address@hidden calc-cnorm
address@hidden cnorm
+The @kbd{V N} (@code{calc-cnorm}) address@hidden command computes
+the column norm, or one-norm, of a vector or matrix. For a plain
+vector, this is the sum of the absolute values of the elements.
+For a matrix, this is the maximum of the column-absolute-value-sums.
+General @expr{k}-norms for @expr{k} other than one or infinity are
+not provided.
+
address@hidden V C
address@hidden calc-cross
address@hidden cross
+The @kbd{V C} (@code{calc-cross}) address@hidden command computes the
+right-handed cross product of two vectors, each of which must have
+exactly three elements.
+
address@hidden
address@hidden &
address@hidden ignore
address@hidden & (matrices)
address@hidden calc-inv (matrices)
address@hidden
address@hidden inv
address@hidden ignore
address@hidden inv (matrices)
+The @kbd{&} (@code{calc-inv}) address@hidden command computes the
+inverse of a square matrix. If the matrix is singular, the inverse
+operation is left in symbolic form. Matrix inverses are recorded so
+that once an inverse (or determinant) of a particular matrix has been
+computed, the inverse and determinant of the matrix can be recomputed
+quickly in the future.
+
+If the argument to @kbd{&} is a plain number @expr{x}, this
+command simply computes @expr{1/x}. This is okay, because the
address@hidden/} operator also does a matrix inversion when dividing one
+by a matrix.
+
address@hidden V D
address@hidden calc-mdet
address@hidden det
+The @kbd{V D} (@code{calc-mdet}) address@hidden command computes the
+determinant of a square matrix.
+
address@hidden V L
address@hidden calc-mlud
address@hidden lud
+The @kbd{V L} (@code{calc-mlud}) address@hidden command computes the
+LU decomposition of a matrix. The result is a list of three matrices
+which, when multiplied together left-to-right, form the original matrix.
+The first is a permutation matrix that arises from pivoting in the
+algorithm, the second is lower-triangular with ones on the diagonal,
+and the third is upper-triangular.
+
address@hidden V T
address@hidden calc-mtrace
address@hidden tr
+The @kbd{V T} (@code{calc-mtrace}) address@hidden command computes the
+trace of a square matrix. This is defined as the sum of the diagonal
+elements of the matrix.
+
address@hidden Set Operations, Statistical Operations, Vector and Matrix
Arithmetic, Matrix Functions
address@hidden Set Operations using Vectors
+
address@hidden
address@hidden Sets, as vectors
+Calc includes several commands which interpret vectors as @dfn{sets} of
+objects. A set is a collection of objects; any given object can appear
+only once in the set. Calc stores sets as vectors of objects in
+sorted order. Objects in a Calc set can be any of the usual things,
+such as numbers, variables, or formulas. Two set elements are considered
+equal if they are identical, except that numerically equal numbers like
+the integer 4 and the float 4.0 are considered equal even though they
+are not ``identical.'' Variables are treated like plain symbols without
+attached values by the set operations; subtracting the set @samp{[b]}
+from @samp{[a, b]} always yields the set @samp{[a]} even though if
+the variables @samp{a} and @samp{b} both equaled 17, you might
+expect the answer @samp{[]}.
+
+If a set contains interval forms, then it is assumed to be a set of
+real numbers. In this case, all set operations require the elements
+of the set to be only things that are allowed in intervals: Real
+numbers, plus and minus infinity, HMS forms, and date forms. If
+there are variables or other non-real objects present in a real set,
+all set operations on it will be left in unevaluated form.
+
+If the input to a set operation is a plain number or interval form
address@hidden, it is treated like the one-element vector @address@hidden
+The result is always a vector, except that if the set consists of a
+single interval, the interval itself is returned instead.
+
address@hidden Operations}, for the @code{in} function which tests if
+a certain value is a member of a given set. To test if the set @expr{A}
+is a subset of the set @expr{B}, use @samp{vdiff(A, B) = []}.
+
address@hidden V +
address@hidden calc-remove-duplicates
address@hidden rdup
+The @kbd{V +} (@code{calc-remove-duplicates}) address@hidden command
+converts an arbitrary vector into set notation. It works by sorting
+the vector as if by @kbd{V S}, then removing duplicates. (For example,
address@hidden, 5, 4, a, 4.0]} is sorted to @samp{[4, 4.0, 5, a, a]} and then
+reduced to @samp{[4, 5, a]}). Overlapping intervals are merged as
+necessary. You rarely need to use @kbd{V +} explicitly, since all the
+other set-based commands apply @kbd{V +} to their inputs before using
+them.
+
address@hidden V V
address@hidden calc-set-union
address@hidden vunion
+The @kbd{V V} (@code{calc-set-union}) address@hidden command computes
+the union of two sets. An object is in the union of two sets if and
+only if it is in either (or both) of the input sets. (You could
+accomplish the same thing by concatenating the sets with @kbd{|},
+then using @kbd{V +}.)
+
address@hidden V ^
address@hidden calc-set-intersect
address@hidden vint
+The @kbd{V ^} (@code{calc-set-intersect}) address@hidden command computes
+the intersection of two sets. An object is in the intersection if
+and only if it is in both of the input sets. Thus if the input
+sets are disjoint, i.e., if they share no common elements, the result
+will be the empty vector @samp{[]}. Note that the characters @kbd{V}
+and @kbd{^} were chosen to be close to the conventional mathematical
+notation for set
address@hidden address@hidden(@math{A \cup B})
address@hidden union
+and
address@hidden address@hidden(@math{A \cap B}).
address@hidden intersection.
+
address@hidden V -
address@hidden calc-set-difference
address@hidden vdiff
+The @kbd{V -} (@code{calc-set-difference}) address@hidden command computes
+the difference between two sets. An object is in the difference
address@hidden - B} if and only if it is in @expr{A} but not in @expr{B}.
+Thus subtracting @samp{[y,z]} from a set will remove the elements
address@hidden and @samp{z} if they are present. You can also think of this
+as a general @dfn{set complement} operator; if @expr{A} is the set of
+all possible values, then @expr{A - B} is the ``complement'' of @expr{B}.
+Obviously this is only practical if the set of all possible values in
+your problem is small enough to list in a Calc vector (or simple
+enough to express in a few intervals).
+
address@hidden V X
address@hidden calc-set-xor
address@hidden vxor
+The @kbd{V X} (@code{calc-set-xor}) address@hidden command computes
+the ``exclusive-or,'' or ``symmetric difference'' of two sets.
+An object is in the symmetric difference of two sets if and only
+if it is in one, but @emph{not} both, of the sets. Objects that
+occur in both sets ``cancel out.''
+
address@hidden V ~
address@hidden calc-set-complement
address@hidden vcompl
+The @kbd{V ~} (@code{calc-set-complement}) address@hidden command
+computes the complement of a set with respect to the real numbers.
+Thus @samp{vcompl(x)} is equivalent to @samp{vdiff([-inf .. inf], x)}.
+For example, @samp{vcompl([2, (3 .. 4]])} evaluates to
address@hidden .. 2), (2 .. 3], (4 .. inf]]}.
+
address@hidden V F
address@hidden calc-set-floor
address@hidden vfloor
+The @kbd{V F} (@code{calc-set-floor}) address@hidden command
+reinterprets a set as a set of integers. Any non-integer values,
+and intervals that do not enclose any integers, are removed. Open
+intervals are converted to equivalent closed intervals. Successive
+integers are converted into intervals of integers. For example, the
+complement of the set @samp{[2, 6, 7, 8]} is messy, but if you wanted
+the complement with respect to the set of integers you could type
address@hidden ~ V F} to get @samp{[[-inf .. 1], [3 .. 5], [9 .. inf]]}.
+
address@hidden V E
address@hidden calc-set-enumerate
address@hidden venum
+The @kbd{V E} (@code{calc-set-enumerate}) address@hidden command
+converts a set of integers into an explicit vector. Intervals in
+the set are expanded out to lists of all integers encompassed by
+the intervals. This only works for finite sets (i.e., sets which
+do not involve @samp{-inf} or @samp{inf}).
+
address@hidden V :
address@hidden calc-set-span
address@hidden vspan
+The @kbd{V :} (@code{calc-set-span}) address@hidden command converts any
+set of reals into an interval form that encompasses all its elements.
+The lower limit will be the smallest element in the set; the upper
+limit will be the largest element. For an empty set, @samp{vspan([])}
+returns the empty interval @address@hidden .. 0)}}.
+
address@hidden V #
address@hidden calc-set-cardinality
address@hidden vcard
+The @kbd{V #} (@code{calc-set-cardinality}) address@hidden command counts
+the number of integers in a set. The result is the length of the vector
+that would be produced by @kbd{V E}, although the computation is much
+more efficient than actually producing that vector.
+
address@hidden Sets, as binary numbers
+Another representation for sets that may be more appropriate in some
+cases is binary numbers. If you are dealing with sets of integers
+in the range 0 to 49, you can use a 50-bit binary number where a
+particular bit is 1 if the corresponding element is in the set.
address@hidden Functions}, for a list of commands that operate on
+binary numbers. Note that many of the above set operations have
+direct equivalents in binary arithmetic: @kbd{b o} (@code{calc-or}),
address@hidden a} (@code{calc-and}), @kbd{b d} (@code{calc-diff}),
address@hidden x} (@code{calc-xor}), and @kbd{b n} (@code{calc-not}),
+respectively. You can use whatever representation for sets is most
+convenient to you.
+
address@hidden b p
address@hidden b u
address@hidden calc-pack-bits
address@hidden calc-unpack-bits
address@hidden vpack
address@hidden vunpack
+The @kbd{b u} (@code{calc-unpack-bits}) address@hidden command
+converts an integer that represents a set in binary into a set
+in vector/interval notation. For example, @samp{vunpack(67)}
+returns @samp{[[0 .. 1], 6]}. If the input is negative, the set
+it represents is semi-infinite: @samp{vunpack(-4) = [2 .. inf)}.
+Use @kbd{V E} afterwards to expand intervals to individual
+values if you wish. Note that this command uses the @kbd{b}
+(binary) prefix key.
+
+The @kbd{b p} (@code{calc-pack-bits}) address@hidden command
+converts the other way, from a vector or interval representing
+a set of nonnegative integers into a binary integer describing
+the same set. The set may include positive infinity, but must
+not include any negative numbers. The input is interpreted as a
+set of integers in the sense of @kbd{V F} (@code{vfloor}). Beware
+that a simple input like @samp{[100]} can result in a huge integer
+representation
address@hidden (@math{2^{100}}, a 31-digit integer, in this case).
address@hidden (@expr{2^100}, a 31-digit integer, in this case).
+
address@hidden Statistical Operations, Reducing and Mapping, Set Operations,
Matrix Functions
address@hidden Statistical Operations on Vectors
+
address@hidden
address@hidden Statistical functions
+The commands in this section take vectors as arguments and compute
+various statistical measures on the data stored in the vectors. The
+references used in the definitions of these functions are Bevington's
address@hidden Reduction and Error Analysis for the Physical Sciences},
+and @emph{Numerical Recipes} by Press, Flannery, Teukolsky and
+Vetterling.
+
+The statistical commands use the @kbd{u} prefix key followed by
+a shifted letter or other character.
+
address@hidden Vectors}, for a description of @kbd{V H}
+(@code{calc-histogram}).
+
address@hidden Fitting}, for the @kbd{a F} command for doing
+least-squares fits to statistical data.
+
address@hidden Distribution Functions}, for several common
+probability distribution functions.
+
address@hidden
+* Single-Variable Statistics::
+* Paired-Sample Statistics::
address@hidden menu
+
address@hidden Single-Variable Statistics, Paired-Sample Statistics,
Statistical Operations, Statistical Operations
address@hidden Single-Variable Statistics
+
address@hidden
+These functions do various statistical computations on single
+vectors. Given a numeric prefix argument, they actually pop
address@hidden objects from the stack and combine them into a data
+vector. Each object may be either a number or a vector; if a
+vector, any sub-vectors inside it are ``flattened'' as if by
address@hidden a 0}; @pxref{Manipulating Vectors}. By default one object
+is popped, which (in order to be useful) is usually a vector.
+
+If an argument is a variable name, and the value stored in that
+variable is a vector, then the stored vector is used. This method
+has the advantage that if your data vector is large, you can avoid
+the slow process of manipulating it directly on the stack.
+
+These functions are left in symbolic form if any of their arguments
+are not numbers or vectors, e.g., if an argument is a formula, or
+a non-vector variable. However, formulas embedded within vector
+arguments are accepted; the result is a symbolic representation
+of the computation, based on the assumption that the formula does
+not itself represent a vector. All varieties of numbers such as
+error forms and interval forms are acceptable.
+
+Some of the functions in this section also accept a single error form
+or interval as an argument. They then describe a property of the
+normal or uniform (respectively) statistical distribution described
+by the argument. The arguments are interpreted in the same way as
+the @var{M} argument of the random number function @kbd{k r}. In
+particular, an interval with integer limits is considered an integer
+distribution, so that @samp{[2 .. 6)} is the same as @samp{[2 .. 5]}.
+An interval with at least one floating-point limit is a continuous
+distribution: @samp{[2.0 .. 6.0)} is @emph{not} the same as
address@hidden .. 5.0]}!
+
address@hidden u #
address@hidden calc-vector-count
address@hidden vcount
+The @kbd{u #} (@code{calc-vector-count}) address@hidden command
+computes the number of data values represented by the inputs.
+For example, @samp{vcount(1, [2, 3], [[4, 5], [], x, y])} returns 7.
+If the argument is a single vector with no sub-vectors, this
+simply computes the length of the vector.
+
address@hidden u +
address@hidden u *
address@hidden calc-vector-sum
address@hidden calc-vector-prod
address@hidden vsum
address@hidden vprod
address@hidden Summations (statistical)
+The @kbd{u +} (@code{calc-vector-sum}) address@hidden command
+computes the sum of the data values. The @kbd{u *}
+(@code{calc-vector-prod}) address@hidden command computes the
+product of the data values. If the input is a single flat vector,
+these are the same as @kbd{V R +} and @kbd{V R *}
+(@pxref{Reducing and Mapping}).
+
address@hidden u X
address@hidden u N
address@hidden calc-vector-max
address@hidden calc-vector-min
address@hidden vmax
address@hidden vmin
+The @kbd{u X} (@code{calc-vector-max}) address@hidden command
+computes the maximum of the data values, and the @kbd{u N}
+(@code{calc-vector-min}) address@hidden command computes the minimum.
+If the argument is an interval, this finds the minimum or maximum
+value in the interval. (Note that @samp{vmax([2..6)) = 5} as
+described above.) If the argument is an error form, this returns
+plus or minus infinity.
+
address@hidden u M
address@hidden calc-vector-mean
address@hidden vmean
address@hidden Mean of data values
+The @kbd{u M} (@code{calc-vector-mean}) address@hidden command
+computes the average (arithmetic mean) of the data values.
+If the inputs are error forms
address@hidden @math{x \pm \sigma},
address@hidden @samp{x +/- s},
+this is the weighted mean of the @expr{x} values with weights
address@hidden @math{1 /\sigma^2}.
address@hidden @expr{1 / s^2}.
address@hidden
+\turnoffactive
+$$ \mu = { \displaystyle \sum { x_i \over \sigma_i^2 } \over
+ \displaystyle \sum { 1 \over \sigma_i^2 } } $$
address@hidden tex
+If the inputs are not error forms, this is simply the sum of the
+values divided by the count of the values.
+
+Note that a plain number can be considered an error form with
+error
address@hidden @math{\sigma = 0}.
address@hidden @expr{s = 0}.
+If the input to @kbd{u M} is a mixture of
+plain numbers and error forms, the result is the mean of the
+plain numbers, ignoring all values with non-zero errors. (By the
+above definitions it's clear that a plain number effectively
+has an infinite weight, next to which an error form with a finite
+weight is completely negligible.)
+
+This function also works for distributions (error forms or
+intervals). The mean of an error form address@hidden @tfn{+/-} @var{b}' is
simply
address@hidden The mean of an interval is the mean of the minimum
+and maximum values of the interval.
+
address@hidden I u M
address@hidden calc-vector-mean-error
address@hidden vmeane
+The @kbd{I u M} (@code{calc-vector-mean-error}) address@hidden
+command computes the mean of the data points expressed as an
+error form. This includes the estimated error associated with
+the mean. If the inputs are error forms, the error is the square
+root of the reciprocal of the sum of the reciprocals of the squares
+of the input errors. (I.e., the variance is the reciprocal of the
+sum of the reciprocals of the variances.)
address@hidden
+\turnoffactive
+$$ \sigma_\mu^2 = {1 \over \displaystyle \sum {1 \over \sigma_i^2}} $$
address@hidden tex
+If the inputs are plain
+numbers, the error is equal to the standard deviation of the values
+divided by the square root of the number of values. (This works
+out to be equivalent to calculating the standard deviation and
+then assuming each value's error is equal to this standard
+deviation.)
address@hidden
+\turnoffactive
+$$ \sigma_\mu^2 = {\sigma^2 \over N} $$
address@hidden tex
+
address@hidden H u M
address@hidden calc-vector-median
address@hidden vmedian
address@hidden Median of data values
+The @kbd{H u M} (@code{calc-vector-median}) address@hidden
+command computes the median of the data values. The values are
+first sorted into numerical order; the median is the middle
+value after sorting. (If the number of data values is even,
+the median is taken to be the average of the two middle values.)
+The median function is different from the other functions in
+this section in that the arguments must all be real numbers;
+variables are not accepted even when nested inside vectors.
+(Otherwise it is not possible to sort the data values.) If
+any of the input values are error forms, their error parts are
+ignored.
+
+The median function also accepts distributions. For both normal
+(error form) and uniform (interval) distributions, the median is
+the same as the mean.
+
address@hidden H I u M
address@hidden calc-vector-harmonic-mean
address@hidden vhmean
address@hidden Harmonic mean
+The @kbd{H I u M} (@code{calc-vector-harmonic-mean}) address@hidden
+command computes the harmonic mean of the data values. This is
+defined as the reciprocal of the arithmetic mean of the reciprocals
+of the values.
address@hidden
+\turnoffactive
+$$ { N \over \displaystyle \sum {1 \over x_i} } $$
address@hidden tex
+
address@hidden u G
address@hidden calc-vector-geometric-mean
address@hidden vgmean
address@hidden Geometric mean
+The @kbd{u G} (@code{calc-vector-geometric-mean}) address@hidden
+command computes the geometric mean of the data values. This
+is the @var{n}th root of the product of the values. This is also
+equal to the @code{exp} of the arithmetic mean of the logarithms
+of the data values.
address@hidden
+\turnoffactive
+$$ \exp \left ( \sum { \ln x_i } \right ) =
+ \left ( \prod { x_i } \right)^{1 / N} $$
address@hidden tex
+
address@hidden H u G
address@hidden agmean
+The @kbd{H u G} address@hidden command computes the ``arithmetic-geometric
+mean'' of two numbers taken from the stack. This is computed by
+replacing the two numbers with their arithmetic mean and geometric
+mean, then repeating until the two values converge.
address@hidden
+\turnoffactive
+$$ a_{i+1} = { a_i + b_i \over 2 } , \qquad b_{i+1} = \sqrt{a_i b_i} $$
address@hidden tex
+
address@hidden Root-mean-square
+Another commonly used mean, the RMS (root-mean-square), can be computed
+for a vector of numbers simply by using the @kbd{A} command.
+
address@hidden u S
address@hidden calc-vector-sdev
address@hidden vsdev
address@hidden Standard deviation
address@hidden Sample statistics
+The @kbd{u S} (@code{calc-vector-sdev}) address@hidden command
+computes the standard
address@hidden address@hidden@math{\sigma}
address@hidden deviation
+of the data values. If the values are error forms, the errors are used
+as weights just as for @kbd{u M}. This is the @emph{sample} standard
+deviation, whose value is the square root of the sum of the squares of
+the differences between the values and the mean of the @expr{N} values,
+divided by @expr{N-1}.
address@hidden
+\turnoffactive
+$$ \sigma^2 = {1 \over N - 1} \sum (x_i - \mu)^2 $$
address@hidden tex
+
+This function also applies to distributions. The standard deviation
+of a single error form is simply the error part. The standard deviation
+of a continuous interval happens to equal the difference between the
+limits, divided by
address@hidden @math{\sqrt{12}}.
address@hidden @expr{sqrt(12)}.
+The standard deviation of an integer interval is the same as the
+standard deviation of a vector of those integers.
+
address@hidden I u S
address@hidden calc-vector-pop-sdev
address@hidden vpsdev
address@hidden Population statistics
+The @kbd{I u S} (@code{calc-vector-pop-sdev}) address@hidden
+command computes the @emph{population} standard deviation.
+It is defined by the same formula as above but dividing
+by @expr{N} instead of by @expr{N-1}. The population standard
+deviation is used when the input represents the entire set of
+data values in the distribution; the sample standard deviation
+is used when the input represents a sample of the set of all
+data values, so that the mean computed from the input is itself
+only an estimate of the true mean.
address@hidden
+\turnoffactive
+$$ \sigma^2 = {1 \over N} \sum (x_i - \mu)^2 $$
address@hidden tex
+
+For error forms and continuous intervals, @code{vpsdev} works
+exactly like @code{vsdev}. For integer intervals, it computes the
+population standard deviation of the equivalent vector of integers.
+
address@hidden H u S
address@hidden H I u S
address@hidden calc-vector-variance
address@hidden calc-vector-pop-variance
address@hidden vvar
address@hidden vpvar
address@hidden Variance of data values
+The @kbd{H u S} (@code{calc-vector-variance}) address@hidden and
address@hidden I u S} (@code{calc-vector-pop-variance}) address@hidden
+commands compute the variance of the data values. The variance
+is the
address@hidden address@hidden@math{\sigma^2}
address@hidden square
+of the standard deviation, i.e., the sum of the
+squares of the deviations of the data values from the mean.
+(This definition also applies when the argument is a distribution.)
+
address@hidden
address@hidden
address@hidden ignore
address@hidden vflat
+The @code{vflat} algebraic function returns a vector of its
+arguments, interpreted in the same way as the other functions
+in this section. For example, @samp{vflat(1, [2, [3, 4]], 5)}
+returns @samp{[1, 2, 3, 4, 5]}.
+
address@hidden Paired-Sample Statistics, , Single-Variable Statistics,
Statistical Operations
address@hidden Paired-Sample Statistics
+
address@hidden
+The functions in this section take two arguments, which must be
+vectors of equal size. The vectors are each flattened in the same
+way as by the single-variable statistical functions. Given a numeric
+prefix argument of 1, these functions instead take one object from
+the stack, which must be an
address@hidden @math{N\times2}
address@hidden Nx2
+matrix of data values. Once again, variable names can be used in place
+of actual vectors and matrices.
+
address@hidden u C
address@hidden calc-vector-covariance
address@hidden vcov
address@hidden Covariance
+The @kbd{u C} (@code{calc-vector-covariance}) address@hidden command
+computes the sample covariance of two vectors. The covariance
+of vectors @var{x} and @var{y} is the sum of the products of the
+differences between the elements of @var{x} and the mean of @var{x}
+times the differences between the corresponding elements of @var{y}
+and the mean of @var{y}, all divided by @expr{N-1}. Note that
+the variance of a vector is just the covariance of the vector
+with itself. Once again, if the inputs are error forms the
+errors are used as weight factors. If both @var{x} and @var{y}
+are composed of error forms, the error for a given data point
+is taken as the square root of the sum of the squares of the two
+input errors.
address@hidden
+\turnoffactive
+$$ \sigma_{x\!y}^2 = {1 \over N-1} \sum (x_i - \mu_x) (y_i - \mu_y) $$
+$$ \sigma_{x\!y}^2 =
+ {\displaystyle {1 \over N-1}
+ \sum {(x_i - \mu_x) (y_i - \mu_y) \over \sigma_i^2}
+ \over \displaystyle {1 \over N} \sum {1 \over \sigma_i^2}}
+$$
address@hidden tex
+
address@hidden I u C
address@hidden calc-vector-pop-covariance
address@hidden vpcov
+The @kbd{I u C} (@code{calc-vector-pop-covariance}) address@hidden
+command computes the population covariance, which is the same as the
+sample covariance computed by @kbd{u C} except dividing by @expr{N}
+instead of @expr{N-1}.
+
address@hidden H u C
address@hidden calc-vector-correlation
address@hidden vcorr
address@hidden Correlation coefficient
address@hidden Linear correlation
+The @kbd{H u C} (@code{calc-vector-correlation}) address@hidden
+command computes the linear correlation coefficient of two vectors.
+This is defined by the covariance of the vectors divided by the
+product of their standard deviations. (There is no difference
+between sample or population statistics here.)
address@hidden
+\turnoffactive
+$$ r_{x\!y} = { \sigma_{x\!y}^2 \over \sigma_x^2 \sigma_y^2 } $$
address@hidden tex
+
address@hidden Reducing and Mapping, Vector and Matrix Formats, Statistical
Operations, Matrix Functions
address@hidden Reducing and Mapping Vectors
+
address@hidden
+The commands in this section allow for more general operations on the
+elements of vectors.
+
address@hidden V A
address@hidden calc-apply
address@hidden apply
+The simplest of these operations is @kbd{V A} (@code{calc-apply})
address@hidden, which applies a given operator to the elements of a vector.
+For example, applying the hypothetical function @code{f} to the vector
address@hidden@samp{[1, 2, 3]}} would produce the function call @samp{f(1, 2,
3)}.
+Applying the @code{+} function to the vector @samp{[a, b]} gives
address@hidden + b}. Applying @code{+} to the vector @samp{[a, b, c]} is an
+error, since the @code{+} function expects exactly two arguments.
+
+While @kbd{V A} is useful in some cases, you will usually find that either
address@hidden R} or @kbd{V M}, described below, is closer to what you want.
+
address@hidden
+* Specifying Operators::
+* Mapping::
+* Reducing::
+* Nesting and Fixed Points::
+* Generalized Products::
address@hidden menu
+
address@hidden Specifying Operators, Mapping, Reducing and Mapping, Reducing
and Mapping
address@hidden Specifying Operators
+
address@hidden
+Commands in this section (like @kbd{V A}) prompt you to press the key
+corresponding to the desired operator. Press @kbd{?} for a partial
+list of the available operators. Generally, an operator is any key or
+sequence of keys that would normally take one or more arguments from
+the stack and replace them with a result. For example, @kbd{V A H C}
+uses the hyperbolic cosine operator, @code{cosh}. (Since @code{cosh}
+expects one argument, @kbd{V A H C} requires a vector with a single
+element as its argument.)
+
+You can press @kbd{x} at the operator prompt to select any algebraic
+function by name to use as the operator. This includes functions you
+have defined yourself using the @kbd{Z F} command. (@xref{Algebraic
+Definitions}.) If you give a name for which no function has been
+defined, the result is left in symbolic form, as in @samp{f(1, 2, 3)}.
+Calc will prompt for the number of arguments the function takes if it
+can't figure it out on its own (say, because you named a function that
+is currently undefined). It is also possible to type a digit key before
+the function name to specify the number of arguments, e.g.,
address@hidden M 3 x f @key{RET}} calls @code{f} with three arguments even if it
+looks like it ought to have only two. This technique may be necessary
+if the function allows a variable number of arguments. For example,
+the @kbd{v e} address@hidden function accepts two or three arguments;
+if you want to map with the three-argument version, you will have to
+type @kbd{V M 3 v e}.
+
+It is also possible to apply any formula to a vector by treating that
+formula as a function. When prompted for the operator to use, press
address@hidden'} (the apostrophe) and type your formula as an algebraic entry.
+You will then be prompted for the argument list, which defaults to a
+list of all variables that appear in the formula, sorted into alphabetic
+order. For example, suppose you enter the formula @address@hidden + 2y^x}}.
+The default argument list would be @samp{(x y)}, which means that if
+this function is applied to the arguments @samp{[3, 10]} the result will
+be @samp{3 + 2*10^3}. (If you plan to use a certain formula in this
+way often, you might consider defining it as a function with @kbd{Z F}.)
+
+Another way to specify the arguments to the formula you enter is with
address@hidden, @kbd{$$}, and so on. For example, @kbd{V A ' $$ + 2$^$$}
+has the same effect as the previous example. The argument list is
+automatically taken to be @samp{($$ $)}. (The order of the arguments
+may seem backwards, but it is analogous to the way normal algebraic
+entry interacts with the stack.)
+
+If you press @kbd{$} at the operator prompt, the effect is similar to
+the apostrophe except that the relevant formula is taken from top-of-stack
+instead. The actual vector arguments of the @kbd{V A $} or related command
+then start at the second-to-top stack position. You will still be
+prompted for an argument list.
+
address@hidden Nameless functions
address@hidden Generic functions
+A function can be written without a name using the notation @samp{<#1 - #2>},
+which means ``a function of two arguments that computes the first
+argument minus the second argument.'' The symbols @samp{#1} and @samp{#2}
+are placeholders for the arguments. You can use any names for these
+placeholders if you wish, by including an argument list followed by a
+colon: @samp{<x, y : x - y>}. When you type @kbd{V A ' $$ + 2$^$$ @key{RET}},
+Calc builds the nameless function @samp{<#1 + 2 #2^#1>} as the function
+to map across the vectors. When you type @kbd{V A ' x + 2y^x @key{RET}
@key{RET}},
+Calc builds the nameless function @address@hidden<x, y : x + 2 y^x>}}. In both
+cases, Calc also writes the nameless function to the Trail so that you
+can get it back later if you wish.
+
+If there is only one argument, you can write @samp{#} in place of @samp{#1}.
+(Note that @samp{< >} notation is also used for date forms. Calc tells
+that @samp{<@var{stuff}>} is a nameless function by the presence of
address@hidden signs inside @var{stuff}, or by the fact that @var{stuff}
+begins with a list of variables followed by a colon.)
+
+You can type a nameless function directly to @kbd{V A '}, or put one on
+the stack and use it with @address@hidden A $}}. Calc will not prompt for an
+argument list in this case, since the nameless function specifies the
+argument list as well as the function itself. In @kbd{V A '}, you can
+omit the @samp{< >} marks if you use @samp{#} notation for the arguments,
+so that @kbd{V A ' #1+#2 @key{RET}} is the same as @kbd{V A ' <#1+#2>
@key{RET}},
+which in turn is the same as @kbd{V A ' $$+$ @key{RET}}.
+
address@hidden Lambda expressions
address@hidden
address@hidden
address@hidden ignore
address@hidden lambda
+The internal format for @samp{<x, y : x + y>} is @samp{lambda(x, y, x + y)}.
+(The word @code{lambda} derives from Lisp notation and the theory of
+functions.) The internal format for @samp{<#1 + #2>} is @samp{lambda(ArgA,
+ArgB, ArgA + ArgB)}. Note that there is no actual Calc function called
address@hidden; the whole point is that the @code{lambda} expression is
+used in its symbolic form, not evaluated for an answer until it is applied
+to specific arguments by a command like @kbd{V A} or @kbd{V M}.
+
+(Actually, @code{lambda} does have one special property: Its arguments
+are never evaluated; for example, putting @samp{<(2/3) #>} on the stack
+will not simplify the @samp{2/3} until the nameless function is actually
+called.)
+
address@hidden add
address@hidden sub
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden mul
address@hidden
address@hidden @null
address@hidden ignore
address@hidden div
address@hidden
address@hidden @null
address@hidden ignore
address@hidden pow
address@hidden
address@hidden @null
address@hidden ignore
address@hidden neg
address@hidden
address@hidden @null
address@hidden ignore
address@hidden mod
address@hidden
address@hidden @null
address@hidden ignore
address@hidden vconcat
+As usual, commands like @kbd{V A} have algebraic function name equivalents.
+For example, @kbd{V A k g} with an argument of @samp{v} is equivalent to
address@hidden(gcd, v)}. The first argument specifies the operator name,
+and is either a variable whose name is the same as the function name,
+or a nameless function like @samp{<#^3+1>}. Operators that are normally
+written as algebraic symbols have the names @code{add}, @code{sub},
address@hidden, @code{div}, @code{pow}, @code{neg}, @code{mod}, and
address@hidden
+
address@hidden
address@hidden
address@hidden ignore
address@hidden call
+The @code{call} function builds a function call out of several arguments:
address@hidden(gcd, x, y)} is the same as @samp{apply(gcd, [x, y])}, which
+in turn is the same as @samp{gcd(x, y)}. The first argument of @code{call},
+like the other functions described here, may be either a variable naming a
+function, or a nameless function (@samp{call(<#1+2#2>, x, y)} is the same
+as @samp{x + 2y}).
+
+(Experts will notice that it's not quite proper to use a variable to name
+a function, since the name @code{gcd} corresponds to the Lisp variable
address@hidden but to the Lisp function @code{calcFunc-gcd}. Calc
+automatically makes this translation, so you don't have to worry
+about it.)
+
address@hidden Mapping, Reducing, Specifying Operators, Reducing and Mapping
address@hidden Mapping
+
address@hidden
address@hidden V M
address@hidden calc-map
address@hidden map
+The @kbd{V M} (@code{calc-map}) address@hidden command applies a given
+operator elementwise to one or more vectors. For example, mapping
address@hidden address@hidden produces a vector of the absolute values of the
+elements in the input vector. Mapping @code{+} pops two vectors from
+the stack, which must be of equal length, and produces a vector of the
+pairwise sums of the elements. If either argument is a non-vector, it
+is duplicated for each element of the other vector. For example,
address@hidden,2,3] 2 V M ^} squares the elements of the specified vector.
+With the 2 listed first, it would have computed a vector of powers of
+two. Mapping a user-defined function pops as many arguments from the
+stack as the function requires. If you give an undefined name, you will
+be prompted for the number of arguments to use.
+
+If any argument to @kbd{V M} is a matrix, the operator is normally mapped
+across all elements of the matrix. For example, given the matrix
address@hidden, -2, 3], [-4, 5, -6]]}, @kbd{V M A} takes six absolute values to
+produce another
address@hidden @math{3\times2}
address@hidden 3x2
+matrix, @expr{[[1, 2, 3], [4, 5, 6]]}.
+
address@hidden mapr
+The command @kbd{V M _} address@hidden (i.e., type an underscore at the
+operator prompt) maps by rows instead. For example, @kbd{V M _ A} views
+the above matrix as a vector of two 3-element row vectors. It produces
+a new vector which contains the absolute values of those row vectors,
+namely @expr{[3.74, 8.77]}. (Recall, the absolute value of a vector is
+defined as the square root of the sum of the squares of the elements.)
+Some operators accept vectors and return new vectors; for example,
address@hidden v} reverses a vector, so @kbd{V M _ v v} would reverse each row
+of the matrix to get a new matrix, @expr{[[3, -2, 1], [-6, 5, -4]]}.
+
+Sometimes a vector of vectors (representing, say, strings, sets, or lists)
+happens to look like a matrix. If so, remember to use @kbd{V M _} if you
+want to map a function across the whole strings or sets rather than across
+their individual elements.
+
address@hidden mapc
+The command @kbd{V M :} address@hidden maps by columns. Basically, it
+transposes the input matrix, maps by rows, and then, if the result is a
+matrix, transposes again. For example, @kbd{V M : A} takes the absolute
+values of the three columns of the matrix, treating each as a 2-vector,
+and @kbd{V M : v v} reverses the columns to get the matrix
address@hidden, 5, -6], [1, -2, 3]]}.
+
+(The symbols @kbd{_} and @kbd{:} were chosen because they had row-like
+and column-like appearances, and were not already taken by useful
+operators. Also, they appear shifted on most keyboards so they are easy
+to type after @kbd{V M}.)
+
+The @kbd{_} and @kbd{:} modifiers have no effect on arguments that are
+not matrices (so if none of the arguments are matrices, they have no
+effect at all). If some of the arguments are matrices and others are
+plain numbers, the plain numbers are held constant for all rows of the
+matrix (so that @kbd{2 V M _ ^} squares every row of a matrix; squaring
+a vector takes a dot product of the vector with itself).
+
+If some of the arguments are vectors with the same lengths as the
+rows (for @kbd{V M _}) or columns (for @kbd{V M :}) of the matrix
+arguments, those vectors are also held constant for every row or
+column.
+
+Sometimes it is useful to specify another mapping command as the operator
+to use with @kbd{V M}. For example, @kbd{V M _ V A +} applies @kbd{V A +}
+to each row of the input matrix, which in turn adds the two values on that
+row. If you give another vector-operator command as the operator for
address@hidden M}, it automatically uses map-by-rows mode if you don't specify
+otherwise; thus @kbd{V M V A +} is equivalent to @kbd{V M _ V A +}. (If
+you really want to map-by-elements another mapping command, you can use
+a triple-nested mapping command: @kbd{V M V M V A +} means to map
address@hidden M V A +} over the rows of the matrix; in turn, @kbd{V A +} is
+mapped over the elements of each row.)
+
address@hidden mapa
address@hidden mapd
+Previous versions of Calc had ``map across'' and ``map down'' modes
+that are now considered obsolete; the old ``map across'' is now simply
address@hidden M V A}, and ``map down'' is now @kbd{V M : V A}. The algebraic
+functions @code{mapa} and @code{mapd} are still supported, though.
+Note also that, while the old mapping modes were persistent (once you
+set the mode, it would apply to later mapping commands until you reset
+it), the new @kbd{:} and @kbd{_} modifiers apply only to the current
+mapping command. The default @kbd{V M} always means map-by-elements.
+
address@hidden Manipulation}, for the @kbd{a M} command, which is like
address@hidden M} but for equations and inequalities instead of vectors.
address@hidden Variables}, for the @kbd{s m} command which modifies a
+variable's stored value using a @kbd{V M}-like operator.
+
address@hidden Reducing, Nesting and Fixed Points, Mapping, Reducing and Mapping
address@hidden Reducing
+
address@hidden
address@hidden V R
address@hidden calc-reduce
address@hidden reduce
+The @kbd{V R} (@code{calc-reduce}) address@hidden command applies a given
+binary operator across all the elements of a vector. A binary operator is
+a function such as @code{+} or @code{max} which takes two arguments. For
+example, reducing @code{+} over a vector computes the sum of the elements
+of the vector. Reducing @code{-} computes the first element minus each of
+the remaining elements. Reducing @code{max} computes the maximum element
+and so on. In general, reducing @code{f} over the vector @samp{[a, b, c, d]}
+produces @samp{f(f(f(a, b), c), d)}.
+
address@hidden I V R
address@hidden rreduce
+The @kbd{I V R} address@hidden command is similar to @kbd{V R} except
+that works from right to left through the vector. For example, plain
address@hidden R -} on the vector @samp{[a, b, c, d]} produces @samp{a - b - c
- d}
+but @kbd{I V R -} on the same vector produces @samp{a - (b - (c - d))},
+or @samp{a - b + c - d}. This ``alternating sum'' occurs frequently
+in power series expansions.
+
address@hidden V U
address@hidden accum
+The @kbd{V U} (@code{calc-accumulate}) address@hidden command does an
+accumulation operation. Here Calc does the corresponding reduction
+operation, but instead of producing only the final result, it produces
+a vector of all the intermediate results. Accumulating @code{+} over
+the vector @samp{[a, b, c, d]} produces the vector
address@hidden, a + b, a + b + c, a + b + c + d]}.
+
address@hidden I V U
address@hidden raccum
+The @kbd{I V U} address@hidden command does a right-to-left accumulation.
+For example, @kbd{I V U -} on the vector @samp{[a, b, c, d]} produces the
+vector @samp{[a - b + c - d, b - c + d, c - d, d]}.
+
address@hidden reducea
address@hidden rreducea
address@hidden reduced
address@hidden rreduced
+As for @kbd{V M}, @kbd{V R} normally reduces a matrix elementwise. For
+example, given the matrix @expr{[[a, b, c], [d, e, f]]}, @kbd{V R +} will
+compute @expr{a + b + c + d + e + f}. You can type @kbd{V R _} or
address@hidden R :} to modify this behavior. The @kbd{V R _} address@hidden
+command reduces ``across'' the matrix; it reduces each row of the matrix
+as a vector, then collects the results. Thus @kbd{V R _ +} of this
+matrix would produce @expr{[a + b + c, d + e + f]}. Similarly, @kbd{V R :}
address@hidden reduces down; @kbd{V R : +} would produce @expr{[a + d,
+b + e, c + f]}.
+
address@hidden reducer
address@hidden rreducer
+There is a third ``by rows'' mode for reduction that is occasionally
+useful; @kbd{V R =} address@hidden simply reduces the operator over
+the rows of the matrix themselves. Thus @kbd{V R = +} on the above
+matrix would get the same result as @kbd{V R : +}, since adding two
+row vectors is equivalent to adding their elements. But @kbd{V R = *}
+would multiply the two rows (to get a single number, their dot product),
+while @kbd{V R : *} would produce a vector of the products of the columns.
+
+These three matrix reduction modes work with @kbd{V R} and @kbd{I V R},
+but they are not currently supported with @kbd{V U} or @kbd{I V U}.
+
address@hidden reducec
address@hidden rreducec
+The obsolete reduce-by-columns function, @code{reducec}, is still
+supported but there is no way to get it through the @kbd{V R} command.
+
+The commands @kbd{C-x * :} and @kbd{C-x * _} are equivalent to typing
address@hidden * r} to grab a rectangle of data into Calc, and then typing
address@hidden R : +} or @kbd{V R _ +}, respectively, to sum the columns or
+rows of the matrix. @xref{Grabbing From Buffers}.
+
address@hidden Nesting and Fixed Points, Generalized Products, Reducing,
Reducing and Mapping
address@hidden Nesting and Fixed Points
+
address@hidden
address@hidden H V R
address@hidden nest
+The @kbd{H V R} address@hidden command applies a function to a given
+argument repeatedly. It takes two values, @samp{a} and @samp{n}, from
+the stack, where @samp{n} must be an integer. It then applies the
+function nested @samp{n} times; if the function is @samp{f} and @samp{n}
+is 3, the result is @samp{f(f(f(a)))}. The number @samp{n} may be
+negative if Calc knows an inverse for the function @samp{f}; for
+example, @samp{nest(sin, a, -2)} returns @samp{arcsin(arcsin(a))}.
+
address@hidden H V U
address@hidden anest
+The @kbd{H V U} address@hidden command is an accumulating version of
address@hidden: It returns a vector of @samp{n+1} values, e.g.,
address@hidden, f(a), f(f(a)), f(f(f(a)))]}. If @samp{n} is negative and
address@hidden is the inverse of @samp{f}, then the result is of the
+form @samp{[a, F(a), F(F(a)), F(F(F(a)))]}.
+
address@hidden H I V R
address@hidden fixp
address@hidden Fixed points
+The @kbd{H I V R} address@hidden command is like @kbd{H V R}, except
+that it takes only an @samp{a} value from the stack; the function is
+applied until it reaches a ``fixed point,'' i.e., until the result
+no longer changes.
+
address@hidden H I V U
address@hidden afixp
+The @kbd{H I V U} address@hidden command is an accumulating @code{fixp}.
+The first element of the return vector will be the initial value @samp{a};
+the last element will be the final result that would have been returned
+by @code{fixp}.
+
+For example, 0.739085 is a fixed point of the cosine function (in radians):
address@hidden(0.739085) = 0.739085}. You can find this value by putting, say,
+1.0 on the stack and typing @kbd{H I V U C}. (We use the accumulating
+version so we can see the intermediate results: @samp{[1, 0.540302, 0.857553,
+0.65329, ...]}. With a precision of six, this command will take 36 steps
+to converge to 0.739085.)
+
+Newton's method for finding roots is a classic example of iteration
+to a fixed point. To find the square root of five starting with an
+initial guess, Newton's method would look for a fixed point of the
+function @samp{(x + 5/x) / 2}. Putting a guess of 1 on the stack
+and typing @kbd{H I V R ' ($ + 5/$)/2 @key{RET}} quickly yields the result
+2.23607. This is equivalent to using the @kbd{a R} (@code{calc-find-root})
+command to find a root of the equation @samp{x^2 = 5}.
+
+These examples used numbers for @samp{a} values. Calc keeps applying
+the function until two successive results are equal to within the
+current precision. For complex numbers, both the real parts and the
+imaginary parts must be equal to within the current precision. If
address@hidden is a formula (say, a variable name), then the function is
+applied until two successive results are exactly the same formula.
+It is up to you to ensure that the function will eventually converge;
+if it doesn't, you may have to press @kbd{C-g} to stop the Calculator.
+
+The algebraic @code{fixp} function takes two optional arguments, @samp{n}
+and @samp{tol}. The first is the maximum number of steps to be allowed,
+and must be either an integer or the symbol @samp{inf} (infinity, the
+default). The second is a convergence tolerance. If a tolerance is
+specified, all results during the calculation must be numbers, not
+formulas, and the iteration stops when the magnitude of the difference
+between two successive results is less than or equal to the tolerance.
+(This implies that a tolerance of zero iterates until the results are
+exactly equal.)
+
+Putting it all together, @samp{fixp(<(# + A/#)/2>, B, 20, 1e-10)}
+computes the square root of @samp{A} given the initial guess @samp{B},
+stopping when the result is correct within the specified tolerance, or
+when 20 steps have been taken, whichever is sooner.
+
address@hidden Generalized Products, , Nesting and Fixed Points, Reducing and
Mapping
address@hidden Generalized Products
+
address@hidden V O
address@hidden calc-outer-product
address@hidden outer
+The @kbd{V O} (@code{calc-outer-product}) address@hidden command applies
+a given binary operator to all possible pairs of elements from two
+vectors, to produce a matrix. For example, @kbd{V O *} with @samp{[a, b]}
+and @samp{[x, y, z]} on the stack produces a multiplication table:
address@hidden x, a y, a z], [b x, b y, b z]]}. Element @var{r},@var{c} of
+the result matrix is obtained by applying the operator to element @var{r}
+of the lefthand vector and element @var{c} of the righthand vector.
+
address@hidden V I
address@hidden calc-inner-product
address@hidden inner
+The @kbd{V I} (@code{calc-inner-product}) address@hidden command computes
+the generalized inner product of two vectors or matrices, given a
+``multiplicative'' operator and an ``additive'' operator. These can each
+actually be any binary operators; if they are @samp{*} and @samp{+},
+respectively, the result is a standard matrix multiplication. Element
address@hidden,@var{c} of the result matrix is obtained by mapping the
+multiplicative operator across row @var{r} of the lefthand matrix and
+column @var{c} of the righthand matrix, and then reducing with the additive
+operator. Just as for the standard @kbd{*} command, this can also do a
+vector-matrix or matrix-vector inner product, or a vector-vector
+generalized dot product.
+
+Since @kbd{V I} requires two operators, it prompts twice. In each case,
+you can use any of the usual methods for entering the operator. If you
+use @kbd{$} twice to take both operator formulas from the stack, the
+first (multiplicative) operator is taken from the top of the stack
+and the second (additive) operator is taken from second-to-top.
+
address@hidden Vector and Matrix Formats, , Reducing and Mapping, Matrix
Functions
address@hidden Vector and Matrix Display Formats
+
address@hidden
+Commands for controlling vector and matrix display use the @kbd{v} prefix
+instead of the usual @kbd{d} prefix. But they are display modes; in
+particular, they are influenced by the @kbd{I} and @kbd{H} prefix keys
+in the same way (@pxref{Display Modes}). Matrix display is also
+influenced by the @kbd{d O} (@code{calc-flat-language}) mode;
address@hidden Language Modes}.
+
address@hidden V <
address@hidden calc-matrix-left-justify
address@hidden V =
address@hidden calc-matrix-center-justify
address@hidden V >
address@hidden calc-matrix-right-justify
+The commands @kbd{v <} (@code{calc-matrix-left-justify}), @kbd{v >}
+(@code{calc-matrix-right-justify}), and @address@hidden =}}
+(@code{calc-matrix-center-justify}) control whether matrix elements
+are justified to the left, right, or center of their columns.
+
address@hidden V [
address@hidden calc-vector-brackets
address@hidden V @{
address@hidden calc-vector-braces
address@hidden V (
address@hidden calc-vector-parens
+The @kbd{v [} (@code{calc-vector-brackets}) command turns the square
+brackets that surround vectors and matrices displayed in the stack on
+and off. The @kbd{v @{} (@code{calc-vector-braces}) and @kbd{v (}
+(@code{calc-vector-parens}) commands use curly braces or parentheses,
+respectively, instead of square brackets. For example, @kbd{v @{} might
+be used in preparation for yanking a matrix into a buffer running
+Mathematica. (In fact, the Mathematica language mode uses this mode;
address@hidden Language Mode}.) Note that, regardless of the
+display mode, either brackets or braces may be used to enter vectors,
+and parentheses may never be used for this purpose.
+
address@hidden V ]
address@hidden calc-matrix-brackets
+The @kbd{v ]} (@code{calc-matrix-brackets}) command controls the
+``big'' style display of matrices. It prompts for a string of code
+letters; currently implemented letters are @code{R}, which enables
+brackets on each row of the matrix; @code{O}, which enables outer
+brackets in opposite corners of the matrix; and @code{C}, which
+enables commas or semicolons at the ends of all rows but the last.
+The default format is @samp{RO}. (Before Calc 2.00, the format
+was fixed at @samp{ROC}.) Here are some example matrices:
+
address@hidden
address@hidden
+[ [ 123, 0, 0 ] [ [ 123, 0, 0 ],
+ [ 0, 123, 0 ] [ 0, 123, 0 ],
+ [ 0, 0, 123 ] ] [ 0, 0, 123 ] ]
+
+ RO ROC
+
address@hidden group
address@hidden example
address@hidden
address@hidden
address@hidden
+ [ 123, 0, 0 [ 123, 0, 0 ;
+ 0, 123, 0 0, 123, 0 ;
+ 0, 0, 123 ] 0, 0, 123 ]
+
+ O OC
+
address@hidden group
address@hidden example
address@hidden
address@hidden
address@hidden
+ [ 123, 0, 0 ] 123, 0, 0
+ [ 0, 123, 0 ] 0, 123, 0
+ [ 0, 0, 123 ] 0, 0, 123
+
+ R @r{blank}
address@hidden group
address@hidden example
+
address@hidden
+Note that of the formats shown here, @samp{RO}, @samp{ROC}, and
address@hidden are all recognized as matrices during reading, while
+the others are useful for display only.
+
address@hidden V ,
address@hidden calc-vector-commas
+The @kbd{v ,} (@code{calc-vector-commas}) command turns commas on and
+off in vector and matrix display.
+
+In vectors of length one, and in all vectors when commas have been
+turned off, Calc adds extra parentheses around formulas that might
+otherwise be ambiguous. For example, @samp{[a b]} could be a vector
+of the one formula @samp{a b}, or it could be a vector of two
+variables with commas turned off. Calc will display the former
+case as @samp{[(a b)]}. You can disable these extra parentheses
+(to make the output less cluttered at the expense of allowing some
+ambiguity) by adding the letter @code{P} to the control string you
+give to @kbd{v ]} (as described above).
+
address@hidden V .
address@hidden calc-full-vectors
+The @kbd{v .} (@code{calc-full-vectors}) command turns abbreviated
+display of long vectors on and off. In this mode, vectors of six
+or more elements, or matrices of six or more rows or columns, will
+be displayed in an abbreviated form that displays only the first
+three elements and the last element: @samp{[a, b, c, ..., z]}.
+When very large vectors are involved this will substantially
+improve Calc's display speed.
+
address@hidden t .
address@hidden calc-full-trail-vectors
+The @kbd{t .} (@code{calc-full-trail-vectors}) command controls a
+similar mode for recording vectors in the Trail. If you turn on
+this mode, vectors of six or more elements and matrices of six or
+more rows or columns will be abbreviated when they are put in the
+Trail. The @kbd{t y} (@code{calc-trail-yank}) command will be
+unable to recover those vectors. If you are working with very
+large vectors, this mode will improve the speed of all operations
+that involve the trail.
+
address@hidden V /
address@hidden calc-break-vectors
+The @kbd{v /} (@code{calc-break-vectors}) command turns multi-line
+vector display on and off. Normally, matrices are displayed with one
+row per line but all other types of vectors are displayed in a single
+line. This mode causes all vectors, whether matrices or not, to be
+displayed with a single element per line. Sub-vectors within the
+vectors will still use the normal linear form.
+
address@hidden Algebra, Units, Matrix Functions, Top
address@hidden Algebra
+
address@hidden
+This section covers the Calc features that help you work with
+algebraic formulas. First, the general sub-formula selection
+mechanism is described; this works in conjunction with any Calc
+commands. Then, commands for specific algebraic operations are
+described. Finally, the flexible @dfn{rewrite rule} mechanism
+is discussed.
+
+The algebraic commands use the @kbd{a} key prefix; selection
+commands use the @kbd{j} (for ``just a letter that wasn't used
+for anything else'') prefix.
+
address@hidden Stack Entries}, to see how to manipulate formulas
+using regular Emacs editing commands.
+
+When doing algebraic work, you may find several of the Calculator's
+modes to be helpful, including Algebraic Simplification mode (@kbd{m A})
+or No-Simplification mode (@kbd{m O}),
+Algebraic entry mode (@kbd{m a}), Fraction mode (@kbd{m f}), and
+Symbolic mode (@kbd{m s}). @xref{Mode Settings}, for discussions
+of these modes. You may also wish to select Big display mode (@kbd{d B}).
address@hidden Language Modes}.
+
address@hidden
+* Selecting Subformulas::
+* Algebraic Manipulation::
+* Simplifying Formulas::
+* Polynomials::
+* Calculus::
+* Solving Equations::
+* Numerical Solutions::
+* Curve Fitting::
+* Summations::
+* Logical Operations::
+* Rewrite Rules::
address@hidden menu
+
address@hidden Selecting Subformulas, Algebraic Manipulation, Algebra, Algebra
address@hidden Selecting Sub-Formulas
+
address@hidden
address@hidden Selections
address@hidden Sub-formulas
address@hidden Parts of formulas
+When working with an algebraic formula it is often necessary to
+manipulate a portion of the formula rather than the formula as a
+whole. Calc allows you to ``select'' a portion of any formula on
+the stack. Commands which would normally operate on that stack
+entry will now operate only on the sub-formula, leaving the
+surrounding part of the stack entry alone.
+
+One common non-algebraic use for selection involves vectors. To work
+on one element of a vector in-place, simply select that element as a
+``sub-formula'' of the vector.
+
address@hidden
+* Making Selections::
+* Changing Selections::
+* Displaying Selections::
+* Operating on Selections::
+* Rearranging with Selections::
address@hidden menu
+
address@hidden Making Selections, Changing Selections, Selecting Subformulas,
Selecting Subformulas
address@hidden Making Selections
+
address@hidden
address@hidden j s
address@hidden calc-select-here
+To select a sub-formula, move the Emacs cursor to any character in that
+sub-formula, and press @address@hidden s}} (@code{calc-select-here}). Calc
will
+highlight the smallest portion of the formula that contains that
+character. By default the sub-formula is highlighted by blanking out
+all of the rest of the formula with dots. Selection works in any
+display mode but is perhaps easiest in Big mode (@kbd{d B}).
+Suppose you enter the following formula:
+
address@hidden
address@hidden
+ 3 ___
+ (a + b) + V c
+1: ---------------
+ 2 x + 1
address@hidden group
address@hidden smallexample
+
address@hidden
+(by typing @kbd{' ((a+b)^3 + sqrt(c)) / (2x+1)}). If you move the
+cursor to the letter @samp{b} and press @address@hidden s}}, the display
changes
+to
+
address@hidden
address@hidden
+ . ...
+ .. . b. . . .
+1* ...............
+ . . . .
address@hidden group
address@hidden smallexample
+
address@hidden
+Every character not part of the sub-formula @samp{b} has been changed
+to a dot. The @samp{*} next to the line number is to remind you that
+the formula has a portion of it selected. (In this case, it's very
+obvious, but it might not always be. If Embedded mode is enabled,
+the word @samp{Sel} also appears in the mode line because the stack
+may not be visible. @pxref{Embedded Mode}.)
+
+If you had instead placed the cursor on the parenthesis immediately to
+the right of the @samp{b}, the selection would have been:
+
address@hidden
address@hidden
+ . ...
+ (a + b) . . .
+1* ...............
+ . . . .
address@hidden group
address@hidden smallexample
+
address@hidden
+The portion selected is always large enough to be considered a complete
+formula all by itself, so selecting the parenthesis selects the whole
+formula that it encloses. Putting the cursor on the @samp{+} sign
+would have had the same effect.
+
+(Strictly speaking, the Emacs cursor is really the manifestation of
+the Emacs ``point,'' which is a position @emph{between} two characters
+in the buffer. So purists would say that Calc selects the smallest
+sub-formula which contains the character to the right of ``point.'')
+
+If you supply a numeric prefix argument @var{n}, the selection is
+expanded to the @var{n}th enclosing sub-formula. Thus, positioning
+the cursor on the @samp{b} and typing @kbd{C-u 1 j s} will select
address@hidden + b}; typing @kbd{C-u 2 j s} will select @samp{(a + b)^3},
+and so on.
+
+If the cursor is not on any part of the formula, or if you give a
+numeric prefix that is too large, the entire formula is selected.
+
+If the cursor is on the @samp{.} line that marks the top of the stack
+(i.e., its normal ``rest position''), this command selects the entire
+formula at stack level 1. Most selection commands similarly operate
+on the formula at the top of the stack if you haven't positioned the
+cursor on any stack entry.
+
address@hidden j a
address@hidden calc-select-additional
+The @kbd{j a} (@code{calc-select-additional}) command enlarges the
+current selection to encompass the cursor. To select the smallest
+sub-formula defined by two different points, move to the first and
+press @kbd{j s}, then move to the other and press @kbd{j a}. This
+is roughly analogous to using @kbd{C-@@} (@code{set-mark-command}) to
+select the two ends of a region of text during normal Emacs editing.
+
address@hidden j o
address@hidden calc-select-once
+The @kbd{j o} (@code{calc-select-once}) command selects a formula in
+exactly the same way as @kbd{j s}, except that the selection will
+last only as long as the next command that uses it. For example,
address@hidden o 1 +} is a handy way to add one to the sub-formula indicated
+by the cursor.
+
+(A somewhat more precise definition: The @kbd{j o} command sets a flag
+such that the next command involving selected stack entries will clear
+the selections on those stack entries afterwards. All other selection
+commands except @kbd{j a} and @kbd{j O} clear this flag.)
+
address@hidden j S
address@hidden j O
address@hidden calc-select-here-maybe
address@hidden calc-select-once-maybe
+The @kbd{j S} (@code{calc-select-here-maybe}) and @kbd{j O}
+(@code{calc-select-once-maybe}) commands are equivalent to @kbd{j s}
+and @kbd{j o}, respectively, except that if the formula already
+has a selection they have no effect. This is analogous to the
+behavior of some commands such as @kbd{j r} (@code{calc-rewrite-selection};
address@hidden with Rewrite Rules}) and is mainly intended to be
+used in keyboard macros that implement your own selection-oriented
+commands.
+
+Selection of sub-formulas normally treats associative terms like
address@hidden + b - c + d} and @samp{x * y * z} as single levels of the
formula.
+If you place the cursor anywhere inside @samp{a + b - c + d} except
+on one of the variable names and use @kbd{j s}, you will select the
+entire four-term sum.
+
address@hidden j b
address@hidden calc-break-selections
+The @kbd{j b} (@code{calc-break-selections}) command controls a mode
+in which the ``deep structure'' of these associative formulas shows
+through. Calc actually stores the above formulas as @samp{((a + b) - c) + d}
+and @samp{x * (y * z)}. (Note that for certain obscure reasons, Calc
+treats multiplication as right-associative.) Once you have enabled
address@hidden b} mode, selecting with the cursor on the @samp{-} sign would
+only select the @samp{a + b - c} portion, which makes sense when the
+deep structure of the sum is considered. There is no way to select
+the @samp{b - c + d} portion; although this might initially look
+like just as legitimate a sub-formula as @samp{a + b - c}, the deep
+structure shows that it isn't. The @kbd{d U} command can be used
+to view the deep structure of any formula (@pxref{Normal Language Modes}).
+
+When @kbd{j b} mode has not been enabled, the deep structure is
+generally hidden by the selection commands---what you see is what
+you get.
+
address@hidden j u
address@hidden calc-unselect
+The @kbd{j u} (@code{calc-unselect}) command unselects the formula
+that the cursor is on. If there was no selection in the formula,
+this command has no effect. With a numeric prefix argument, it
+unselects the @var{n}th stack element rather than using the cursor
+position.
+
address@hidden j c
address@hidden calc-clear-selections
+The @kbd{j c} (@code{calc-clear-selections}) command unselects all
+stack elements.
+
address@hidden Changing Selections, Displaying Selections, Making Selections,
Selecting Subformulas
address@hidden Changing Selections
+
address@hidden
address@hidden j m
address@hidden calc-select-more
+Once you have selected a sub-formula, you can expand it using the
address@hidden@kbd{j m}} (@code{calc-select-more}) command. If @samp{a + b} is
+selected, pressing @address@hidden m}} repeatedly works as follows:
+
address@hidden
address@hidden
+ 3 ... 3 ___ 3 ___
+ (a + b) . . . (a + b) + V c (a + b) + V c
+1* ............... 1* ............... 1* ---------------
+ . . . . . . . . 2 x + 1
address@hidden group
address@hidden smallexample
+
address@hidden
+In the last example, the entire formula is selected. This is roughly
+the same as having no selection at all, but because there are subtle
+differences the @samp{*} character is still there on the line number.
+
+With a numeric prefix argument @var{n}, @kbd{j m} expands @var{n}
+times (or until the entire formula is selected). Note that @kbd{j s}
+with argument @var{n} is equivalent to plain @kbd{j s} followed by
address@hidden m} with argument @var{n}. If @address@hidden m}} is used when
there
+is no current selection, it is equivalent to @address@hidden s}}.
+
+Even though @kbd{j m} does not explicitly use the location of the
+cursor within the formula, it nevertheless uses the cursor to determine
+which stack element to operate on. As usual, @kbd{j m} when the cursor
+is not on any stack element operates on the top stack element.
+
address@hidden j l
address@hidden calc-select-less
+The @kbd{j l} (@code{calc-select-less}) command reduces the current
+selection around the cursor position. That is, it selects the
+immediate sub-formula of the current selection which contains the
+cursor, the opposite of @kbd{j m}. If the cursor is not inside the
+current selection, the command de-selects the formula.
+
address@hidden j 1-9
address@hidden calc-select-part
+The @kbd{j 1} through @kbd{j 9} (@code{calc-select-part}) commands
+select the @var{n}th sub-formula of the current selection. They are
+like @kbd{j l} (@code{calc-select-less}) except they use counting
+rather than the cursor position to decide which sub-formula to select.
+For example, if the current selection is @kbd{a + b + c} or
address@hidden(a, b, c)} or @kbd{[a, b, c]}, then @kbd{j 1} selects @samp{a},
address@hidden 2} selects @samp{b}, and @kbd{j 3} selects @samp{c}; in each of
+these cases, @kbd{j 4} through @kbd{j 9} would be errors.
+
+If there is no current selection, @kbd{j 1} through @kbd{j 9} select
+the @var{n}th top-level sub-formula. (In other words, they act as if
+the entire stack entry were selected first.) To select the @var{n}th
+sub-formula where @var{n} is greater than nine, you must instead invoke
address@hidden@kbd{j 1}} with @var{n} as a numeric prefix argument.
+
address@hidden j n
address@hidden j p
address@hidden calc-select-next
address@hidden calc-select-previous
+The @kbd{j n} (@code{calc-select-next}) and @kbd{j p}
+(@code{calc-select-previous}) commands change the current selection
+to the next or previous sub-formula at the same level. For example,
+if @samp{b} is selected in @address@hidden + a*b*c + x}}, then @kbd{j n}
+selects @samp{c}. Further @kbd{j n} commands would be in error because,
+even though there is something to the right of @samp{c} (namely, @samp{x}),
+it is not at the same level; in this case, it is not a term of the
+same product as @samp{b} and @samp{c}. However, @kbd{j m} (to select
+the whole product @samp{a*b*c} as a term of the sum) followed by
address@hidden@kbd{j n}} would successfully select the @samp{x}.
+
+Similarly, @kbd{j p} moves the selection from the @samp{b} in this
+sample formula to the @samp{a}. Both commands accept numeric prefix
+arguments to move several steps at a time.
+
+It is interesting to compare Calc's selection commands with the
+Emacs Info system's commands for navigating through hierarchically
+organized documentation. Calc's @kbd{j n} command is completely
+analogous to Info's @kbd{n} command. Likewise, @kbd{j p} maps to
address@hidden, @kbd{j 2} maps to @kbd{2}, and Info's @kbd{u} is like @kbd{j m}.
+(Note that @kbd{j u} stands for @code{calc-unselect}, not ``up''.)
+The Info @kbd{m} command is somewhat similar to Calc's @kbd{j s} and
address@hidden l}; in each case, you can jump directly to a sub-component
+of the hierarchy simply by pointing to it with the cursor.
+
address@hidden Displaying Selections, Operating on Selections, Changing
Selections, Selecting Subformulas
address@hidden Displaying Selections
+
address@hidden
address@hidden j d
address@hidden calc-show-selections
+The @kbd{j d} (@code{calc-show-selections}) command controls how
+selected sub-formulas are displayed. One of the alternatives is
+illustrated in the above examples; if we press @kbd{j d} we switch
+to the other style in which the selected portion itself is obscured
+by @samp{#} signs:
+
address@hidden
address@hidden
+ 3 ... # ___
+ (a + b) . . . ## # ## + V c
+1* ............... 1* ---------------
+ . . . . 2 x + 1
address@hidden group
address@hidden smallexample
+
address@hidden Operating on Selections, Rearranging with Selections, Displaying
Selections, Selecting Subformulas
address@hidden Operating on Selections
+
address@hidden
+Once a selection is made, all Calc commands that manipulate items
+on the stack will operate on the selected portions of the items
+instead. (Note that several stack elements may have selections
+at once, though there can be only one selection at a time in any
+given stack element.)
+
address@hidden j e
address@hidden calc-enable-selections
+The @kbd{j e} (@code{calc-enable-selections}) command disables the
+effect that selections have on Calc commands. The current selections
+still exist, but Calc commands operate on whole stack elements anyway.
+This mode can be identified by the fact that the @samp{*} markers on
+the line numbers are gone, even though selections are visible. To
+reactivate the selections, press @kbd{j e} again.
+
+To extract a sub-formula as a new formula, simply select the
+sub-formula and press @key{RET}. This normally duplicates the top
+stack element; here it duplicates only the selected portion of that
+element.
+
+To replace a sub-formula with something different, you can enter the
+new value onto the stack and press @key{TAB}. This normally exchanges
+the top two stack elements; here it swaps the value you entered into
+the selected portion of the formula, returning the old selected
+portion to the top of the stack.
+
address@hidden
address@hidden
+ 3 ... ... ___
+ (a + b) . . . 17 x y . . . 17 x y + V c
+2* ............... 2* ............. 2: -------------
+ . . . . . . . . 2 x + 1
+
+ 3 3
+1: 17 x y 1: (a + b) 1: (a + b)
address@hidden group
address@hidden smallexample
+
+In this example we select a sub-formula of our original example,
+enter a new formula, @key{TAB} it into place, then deselect to see
+the complete, edited formula.
+
+If you want to swap whole formulas around even though they contain
+selections, just use @kbd{j e} before and after.
+
address@hidden j '
address@hidden calc-enter-selection
+The @kbd{j '} (@code{calc-enter-selection}) command is another way
+to replace a selected sub-formula. This command does an algebraic
+entry just like the regular @kbd{'} key. When you press @key{RET},
+the formula you type replaces the original selection. You can use
+the @samp{$} symbol in the formula to refer to the original
+selection. If there is no selection in the formula under the cursor,
+the cursor is used to make a temporary selection for the purposes of
+the command. Thus, to change a term of a formula, all you have to
+do is move the Emacs cursor to that term and press @kbd{j '}.
+
address@hidden j `
address@hidden calc-edit-selection
+The @kbd{j `} (@code{calc-edit-selection}) command is a similar
+analogue of the @kbd{`} (@code{calc-edit}) command. It edits the
+selected sub-formula in a separate buffer. If there is no
+selection, it edits the sub-formula indicated by the cursor.
+
+To delete a sub-formula, press @key{DEL}. This generally replaces
+the sub-formula with the constant zero, but in a few suitable contexts
+it uses the constant one instead. The @key{DEL} key automatically
+deselects and re-simplifies the entire formula afterwards. Thus:
+
address@hidden
address@hidden
+ ###
+ 17 x y + # # 17 x y 17 # y 17 y
+1* ------------- 1: ------- 1* ------- 1: -------
+ 2 x + 1 2 x + 1 2 x + 1 2 x + 1
address@hidden group
address@hidden smallexample
+
+In this example, we first delete the @samp{sqrt(c)} term; Calc
+accomplishes this by replacing @samp{sqrt(c)} with zero and
+resimplifying. We then delete the @kbd{x} in the numerator;
+since this is part of a product, Calc replaces it with @samp{1}
+and resimplifies.
+
+If you select an element of a vector and press @key{DEL}, that
+element is deleted from the vector. If you delete one side of
+an equation or inequality, only the opposite side remains.
+
address@hidden j @key{DEL}
address@hidden calc-del-selection
+The @kbd{j @key{DEL}} (@code{calc-del-selection}) command is like
address@hidden but with the auto-selecting behavior of @kbd{j '} and
address@hidden `}. It deletes the selected portion of the formula
+indicated by the cursor, or, in the absence of a selection, it
+deletes the sub-formula indicated by the cursor position.
+
address@hidden j @key{RET}
address@hidden calc-grab-selection
+(There is also an auto-selecting @kbd{j @key{RET}} (@code{calc-copy-selection})
+command.)
+
+Normal arithmetic operations also apply to sub-formulas. Here we
+select the denominator, press @kbd{5 -} to subtract five from the
+denominator, press @kbd{n} to negate the denominator, then
+press @kbd{Q} to take the square root.
+
address@hidden
address@hidden
+ .. . .. . .. . .. .
+1* ....... 1* ....... 1* ....... 1* ..........
+ 2 x + 1 2 x - 4 4 - 2 x _________
+ V 4 - 2 x
address@hidden group
address@hidden smallexample
+
+Certain types of operations on selections are not allowed. For
+example, for an arithmetic function like @kbd{-} no more than one of
+the arguments may be a selected sub-formula. (As the above example
+shows, the result of the subtraction is spliced back into the argument
+which had the selection; if there were more than one selection involved,
+this would not be well-defined.) If you try to subtract two selections,
+the command will abort with an error message.
+
+Operations on sub-formulas sometimes leave the formula as a whole
+in an ``un-natural'' state. Consider negating the @samp{2 x} term
+of our sample formula by selecting it and pressing @kbd{n}
+(@code{calc-change-sign}).
+
address@hidden
address@hidden
+ .. . .. .
+1* .......... 1* ...........
+ ......... ..........
+ . . . 2 x . . . -2 x
address@hidden group
address@hidden smallexample
+
+Unselecting the sub-formula reveals that the minus sign, which would
+normally have cancelled out with the subtraction automatically, has
+not been able to do so because the subtraction was not part of the
+selected portion. Pressing @kbd{=} (@code{calc-evaluate}) or doing
+any other mathematical operation on the whole formula will cause it
+to be simplified.
+
address@hidden
address@hidden
+ 17 y 17 y
+1: ----------- 1: ----------
+ __________ _________
+ V 4 - -2 x V 4 + 2 x
address@hidden group
address@hidden smallexample
+
address@hidden Rearranging with Selections, , Operating on Selections,
Selecting Subformulas
address@hidden Rearranging Formulas using Selections
+
address@hidden
address@hidden j R
address@hidden calc-commute-right
+The @kbd{j R} (@code{calc-commute-right}) command moves the selected
+sub-formula to the right in its surrounding formula. Generally the
+selection is one term of a sum or product; the sum or product is
+rearranged according to the commutative laws of algebra.
+
+As with @kbd{j '} and @kbd{j @key{DEL}}, the term under the cursor is used
+if there is no selection in the current formula. All commands described
+in this section share this property. In this example, we place the
+cursor on the @samp{a} and type @kbd{j R}, then repeat.
+
address@hidden
+1: a + b - c 1: b + a - c 1: b - c + a
address@hidden smallexample
+
address@hidden
+Note that in the final step above, the @samp{a} is switched with
+the @samp{c} but the signs are adjusted accordingly. When moving
+terms of sums and products, @kbd{j R} will never change the
+mathematical meaning of the formula.
+
+The selected term may also be an element of a vector or an argument
+of a function. The term is exchanged with the one to its right.
+In this case, the ``meaning'' of the vector or function may of
+course be drastically changed.
+
address@hidden
+1: [a, b, c] 1: [b, a, c] 1: [b, c, a]
+
+1: f(a, b, c) 1: f(b, a, c) 1: f(b, c, a)
address@hidden smallexample
+
address@hidden j L
address@hidden calc-commute-left
+The @kbd{j L} (@code{calc-commute-left}) command is like @kbd{j R}
+except that it swaps the selected term with the one to its left.
+
+With numeric prefix arguments, these commands move the selected
+term several steps at a time. It is an error to try to move a
+term left or right past the end of its enclosing formula.
+With numeric prefix arguments of zero, these commands move the
+selected term as far as possible in the given direction.
+
address@hidden j D
address@hidden calc-sel-distribute
+The @kbd{j D} (@code{calc-sel-distribute}) command mixes the selected
+sum or product into the surrounding formula using the distributive
+law. For example, in @samp{a * (b - c)} with the @samp{b - c}
+selected, the result is @samp{a b - a c}. This also distributes
+products or quotients into surrounding powers, and can also do
+transformations like @samp{exp(a + b)} to @samp{exp(a) exp(b)},
+where @samp{a + b} is the selected term, and @samp{ln(a ^ b)}
+to @samp{ln(a) b}, where @samp{a ^ b} is the selected term.
+
+For multiple-term sums or products, @kbd{j D} takes off one term
+at a time: @samp{a * (b + c - d)} goes to @samp{a * (c - d) + a b}
+with the @samp{c - d} selected so that you can type @kbd{j D}
+repeatedly to expand completely. The @kbd{j D} command allows a
+numeric prefix argument which specifies the maximum number of
+times to expand at once; the default is one time only.
+
address@hidden DistribRules
+The @kbd{j D} command is implemented using rewrite rules.
address@hidden with Rewrite Rules}. The rules are stored in
+the Calc variable @code{DistribRules}. A convenient way to view
+these rules is to use @kbd{s e} (@code{calc-edit-variable}) which
+displays and edits the stored value of a variable. Press @kbd{C-c C-c}
+to return from editing mode; be careful not to make any actual changes
+or else you will affect the behavior of future @kbd{j D} commands!
+
+To extend @kbd{j D} to handle new cases, just edit @code{DistribRules}
+as described above. You can then use the @kbd{s p} command to save
+this variable's value permanently for future Calc sessions.
address@hidden on Variables}.
+
address@hidden j M
address@hidden calc-sel-merge
address@hidden MergeRules
+The @kbd{j M} (@code{calc-sel-merge}) command is the complement
+of @kbd{j D}; given @samp{a b - a c} with either @samp{a b} or
address@hidden c} selected, the result is @samp{a * (b - c)}. Once
+again, @kbd{j M} can also merge calls to functions like @code{exp}
+and @code{ln}; examine the variable @code{MergeRules} to see all
+the relevant rules.
+
address@hidden j C
address@hidden calc-sel-commute
address@hidden CommuteRules
+The @kbd{j C} (@code{calc-sel-commute}) command swaps the arguments
+of the selected sum, product, or equation. It always behaves as
+if @kbd{j b} mode were in effect, i.e., the sum @samp{a + b + c} is
+treated as the nested sums @samp{(a + b) + c} by this command.
+If you put the cursor on the first @samp{+}, the result is
address@hidden(b + a) + c}; if you put the cursor on the second @samp{+}, the
+result is @samp{c + (a + b)} (which the default simplifications
+will rearrange to @samp{(c + a) + b}). The relevant rules are stored
+in the variable @code{CommuteRules}.
+
+You may need to turn default simplifications off (with the @kbd{m O}
+command) in order to get the full benefit of @kbd{j C}. For example,
+commuting @samp{a - b} produces @samp{-b + a}, but the default
+simplifications will ``simplify'' this right back to @samp{a - b} if
+you don't turn them off. The same is true of some of the other
+manipulations described in this section.
+
address@hidden j N
address@hidden calc-sel-negate
address@hidden NegateRules
+The @kbd{j N} (@code{calc-sel-negate}) command replaces the selected
+term with the negative of that term, then adjusts the surrounding
+formula in order to preserve the meaning. For example, given
address@hidden(a - b)} where @samp{a - b} is selected, the result is
address@hidden / exp(b - a)}. By contrast, selecting a term and using the
+regular @kbd{n} (@code{calc-change-sign}) command negates the
+term without adjusting the surroundings, thus changing the meaning
+of the formula as a whole. The rules variable is @code{NegateRules}.
+
address@hidden j &
address@hidden calc-sel-invert
address@hidden InvertRules
+The @kbd{j &} (@code{calc-sel-invert}) command is similar to @kbd{j N}
+except it takes the reciprocal of the selected term. For example,
+given @samp{a - ln(b)} with @samp{b} selected, the result is
address@hidden + ln(1/b)}. The rules variable is @code{InvertRules}.
+
address@hidden j E
address@hidden calc-sel-jump-equals
address@hidden JumpRules
+The @kbd{j E} (@code{calc-sel-jump-equals}) command moves the
+selected term from one side of an equation to the other. Given
address@hidden + b = c + d} with @samp{c} selected, the result is
address@hidden + b - c = d}. This command also works if the selected
+term is part of a @samp{*}, @samp{/}, or @samp{^} formula. The
+relevant rules variable is @code{JumpRules}.
+
address@hidden j I
address@hidden H j I
address@hidden calc-sel-isolate
+The @kbd{j I} (@code{calc-sel-isolate}) command isolates the
+selected term on its side of an equation. It uses the @kbd{a S}
+(@code{calc-solve-for}) command to solve the equation, and the
+Hyperbolic flag affects it in the same way. @xref{Solving Equations}.
+When it applies, @kbd{j I} is often easier to use than @kbd{j E}.
+It understands more rules of algebra, and works for inequalities
+as well as equations.
+
address@hidden j *
address@hidden j /
address@hidden calc-sel-mult-both-sides
address@hidden calc-sel-div-both-sides
+The @kbd{j *} (@code{calc-sel-mult-both-sides}) command prompts for a
+formula using algebraic entry, then multiplies both sides of the
+selected quotient or equation by that formula. It simplifies each
+side with @kbd{a s} (@code{calc-simplify}) before re-forming the
+quotient or equation. You can suppress this simplification by
+providing any numeric prefix argument. There is also a @kbd{j /}
+(@code{calc-sel-div-both-sides}) which is similar to @kbd{j *} but
+dividing instead of multiplying by the factor you enter.
+
+As a special feature, if the numerator of the quotient is 1, then
+the denominator is expanded at the top level using the distributive
+law (i.e., using the @kbd{C-u -1 a x} command). Suppose the
+formula on the stack is @samp{1 / (sqrt(a) + 1)}, and you wish
+to eliminate the square root in the denominator by multiplying both
+sides by @samp{sqrt(a) - 1}. Calc's default simplifications would
+change the result @samp{(sqrt(a) - 1) / (sqrt(a) - 1) (sqrt(a) + 1)}
+right back to the original form by cancellation; Calc expands the
+denominator to @samp{sqrt(a) (sqrt(a) - 1) + sqrt(a) - 1} to prevent
+this. (You would now want to use an @kbd{a x} command to expand
+the rest of the way, whereupon the denominator would cancel out to
+the desired form, @samp{a - 1}.) When the numerator is not 1, this
+initial expansion is not necessary because Calc's default
+simplifications will not notice the potential cancellation.
+
+If the selection is an inequality, @kbd{j *} and @kbd{j /} will
+accept any factor, but will warn unless they can prove the factor
+is either positive or negative. (In the latter case the direction
+of the inequality will be switched appropriately.) @xref{Declarations},
+for ways to inform Calc that a given variable is positive or
+negative. If Calc can't tell for sure what the sign of the factor
+will be, it will assume it is positive and display a warning
+message.
+
+For selections that are not quotients, equations, or inequalities,
+these commands pull out a multiplicative factor: They divide (or
+multiply) by the entered formula, simplify, then multiply (or divide)
+back by the formula.
+
address@hidden j +
address@hidden j -
address@hidden calc-sel-add-both-sides
address@hidden calc-sel-sub-both-sides
+The @kbd{j +} (@code{calc-sel-add-both-sides}) and @kbd{j -}
+(@code{calc-sel-sub-both-sides}) commands analogously add to or
+subtract from both sides of an equation or inequality. For other
+types of selections, they extract an additive factor. A numeric
+prefix argument suppresses simplification of the intermediate
+results.
+
address@hidden j U
address@hidden calc-sel-unpack
+The @kbd{j U} (@code{calc-sel-unpack}) command replaces the
+selected function call with its argument. For example, given
address@hidden + sin(x^2)} with @samp{sin(x^2)} selected, the result
+is @samp{a + x^2}. (The @samp{x^2} will remain selected; if you
+wanted to change the @code{sin} to @code{cos}, just press @kbd{C}
+now to take the cosine of the selected part.)
+
address@hidden j v
address@hidden calc-sel-evaluate
+The @kbd{j v} (@code{calc-sel-evaluate}) command performs the
+normal default simplifications on the selected sub-formula.
+These are the simplifications that are normally done automatically
+on all results, but which may have been partially inhibited by
+previous selection-related operations, or turned off altogether
+by the @kbd{m O} command. This command is just an auto-selecting
+version of the @address@hidden v}} command (@pxref{Algebraic Manipulation}).
+
+With a numeric prefix argument of 2, @kbd{C-u 2 j v} applies
+the @kbd{a s} (@code{calc-simplify}) command to the selected
+sub-formula. With a prefix argument of 3 or more, e.g., @kbd{C-u j v}
+applies the @kbd{a e} (@code{calc-simplify-extended}) command.
address@hidden Formulas}. With a negative prefix argument
+it simplifies at the top level only, just as with @kbd{a v}.
+Here the ``top'' level refers to the top level of the selected
+sub-formula.
+
address@hidden j "
address@hidden calc-sel-expand-formula
+The @kbd{j "} (@code{calc-sel-expand-formula}) command is to @kbd{a "}
+(@pxref{Algebraic Manipulation}) what @kbd{j v} is to @kbd{a v}.
+
+You can use the @kbd{j r} (@code{calc-rewrite-selection}) command
+to define other algebraic operations on sub-formulas. @xref{Rewrite Rules}.
+
address@hidden Algebraic Manipulation, Simplifying Formulas, Selecting
Subformulas, Algebra
address@hidden Algebraic Manipulation
+
address@hidden
+The commands in this section perform general-purpose algebraic
+manipulations. They work on the whole formula at the top of the
+stack (unless, of course, you have made a selection in that
+formula).
+
+Many algebra commands prompt for a variable name or formula. If you
+answer the prompt with a blank line, the variable or formula is taken
+from top-of-stack, and the normal argument for the command is taken
+from the second-to-top stack level.
+
address@hidden a v
address@hidden calc-alg-evaluate
+The @kbd{a v} (@code{calc-alg-evaluate}) command performs the normal
+default simplifications on a formula; for example, @samp{a - -b} is
+changed to @samp{a + b}. These simplifications are normally done
+automatically on all Calc results, so this command is useful only if
+you have turned default simplifications off with an @kbd{m O}
+command. @xref{Simplification Modes}.
+
+It is often more convenient to type @kbd{=}, which is like @kbd{a v}
+but which also substitutes stored values for variables in the formula.
+Use @kbd{a v} if you want the variables to ignore their stored values.
+
+If you give a numeric prefix argument of 2 to @kbd{a v}, it simplifies
+as if in Algebraic Simplification mode. This is equivalent to typing
address@hidden s}; @pxref{Simplifying Formulas}. If you give a numeric prefix
+of 3 or more, it uses Extended Simplification mode (@kbd{a e}).
+
+If you give a negative prefix argument @mathit{-1}, @mathit{-2}, or
@mathit{-3},
+it simplifies in the corresponding mode but only works on the top-level
+function call of the formula. For example, @samp{(2 + 3) * (2 + 3)} will
+simplify to @samp{(2 + 3)^2}, without simplifying the sub-formulas
address@hidden + 3}. As another example, typing @kbd{V R +} to sum the vector
address@hidden, 2, 3, 4]} produces the formula @samp{reduce(add, [1, 2, 3, 4])}
+in No-Simplify mode. Using @kbd{a v} will evaluate this all the way to
+10; using @kbd{C-u - a v} will evaluate it only to @samp{1 + 2 + 3 + 4}.
+(@xref{Reducing and Mapping}.)
+
address@hidden evalv
address@hidden evalvn
+The @kbd{=} command corresponds to the @code{evalv} function, and
+the related @kbd{N} command, which is like @kbd{=} but temporarily
+disables Symbolic mode (@kbd{m s}) during the evaluation, corresponds
+to the @code{evalvn} function. (These commands interpret their prefix
+arguments differently than @kbd{a v}; @kbd{=} treats the prefix as
+the number of stack elements to evaluate at once, and @kbd{N} treats
+it as a temporary different working precision.)
+
+The @code{evalvn} function can take an alternate working precision
+as an optional second argument. This argument can be either an
+integer, to set the precision absolutely, or a vector containing
+a single integer, to adjust the precision relative to the current
+precision. Note that @code{evalvn} with a larger than current
+precision will do the calculation at this higher precision, but the
+result will as usual be rounded back down to the current precision
+afterward. For example, @samp{evalvn(pi - 3.1415)} at a precision
+of 12 will return @samp{9.265359e-5}; @samp{evalvn(pi - 3.1415, 30)}
+will return @samp{9.26535897932e-5} (computing a 25-digit result which
+is then rounded down to 12); and @samp{evalvn(pi - 3.1415, [-2])}
+will return @samp{9.2654e-5}.
+
address@hidden a "
address@hidden calc-expand-formula
+The @kbd{a "} (@code{calc-expand-formula}) command expands functions
+into their defining formulas wherever possible. For example,
address@hidden(x^2)} is changed to @samp{180 x^2 / pi}. Most functions,
+like @code{sin} and @code{gcd}, are not defined by simple formulas
+and so are unaffected by this command. One important class of
+functions which @emph{can} be expanded is the user-defined functions
+created by the @kbd{Z F} command. @xref{Algebraic Definitions}.
+Other functions which @kbd{a "} can expand include the probability
+distribution functions, most of the financial functions, and the
+hyperbolic and inverse hyperbolic functions. A numeric prefix argument
+affects @kbd{a "} in the same way as it does @kbd{a v}: A positive
+argument expands all functions in the formula and then simplifies in
+various ways; a negative argument expands and simplifies only the
+top-level function call.
+
address@hidden a M
address@hidden calc-map-equation
address@hidden mapeq
+The @kbd{a M} (@code{calc-map-equation}) address@hidden command applies
+a given function or operator to one or more equations. It is analogous
+to @kbd{V M}, which operates on vectors instead of equations.
address@hidden and Mapping}. For example, @kbd{a M S} changes
address@hidden = y+1} to @samp{sin(x) = sin(y+1)}, and @kbd{a M +} with
address@hidden = y+1} and @expr{6} on the stack produces @samp{x+6 = y+7}.
+With two equations on the stack, @kbd{a M +} would add the lefthand
+sides together and the righthand sides together to get the two
+respective sides of a new equation.
+
+Mapping also works on inequalities. Mapping two similar inequalities
+produces another inequality of the same type. Mapping an inequality
+with an equation produces an inequality of the same type. Mapping a
address@hidden<=} with a @samp{<} or @samp{!=} (not-equal) produces a @samp{<}.
+If inequalities with opposite direction (e.g., @samp{<} and @samp{>})
+are mapped, the direction of the second inequality is reversed to
+match the first: Using @kbd{a M +} on @samp{a < b} and @samp{a > 2}
+reverses the latter to get @samp{2 < a}, which then allows the
+combination @samp{a + 2 < b + a}, which the @kbd{a s} command can
+then simplify to get @samp{2 < b}.
+
+Using @kbd{a M *}, @kbd{a M /}, @kbd{a M n}, or @kbd{a M &} to negate
+or invert an inequality will reverse the direction of the inequality.
+Other adjustments to inequalities are @emph{not} done automatically;
address@hidden M S} will change @address@hidden < y}} to @samp{sin(x) < sin(y)}
even
+though this is not true for all values of the variables.
+
address@hidden H a M
address@hidden mapeqp
+With the Hyperbolic flag, @kbd{H a M} address@hidden does a plain
+mapping operation without reversing the direction of any inequalities.
+Thus, @kbd{H a M &} would change @kbd{x > 2} to @kbd{1/x > 0.5}.
+(This change is mathematically incorrect, but perhaps you were
+fixing an inequality which was already incorrect.)
+
address@hidden I a M
address@hidden mapeqr
+With the Inverse flag, @kbd{I a M} address@hidden always reverses
+the direction of the inequality. You might use @kbd{I a M C} to
+change @samp{x < y} to @samp{cos(x) > cos(y)} if you know you are
+working with small positive angles.
+
address@hidden a b
address@hidden calc-substitute
address@hidden subst
+The @kbd{a b} (@code{calc-substitute}) address@hidden command substitutes
+all occurrences
+of some variable or sub-expression of an expression with a new
+sub-expression. For example, substituting @samp{sin(x)} with @samp{cos(y)}
+in @samp{2 sin(x)^2 + x sin(x) + sin(2 x)} produces
address@hidden cos(y)^2 + x cos(y) + @w{sin(2 x)}}.
+Note that this is a purely structural substitution; the lone @samp{x} and
+the @samp{sin(2 x)} stayed the same because they did not look like
address@hidden(x)}. @xref{Rewrite Rules}, for a more general method for
+doing substitutions.
+
+The @kbd{a b} command normally prompts for two formulas, the old
+one and the new one. If you enter a blank line for the first
+prompt, all three arguments are taken from the stack (new, then old,
+then target expression). If you type an old formula but then enter a
+blank line for the new one, the new formula is taken from top-of-stack
+and the target from second-to-top. If you answer both prompts, the
+target is taken from top-of-stack as usual.
+
+Note that @kbd{a b} has no understanding of commutativity or
+associativity. The pattern @samp{x+y} will not match the formula
address@hidden Also, @samp{y+z} will not match inside the formula @samp{x+y+z}
+because the @samp{+} operator is left-associative, so the ``deep
+structure'' of that formula is @samp{(x+y) + z}. Use @kbd{d U}
+(@code{calc-unformatted-language}) mode to see the true structure of
+a formula. The rewrite rule mechanism, discussed later, does not have
+these limitations.
+
+As an algebraic function, @code{subst} takes three arguments:
+Target expression, old, new. Note that @code{subst} is always
+evaluated immediately, even if its arguments are variables, so if
+you wish to put a call to @code{subst} onto the stack you must
+turn the default simplifications off first (with @kbd{m O}).
+
address@hidden Simplifying Formulas, Polynomials, Algebraic Manipulation,
Algebra
address@hidden Simplifying Formulas
+
address@hidden
address@hidden a s
address@hidden calc-simplify
address@hidden simplify
+The @kbd{a s} (@code{calc-simplify}) address@hidden command applies
+various algebraic rules to simplify a formula. This includes rules which
+are not part of the default simplifications because they may be too slow
+to apply all the time, or may not be desirable all of the time. For
+example, non-adjacent terms of sums are combined, as in @samp{a + b + 2 a}
+to @samp{b + 3 a}, and some formulas like @samp{sin(arcsin(x))} are
+simplified to @samp{x}.
+
+The sections below describe all the various kinds of algebraic
+simplifications Calc provides in full detail. None of Calc's
+simplification commands are designed to pull rabbits out of hats;
+they simply apply certain specific rules to put formulas into
+less redundant or more pleasing forms. Serious algebra in Calc
+must be done manually, usually with a combination of selections
+and rewrite rules. @xref{Rearranging with Selections}.
address@hidden Rules}.
+
address@hidden Modes}, for commands to control what level of
+simplification occurs automatically. Normally only the ``default
+simplifications'' occur.
+
address@hidden
+* Default Simplifications::
+* Algebraic Simplifications::
+* Unsafe Simplifications::
+* Simplification of Units::
address@hidden menu
+
address@hidden Default Simplifications, Algebraic Simplifications, Simplifying
Formulas, Simplifying Formulas
address@hidden Default Simplifications
+
address@hidden
address@hidden Default simplifications
+This section describes the ``default simplifications,'' those which are
+normally applied to all results. For example, if you enter the variable
address@hidden on the stack twice and push @kbd{+}, Calc's default
+simplifications automatically change @expr{x + x} to @expr{2 x}.
+
+The @kbd{m O} command turns off the default simplifications, so that
address@hidden + x} will remain in this form unless you give an explicit
+``simplify'' command like @kbd{=} or @kbd{a v}. @xref{Algebraic
+Manipulation}. The @kbd{m D} command turns the default simplifications
+back on.
+
+The most basic default simplification is the evaluation of functions.
+For example, @expr{2 + 3} is evaluated to @expr{5}, and @address@hidden(9)}
+is evaluated to @expr{3}. Evaluation does not occur if the arguments
+to a function are somehow of the wrong type @address@hidden([2,3,4])}),
+range (@address@hidden(90)}), or number (@address@hidden(3,5)}),
+or if the function name is not recognized (@address@hidden(5)}), or if
+Symbolic mode (@pxref{Symbolic Mode}) prevents evaluation
+(@address@hidden(2)}).
+
+Calc simplifies (evaluates) the arguments to a function before it
+simplifies the function itself. Thus @address@hidden(5+4)} is
+simplified to @address@hidden(9)} before the @code{sqrt} function
+itself is applied. There are very few exceptions to this rule:
address@hidden, @code{lambda}, and @code{condition} (the @code{::}
+operator) do not evaluate their arguments, @code{if} (the @code{? :}
+operator) does not evaluate all of its arguments, and @code{evalto}
+does not evaluate its lefthand argument.
+
+Most commands apply the default simplifications to all arguments they
+take from the stack, perform a particular operation, then simplify
+the result before pushing it back on the stack. In the common special
+case of regular arithmetic commands like @kbd{+} and @kbd{Q} address@hidden,
+the arguments are simply popped from the stack and collected into a
+suitable function call, which is then simplified (the arguments being
+simplified first as part of the process, as described above).
+
+The default simplifications are too numerous to describe completely
+here, but this section will describe the ones that apply to the
+major arithmetic operators. This list will be rather technical in
+nature, and will probably be interesting to you only if you are
+a serious user of Calc's algebra facilities.
+
address@hidden
+\bigskip
address@hidden tex
+
+As well as the simplifications described here, if you have stored
+any rewrite rules in the variable @code{EvalRules} then these rules
+will also be applied before any built-in default simplifications.
address@hidden Rewrites}, for details.
+
address@hidden
+\bigskip
address@hidden tex
+
+And now, on with the default simplifications:
+
+Arithmetic operators like @kbd{+} and @kbd{*} always take two
+arguments in Calc's internal form. Sums and products of three or
+more terms are arranged by the associative law of algebra into
+a left-associative form for sums, @expr{((a + b) + c) + d}, and
+a right-associative form for products, @expr{a * (b * (c * d))}.
+Formulas like @expr{(a + b) + (c + d)} are rearranged to
+left-associative form, though this rarely matters since Calc's
+algebra commands are designed to hide the inner structure of
+sums and products as much as possible. Sums and products in
+their proper associative form will be written without parentheses
+in the examples below.
+
+Sums and products are @emph{not} rearranged according to the
+commutative law (@expr{a + b} to @expr{b + a}) except in a few
+special cases described below. Some algebra programs always
+rearrange terms into a canonical order, which enables them to
+see that @expr{a b + b a} can be simplified to @expr{2 a b}.
+Calc assumes you have put the terms into the order you want
+and generally leaves that order alone, with the consequence
+that formulas like the above will only be simplified if you
+explicitly give the @kbd{a s} command. @xref{Algebraic
+Simplifications}.
+
+Differences @expr{a - b} are treated like sums @expr{a + (-b)}
+for purposes of simplification; one of the default simplifications
+is to rewrite @expr{a + (-b)} or @expr{(-b) + a}, where @expr{-b}
+represents a ``negative-looking'' term, into @expr{a - b} form.
+``Negative-looking'' means negative numbers, negated formulas like
address@hidden, and products or quotients in which either term is
+negative-looking.
+
+Other simplifications involving negation are @expr{-(-x)} to @expr{x};
address@hidden(a b)} or @expr{-(a/b)} where either @expr{a} or @expr{b} is
+negative-looking, simplified by negating that term, or else where
address@hidden or @expr{b} is any number, by negating that number;
address@hidden(a + b)} to @expr{-a - b}, and @expr{-(b - a)} to @expr{a - b}.
+(This, and rewriting @expr{(-b) + a} to @expr{a - b}, are the only
+cases where the order of terms in a sum is changed by the default
+simplifications.)
+
+The distributive law is used to simplify sums in some cases:
address@hidden x + b x} to @expr{(a + b) x}, where @expr{a} represents
+a number or an implicit 1 or @mathit{-1} (as in @expr{x} or @expr{-x})
+and similarly for @expr{b}. Use the @kbd{a c}, @address@hidden f}}, or
address@hidden M} commands to merge sums with non-numeric coefficients
+using the distributive law.
+
+The distributive law is only used for sums of two terms, or
+for adjacent terms in a larger sum. Thus @expr{a + b + b + c}
+is simplified to @expr{a + 2 b + c}, but @expr{a + b + c + b}
+is not simplified. The reason is that comparing all terms of a
+sum with one another would require time proportional to the
+square of the number of terms; Calc relegates potentially slow
+operations like this to commands that have to be invoked
+explicitly, like @kbd{a s}.
+
+Finally, @expr{a + 0} and @expr{0 + a} are simplified to @expr{a}.
+A consequence of the above rules is that @expr{0 - a} is simplified
+to @expr{-a}.
+
address@hidden
+\bigskip
address@hidden tex
+
+The products @expr{1 a} and @expr{a 1} are simplified to @expr{a};
address@hidden(-1) a} and @expr{a (-1)} are simplified to @expr{-a};
address@hidden a} and @expr{a 0} are simplified to @expr{0}, except that
+in Matrix mode where @expr{a} is not provably scalar the result
+is the generic zero matrix @samp{idn(0)}, and that if @expr{a} is
+infinite the result is @samp{nan}.
+
+Also, @expr{(-a) b} and @expr{a (-b)} are simplified to @expr{-(a b)},
+where this occurs for negated formulas but not for regular negative
+numbers.
+
+Products are commuted only to move numbers to the front:
address@hidden b 2} is commuted to @expr{2 a b}.
+
+The product @expr{a (b + c)} is distributed over the sum only if
address@hidden and at least one of @expr{b} and @expr{c} are numbers:
address@hidden (x + 3)} goes to @expr{2 x + 6}. The formula
address@hidden(-a) (b - c)}, where @expr{-a} is a negative number, is
+rewritten to @expr{a (c - b)}.
+
+The distributive law of products and powers is used for adjacent
+terms of the product: @expr{x^a x^b} goes to
address@hidden @math{x^{a+b}}
address@hidden @expr{x^(a+b)}
+where @expr{a} is a number, or an implicit 1 (as in @expr{x}),
+or the implicit one-half of @address@hidden(x)}, and similarly for
address@hidden The result is written using @samp{sqrt} or @samp{1/sqrt}
+if the sum of the powers is @expr{1/2} or @expr{-1/2}, respectively.
+If the sum of the powers is zero, the product is simplified to
address@hidden or to @samp{idn(1)} if Matrix mode is enabled.
+
+The product of a negative power times anything but another negative
+power is changed to use division:
address@hidden @math{x^{-2} y}
address@hidden @expr{x^(-2) y}
+goes to @expr{y / x^2} unless Matrix mode is
+in effect and neither @expr{x} nor @expr{y} are scalar (in which
+case it is considered unsafe to rearrange the order of the terms).
+
+Finally, @expr{a (b/c)} is rewritten to @expr{(a b)/c}, and also
address@hidden(a/b) c} is changed to @expr{(a c)/b} unless in Matrix mode.
+
address@hidden
+\bigskip
address@hidden tex
+
+Simplifications for quotients are analogous to those for products.
+The quotient @expr{0 / x} is simplified to @expr{0}, with the same
+exceptions that were noted for @expr{0 x}. Likewise, @expr{x / 1}
+and @expr{x / (-1)} are simplified to @expr{x} and @expr{-x},
+respectively.
+
+The quotient @expr{x / 0} is left unsimplified or changed to an
+infinite quantity, as directed by the current infinite mode.
address@hidden Mode}.
+
+The expression
address@hidden @math{a / b^{-c}}
address@hidden @expr{a / b^(-c)}
+is changed to @expr{a b^c}, where @expr{-c} is any negative-looking
+power. Also, @expr{1 / b^c} is changed to
address@hidden @math{b^{-c}}
address@hidden @expr{b^(-c)}
+for any power @expr{c}.
+
+Also, @expr{(-a) / b} and @expr{a / (-b)} go to @expr{-(a/b)};
address@hidden(a/b) / c} goes to @expr{a / (b c)}; and @expr{a / (b/c)}
+goes to @expr{(a c) / b} unless Matrix mode prevents this
+rearrangement. Similarly, @expr{a / (b:c)} is simplified to
address@hidden(c:b) a} for any fraction @expr{b:c}.
+
+The distributive law is applied to @expr{(a + b) / c} only if
address@hidden and at least one of @expr{a} and @expr{b} are numbers.
+Quotients of powers and square roots are distributed just as
+described for multiplication.
+
+Quotients of products cancel only in the leading terms of the
+numerator and denominator. In other words, @expr{a x b / a y b}
+is cancelled to @expr{x b / y b} but not to @expr{x / y}. Once
+again this is because full cancellation can be slow; use @kbd{a s}
+to cancel all terms of the quotient.
+
+Quotients of negative-looking values are simplified according
+to @expr{(-a) / (-b)} to @expr{a / b}, @expr{(-a) / (b - c)}
+to @expr{a / (c - b)}, and @expr{(a - b) / (-c)} to @expr{(b - a) / c}.
+
address@hidden
+\bigskip
address@hidden tex
+
+The formula @expr{x^0} is simplified to @expr{1}, or to @samp{idn(1)}
+in Matrix mode. The formula @expr{0^x} is simplified to @expr{0}
+unless @expr{x} is a negative number, complex number or zero.
+If @expr{x} is negative, complex or @expr{0.0}, @expr{0^x} is an
+infinity or an unsimplified formula according to the current infinite
+mode. The expression @expr{0^0} is simplified to @expr{1}.
+
+Powers of products or quotients @expr{(a b)^c}, @expr{(a/b)^c}
+are distributed to @expr{a^c b^c}, @expr{a^c / b^c} only if @expr{c}
+is an integer, or if either @expr{a} or @expr{b} are nonnegative
+real numbers. Powers of powers @expr{(a^b)^c} are simplified to
address@hidden @math{a^{b c}}
address@hidden @expr{a^(b c)}
+only when @expr{c} is an integer and @expr{b c} also
+evaluates to an integer. Without these restrictions these simplifications
+would not be safe because of problems with principal values.
+(In other words,
address@hidden @math{((-3)^{1/2})^2}
address@hidden @expr{((-3)^1:2)^2}
+is safe to simplify, but
address@hidden @math{((-3)^2)^{1/2}}
address@hidden @expr{((-3)^2)^1:2}
+is not.) @xref{Declarations}, for ways to inform Calc that your
+variables satisfy these requirements.
+
+As a special case of this rule, @address@hidden(x)^n} is simplified to
address@hidden @math{x^{n/2}}
address@hidden @expr{x^(n/2)}
+only for even integers @expr{n}.
+
+If @expr{a} is known to be real, @expr{b} is an even integer, and
address@hidden is a half- or quarter-integer, then @expr{(a^b)^c} is
+simplified to @address@hidden(a^(b c))}.
+
+Also, @expr{(-a)^b} is simplified to @expr{a^b} if @expr{b} is an
+even integer, or to @expr{-(a^b)} if @expr{b} is an odd integer,
+for any negative-looking expression @expr{-a}.
+
+Square roots @address@hidden(x)} generally act like one-half powers
address@hidden @math{x^{1:2}}
address@hidden @expr{x^1:2}
+for the purposes of the above-listed simplifications.
+
+Also, note that
address@hidden @math{1 / x^{1:2}}
address@hidden @expr{1 / x^1:2}
+is changed to
address@hidden @math{x^{-1:2}},
address@hidden @expr{x^(-1:2)},
+but @expr{1 / @tfn{sqrt}(x)} is left alone.
+
address@hidden
+\bigskip
address@hidden tex
+
+Generic identity matrices (@pxref{Matrix Mode}) are simplified by the
+following rules: @address@hidden(a) + b} to @expr{a + b} if @expr{b}
+is provably scalar, or expanded out if @expr{b} is a matrix;
address@hidden@tfn{idn}(a) + @tfn{idn}(b)} to @address@hidden(a + b)};
address@hidden@tfn{idn}(a)} to @address@hidden(-a)}; @expr{a @tfn{idn}(b)} to
address@hidden@tfn{idn}(a b)} if @expr{a} is provably scalar, or to @expr{a b}
+if @expr{a} is provably non-scalar; @address@hidden(a) @tfn{idn}(b)} to
address@hidden@tfn{idn}(a b)}; analogous simplifications for quotients involving
address@hidden; and @address@hidden(a)^n} to @address@hidden(a^n)} where
address@hidden is an integer.
+
address@hidden
+\bigskip
address@hidden tex
+
+The @code{floor} function and other integer truncation functions
+vanish if the argument is provably integer-valued, so that
address@hidden@tfn{floor}(@tfn{round}(x))} simplifies to @address@hidden(x)}.
+Also, combinations of @code{float}, @code{floor} and its friends,
+and @code{ffloor} and its friends, are simplified in appropriate
+ways. @xref{Integer Truncation}.
+
+The expression @address@hidden(-x)} changes to @address@hidden(x)}.
+The expression @address@hidden(@tfn{abs}(x))} changes to
address@hidden@tfn{abs}(x)}; in fact, @address@hidden(x)} changes to @expr{x}
or
address@hidden if @expr{x} is provably nonnegative or nonpositive
+(@pxref{Declarations}).
+
+While most functions do not recognize the variable @code{i} as an
+imaginary number, the @code{arg} function does handle the two cases
address@hidden@tfn{arg}(@tfn{i})} and @address@hidden(address@hidden)} just for
convenience.
+
+The expression @address@hidden(@tfn{conj}(x))} simplifies to @expr{x}.
+Various other expressions involving @code{conj}, @code{re}, and
address@hidden are simplified, especially if some of the arguments are
+provably real or involve the constant @code{i}. For example,
address@hidden@tfn{conj}(a + b i)} is changed to
address@hidden@tfn{conj}(a) - @tfn{conj}(b) i}, or to @expr{a - b i} if
@expr{a}
+and @expr{b} are known to be real.
+
+Functions like @code{sin} and @code{arctan} generally don't have
+any default simplifications beyond simply evaluating the functions
+for suitable numeric arguments and infinity. The @kbd{a s} command
+described in the next section does provide some simplifications for
+these functions, though.
+
+One important simplification that does occur is that
address@hidden@tfn{ln}(@tfn{e})} is simplified to 1, and
@address@hidden(@tfn{e}^x)} is
+simplified to @expr{x} for any @expr{x}. This occurs even if you have
+stored a different value in the Calc variable @samp{e}; but this would
+be a bad idea in any case if you were also using natural logarithms!
+
+Among the logical functions, @tfn{!(@var{a} <= @var{b})} changes to
address@hidden@var{a} > @var{b}} and so on. Equations and inequalities where
both sides
+are either negative-looking or zero are simplified by negating both sides
+and reversing the inequality. While it might seem reasonable to simplify
address@hidden to @expr{x}, this would not be valid in general because
address@hidden is 1, not 2.
+
+Most other Calc functions have few if any default simplifications
+defined, aside of course from evaluation when the arguments are
+suitable numbers.
+
address@hidden Algebraic Simplifications, Unsafe Simplifications, Default
Simplifications, Simplifying Formulas
address@hidden Algebraic Simplifications
+
address@hidden
address@hidden Algebraic simplifications
+The @kbd{a s} command makes simplifications that may be too slow to
+do all the time, or that may not be desirable all of the time.
+If you find these simplifications are worthwhile, you can type
address@hidden A} to have Calc apply them automatically.
+
+This section describes all simplifications that are performed by
+the @kbd{a s} command. Note that these occur in addition to the
+default simplifications; even if the default simplifications have
+been turned off by an @kbd{m O} command, @kbd{a s} will turn them
+back on temporarily while it simplifies the formula.
+
+There is a variable, @code{AlgSimpRules}, in which you can put rewrites
+to be applied by @kbd{a s}. Its use is analogous to @code{EvalRules},
+but without the special restrictions. Basically, the simplifier does
address@hidden@w{a r} AlgSimpRules} with an infinite repeat count on the whole
+expression being simplified, then it traverses the expression applying
+the built-in rules described below. If the result is different from
+the original expression, the process repeats with the default
+simplifications (including @code{EvalRules}), then @code{AlgSimpRules},
+then the built-in simplifications, and so on.
+
address@hidden
+\bigskip
address@hidden tex
+
+Sums are simplified in two ways. Constant terms are commuted to the
+end of the sum, so that @expr{a + 2 + b} changes to @expr{a + b + 2}.
+The only exception is that a constant will not be commuted away
+from the first position of a difference, i.e., @expr{2 - x} is not
+commuted to @expr{-x + 2}.
+
+Also, terms of sums are combined by the distributive law, as in
address@hidden + y + 2 x} to @expr{y + 3 x}. This always occurs for
+adjacent terms, but @kbd{a s} compares all pairs of terms including
+non-adjacent ones.
+
address@hidden
+\bigskip
address@hidden tex
+
+Products are sorted into a canonical order using the commutative
+law. For example, @expr{b c a} is commuted to @expr{a b c}.
+This allows easier comparison of products; for example, the default
+simplifications will not change @expr{x y + y x} to @expr{2 x y},
+but @kbd{a s} will; it first rewrites the sum to @expr{x y + x y},
+and then the default simplifications are able to recognize a sum
+of identical terms.
+
+The canonical ordering used to sort terms of products has the
+property that real-valued numbers, interval forms and infinities
+come first, and are sorted into increasing order. The @kbd{V S}
+command uses the same ordering when sorting a vector.
+
+Sorting of terms of products is inhibited when Matrix mode is
+turned on; in this case, Calc will never exchange the order of
+two terms unless it knows at least one of the terms is a scalar.
+
+Products of powers are distributed by comparing all pairs of
+terms, using the same method that the default simplifications
+use for adjacent terms of products.
+
+Even though sums are not sorted, the commutative law is still
+taken into account when terms of a product are being compared.
+Thus @expr{(x + y) (y + x)} will be simplified to @expr{(x + y)^2}.
+A subtle point is that @expr{(x - y) (y - x)} will @emph{not}
+be simplified to @expr{-(x - y)^2}; Calc does not notice that
+one term can be written as a constant times the other, even if
+that constant is @mathit{-1}.
+
+A fraction times any expression, @expr{(a:b) x}, is changed to
+a quotient involving integers: @expr{a x / b}. This is not
+done for floating-point numbers like @expr{0.5}, however. This
+is one reason why you may find it convenient to turn Fraction mode
+on while doing algebra; @pxref{Fraction Mode}.
+
address@hidden
+\bigskip
address@hidden tex
+
+Quotients are simplified by comparing all terms in the numerator
+with all terms in the denominator for possible cancellation using
+the distributive law. For example, @expr{a x^2 b / c x^3 d} will
+cancel @expr{x^2} from the top and bottom to get @expr{a b / c x d}.
+(The terms in the denominator will then be rearranged to @expr{c d x}
+as described above.) If there is any common integer or fractional
+factor in the numerator and denominator, it is cancelled out;
+for example, @expr{(4 x + 6) / 8 x} simplifies to @expr{(2 x + 3) / 4 x}.
+
+Non-constant common factors are not found even by @kbd{a s}. To
+cancel the factor @expr{a} in @expr{(a x + a) / a^2} you could first
+use @kbd{j M} on the product @expr{a x} to Merge the numerator to
address@hidden (1+x)}, which can then be simplified successfully.
+
address@hidden
+\bigskip
address@hidden tex
+
+Integer powers of the variable @code{i} are simplified according
+to the identity @expr{i^2 = -1}. If you store a new value other
+than the complex number @expr{(0,1)} in @code{i}, this simplification
+will no longer occur. This is done by @kbd{a s} instead of by default
+in case someone (unwisely) uses the name @code{i} for a variable
+unrelated to complex numbers; it would be unfortunate if Calc
+quietly and automatically changed this formula for reasons the
+user might not have been thinking of.
+
+Square roots of integer or rational arguments are simplified in
+several ways. (Note that these will be left unevaluated only in
+Symbolic mode.) First, square integer or rational factors are
+pulled out so that @address@hidden(8)} is rewritten as
address@hidden @math{2\,@tfn{sqrt}(2)}.
address@hidden @expr{2 sqrt(2)}.
+Conceptually speaking this implies factoring the argument into primes
+and moving pairs of primes out of the square root, but for reasons of
+efficiency Calc only looks for primes up to 29.
+
+Square roots in the denominator of a quotient are moved to the
+numerator: @expr{1 / @tfn{sqrt}(3)} changes to @address@hidden(3) / 3}.
+The same effect occurs for the square root of a fraction:
address@hidden@tfn{sqrt}(2:3)} changes to @address@hidden(6) / 3}.
+
address@hidden
+\bigskip
address@hidden tex
+
+The @code{%} (modulo) operator is simplified in several ways
+when the modulus @expr{M} is a positive real number. First, if
+the argument is of the form @expr{x + n} for some real number
address@hidden, then @expr{n} is itself reduced modulo @expr{M}. For
+example, @samp{(x - 23) % 10} is simplified to @samp{(x + 7) % 10}.
+
+If the argument is multiplied by a constant, and this constant
+has a common integer divisor with the modulus, then this factor is
+cancelled out. For example, @samp{12 x % 15} is changed to
address@hidden (4 x % 5)} by factoring out 3. Also, @samp{(12 x + 1) % 15}
+is changed to @samp{3 ((4 x + 1:3) % 5)}. While these forms may
+not seem ``simpler,'' they allow Calc to discover useful information
+about modulo forms in the presence of declarations.
+
+If the modulus is 1, then Calc can use @code{int} declarations to
+evaluate the expression. For example, the idiom @samp{x % 2} is
+often used to check whether a number is odd or even. As described
+above, @address@hidden n % 2}} and @samp{(2 n + 1) % 2} are simplified to
address@hidden (n % 1)} and @samp{2 ((n + 1:2) % 1)}, respectively; Calc
+can simplify these to 0 and 1 (respectively) if @code{n} has been
+declared to be an integer.
+
address@hidden
+\bigskip
address@hidden tex
+
+Trigonometric functions are simplified in several ways. Whenever a
+products of two trigonometric functions can be replaced by a single
+function, the replacement is made; for example,
address@hidden@tfn{tan}(x) @tfn{cos}(x)} is simplified to @address@hidden(x)}.
+Reciprocals of trigonometric functions are replaced by their reciprocal
+function; for example, @expr{1/@tfn{sec}(x)} is simplified to
address@hidden@tfn{cos}(x)}. The corresponding simplifications for the
+hyperbolic functions are also handled.
+
+Trigonometric functions of their inverse functions are
+simplified. The expression @address@hidden(@tfn{arcsin}(x))} is
+simplified to @expr{x}, and similarly for @code{cos} and @code{tan}.
+Trigonometric functions of inverses of different trigonometric
+functions can also be simplified, as in @address@hidden(@tfn{arccos}(x))}
+to @address@hidden(1 - x^2)}.
+
+If the argument to @code{sin} is negative-looking, it is simplified to
address@hidden@tfn{sin}(x)}, and similarly for @code{cos} and @code{tan}.
+Finally, certain special values of the argument are recognized;
address@hidden and Hyperbolic Functions}.
+
+Hyperbolic functions of their inverses and of negative-looking
+arguments are also handled, as are exponentials of inverse
+hyperbolic functions.
+
+No simplifications for inverse trigonometric and hyperbolic
+functions are known, except for negative arguments of @code{arcsin},
address@hidden, @code{arcsinh}, and @code{arctanh}. Note that
address@hidden@tfn{arcsin}(@tfn{sin}(x))} can @emph{not} safely change to
address@hidden, since this only correct within an integer multiple of
address@hidden @math{2 \pi}
address@hidden @expr{2 pi}
+radians or 360 degrees. However, @address@hidden(@tfn{sinh}(x))} is
+simplified to @expr{x} if @expr{x} is known to be real.
+
+Several simplifications that apply to logarithms and exponentials
+are that @address@hidden(@tfn{ln}(x))},
address@hidden @address@hidden(x)}},
address@hidden @address@hidden(x)},
+and
address@hidden @math{10^{{\rm log10}(x)}}
address@hidden @address@hidden(x)}
+all reduce to @expr{x}. Also, @address@hidden(@tfn{exp}(x))}, etc., can
+reduce to @expr{x} if @expr{x} is provably real. The form
address@hidden@tfn{exp}(x)^y} is simplified to @address@hidden(x y)}. If
@expr{x}
+is a suitable multiple of
address@hidden @math{\pi i}
address@hidden @expr{pi i}
+(as described above for the trigonometric functions), then
address@hidden@tfn{exp}(x)} or @expr{e^x} will be expanded. Finally,
address@hidden@tfn{ln}(x)} is simplified to a form involving @code{pi} and
address@hidden where @expr{x} is provably negative, positive imaginary, or
+negative imaginary.
+
+The error functions @code{erf} and @code{erfc} are simplified when
+their arguments are negative-looking or are calls to the @code{conj}
+function.
+
address@hidden
+\bigskip
address@hidden tex
+
+Equations and inequalities are simplified by cancelling factors
+of products, quotients, or sums on both sides. Inequalities
+change sign if a negative multiplicative factor is cancelled.
+Non-constant multiplicative factors as in @expr{a b = a c} are
+cancelled from equations only if they are provably nonzero (generally
+because they were declared so; @pxref{Declarations}). Factors
+are cancelled from inequalities only if they are nonzero and their
+sign is known.
+
+Simplification also replaces an equation or inequality with
+1 or 0 (``true'' or ``false'') if it can through the use of
+declarations. If @expr{x} is declared to be an integer greater
+than 5, then @expr{x < 3}, @expr{x = 3}, and @expr{x = 7.5} are
+all simplified to 0, but @expr{x > 3} is simplified to 1.
+By a similar analysis, @expr{abs(x) >= 0} is simplified to 1,
+as is @expr{x^2 >= 0} if @expr{x} is known to be real.
+
address@hidden Unsafe Simplifications, Simplification of Units, Algebraic
Simplifications, Simplifying Formulas
address@hidden ``Unsafe'' Simplifications
+
address@hidden
address@hidden Unsafe simplifications
address@hidden Extended simplification
address@hidden a e
address@hidden calc-simplify-extended
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden esimplify
+The @kbd{a e} (@code{calc-simplify-extended}) address@hidden command
+is like @kbd{a s}
+except that it applies some additional simplifications which are not
+``safe'' in all cases. Use this only if you know the values in your
+formula lie in the restricted ranges for which these simplifications
+are valid. The symbolic integrator uses @kbd{a e};
+one effect of this is that the integrator's results must be used with
+caution. Where an integral table will often attach conditions like
+``for positive @expr{a} only,'' Calc (like most other symbolic
+integration programs) will simply produce an unqualified result.
+
+Because @kbd{a e}'s simplifications are unsafe, it is sometimes better
+to type @kbd{C-u -3 a v}, which does extended simplification only
+on the top level of the formula without affecting the sub-formulas.
+In fact, @kbd{C-u -3 j v} allows you to target extended simplification
+to any specific part of a formula.
+
+The variable @code{ExtSimpRules} contains rewrites to be applied by
+the @kbd{a e} command. These are applied in addition to
address@hidden and @code{AlgSimpRules}. (The @kbd{a r AlgSimpRules}
+step described above is simply followed by an @kbd{a r ExtSimpRules} step.)
+
+Following is a complete list of ``unsafe'' simplifications performed
+by @kbd{a e}.
+
address@hidden
+\bigskip
address@hidden tex
+
+Inverse trigonometric or hyperbolic functions, called with their
+corresponding non-inverse functions as arguments, are simplified
+by @kbd{a e}. For example, @address@hidden(@tfn{sin}(x))} changes
+to @expr{x}. Also, @address@hidden(@tfn{cos}(x))} and
address@hidden@tfn{arccos}(@tfn{sin}(x))} both change to @address@hidden/2 - x}.
+These simplifications are unsafe because they are valid only for
+values of @expr{x} in a certain range; outside that range, values
+are folded down to the 360-degree range that the inverse trigonometric
+functions always produce.
+
+Powers of powers @expr{(x^a)^b} are simplified to
address@hidden @math{x^{a b}}
address@hidden @expr{x^(a b)}
+for all @expr{a} and @expr{b}. These results will be valid only
+in a restricted range of @expr{x}; for example, in
address@hidden @math{(x^2)^{1:2}}
address@hidden @expr{(x^2)^1:2}
+the powers cancel to get @expr{x}, which is valid for positive values
+of @expr{x} but not for negative or complex values.
+
+Similarly, @address@hidden(x^a)} and @address@hidden(x)^a} are both
+simplified (possibly unsafely) to
address@hidden @math{x^{a/2}}.
address@hidden @expr{x^(a/2)}.
+
+Forms like @address@hidden(1 - sin(x)^2)} are simplified to, e.g.,
address@hidden@tfn{cos}(x)}. Calc has identities of this sort for @code{sin},
address@hidden, @code{tan}, @code{sinh}, and @code{cosh}.
+
+Arguments of square roots are partially factored to look for
+squared terms that can be extracted. For example,
address@hidden@tfn{sqrt}(a^2 b^3 + a^3 b^2)} simplifies to
address@hidden b @tfn{sqrt}(a+b)}.
+
+The simplifications of @address@hidden(@tfn{exp}(x))},
address@hidden@tfn{ln}(@tfn{e}^x)}, and @address@hidden(10^x)} to @expr{x} are
also
+unsafe because of problems with principal values (although these
+simplifications are safe if @expr{x} is known to be real).
+
+Common factors are cancelled from products on both sides of an
+equation, even if those factors may be zero: @expr{a x / b x}
+to @expr{a / b}. Such factors are never cancelled from
+inequalities: Even @kbd{a e} is not bold enough to reduce
address@hidden x < b x} to @expr{a < b} (or @expr{a > b}, depending
+on whether you believe @expr{x} is positive or negative).
+The @kbd{a M /} command can be used to divide a factor out of
+both sides of an inequality.
+
address@hidden Simplification of Units, , Unsafe Simplifications, Simplifying
Formulas
address@hidden Simplification of Units
+
address@hidden
+The simplifications described in this section are applied by the
address@hidden s} (@code{calc-simplify-units}) command. These are in addition
+to the regular @kbd{a s} (but not @kbd{a e}) simplifications described
+earlier. @xref{Basic Operations on Units}.
+
+The variable @code{UnitSimpRules} contains rewrites to be applied by
+the @kbd{u s} command. These are applied in addition to @code{EvalRules}
+and @code{AlgSimpRules}.
+
+Scalar mode is automatically put into effect when simplifying units.
address@hidden Mode}.
+
+Sums @expr{a + b} involving units are simplified by extracting the
+units of @expr{a} as if by the @kbd{u x} command (call the result
address@hidden), then simplifying the expression @expr{b / u_a}
+using @kbd{u b} and @kbd{u s}. If the result has units then the sum
+is inconsistent and is left alone. Otherwise, it is rewritten
+in terms of the units @expr{u_a}.
+
+If units auto-ranging mode is enabled, products or quotients in
+which the first argument is a number which is out of range for the
+leading unit are modified accordingly.
+
+When cancelling and combining units in products and quotients,
+Calc accounts for unit names that differ only in the prefix letter.
+For example, @samp{2 km m} is simplified to @samp{2000 m^2}.
+However, compatible but different units like @code{ft} and @code{in}
+are not combined in this way.
+
+Quotients @expr{a / b} are simplified in three additional ways. First,
+if @expr{b} is a number or a product beginning with a number, Calc
+computes the reciprocal of this number and moves it to the numerator.
+
+Second, for each pair of unit names from the numerator and denominator
+of a quotient, if the units are compatible (e.g., they are both
+units of area) then they are replaced by the ratio between those
+units. For example, in @samp{3 s in N / kg cm} the units
address@hidden / cm} will be replaced by @expr{2.54}.
+
+Third, if the units in the quotient exactly cancel out, so that
+a @kbd{u b} command on the quotient would produce a dimensionless
+number for an answer, then the quotient simplifies to that number.
+
+For powers and square roots, the ``unsafe'' simplifications
address@hidden(a b)^c} to @expr{a^c b^c}, @expr{(a/b)^c} to @expr{a^c / b^c},
+and @expr{(a^b)^c} to
address@hidden @math{a^{b c}}
address@hidden @expr{a^(b c)}
+are done if the powers are real numbers. (These are safe in the context
+of units because all numbers involved can reasonably be assumed to be
+real.)
+
+Also, if a unit name is raised to a fractional power, and the
+base units in that unit name all occur to powers which are a
+multiple of the denominator of the power, then the unit name
+is expanded out into its base units, which can then be simplified
+according to the previous paragraph. For example, @samp{acre^1.5}
+is simplified by noting that @expr{1.5 = 3:2}, that @samp{acre}
+is defined in terms of @samp{m^2}, and that the 2 in the power of
address@hidden is a multiple of 2 in @expr{3:2}. Thus, @code{acre^1.5} is
+replaced by approximately
address@hidden @math{(4046 m^2)^{1.5}}
address@hidden @expr{(4046 m^2)^1.5},
+which is then changed to
address@hidden @math{4046^{1.5} \, (m^2)^{1.5}},
address@hidden @expr{4046^1.5 (m^2)^1.5},
+then to @expr{257440 m^3}.
+
+The functions @code{float}, @code{frac}, @code{clean}, @code{abs},
+as well as @code{floor} and the other integer truncation functions,
+applied to unit names or products or quotients involving units, are
+simplified. For example, @samp{round(1.6 in)} is changed to
address@hidden(1.6) round(in)}; the lefthand term evaluates to 2,
+and the righthand term simplifies to @code{in}.
+
+The functions @code{sin}, @code{cos}, and @code{tan} with arguments
+that have angular units like @code{rad} or @code{arcmin} are
+simplified by converting to base units (radians), then evaluating
+with the angular mode temporarily set to radians.
+
address@hidden Polynomials, Calculus, Simplifying Formulas, Algebra
address@hidden Polynomials
+
+A @dfn{polynomial} is a sum of terms which are coefficients times
+various powers of a ``base'' variable. For example, @expr{2 x^2 + 3 x - 4}
+is a polynomial in @expr{x}. Some formulas can be considered
+polynomials in several different variables: @expr{1 + 2 x + 3 y + 4 x y^2}
+is a polynomial in both @expr{x} and @expr{y}. Polynomial coefficients
+are often numbers, but they may in general be any formulas not
+involving the base variable.
+
address@hidden a f
address@hidden calc-factor
address@hidden factor
+The @kbd{a f} (@code{calc-factor}) address@hidden command factors a
+polynomial into a product of terms. For example, the polynomial
address@hidden + 2 x^2 + x} is factored into @samp{x*(x+1)^2}. As another
+example, @expr{a c + b d + b c + a d} is factored into the product
address@hidden(a + b) (c + d)}.
+
+Calc currently has three algorithms for factoring. Formulas which are
+linear in several variables, such as the second example above, are
+merged according to the distributive law. Formulas which are
+polynomials in a single variable, with constant integer or fractional
+coefficients, are factored into irreducible linear and/or quadratic
+terms. The first example above factors into three linear terms
+(@expr{x}, @expr{x+1}, and @expr{x+1} again). Finally, formulas
+which do not fit the above criteria are handled by the algebraic
+rewrite mechanism.
+
+Calc's polynomial factorization algorithm works by using the general
+root-finding command (@address@hidden P}}) to solve for the roots of the
+polynomial. It then looks for roots which are rational numbers
+or complex-conjugate pairs, and converts these into linear and
+quadratic terms, respectively. Because it uses floating-point
+arithmetic, it may be unable to find terms that involve large
+integers (whose number of digits approaches the current precision).
+Also, irreducible factors of degree higher than quadratic are not
+found, and polynomials in more than one variable are not treated.
+(A more robust factorization algorithm may be included in a future
+version of Calc.)
+
address@hidden FactorRules
address@hidden
address@hidden
address@hidden ignore
address@hidden thecoefs
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden thefactors
+The rewrite-based factorization method uses rules stored in the variable
address@hidden @xref{Rewrite Rules}, for a discussion of the
+operation of rewrite rules. The default @code{FactorRules} are able
+to factor quadratic forms symbolically into two linear terms,
address@hidden(a x + b) (c x + d)}. You can edit these rules to include other
+cases if you wish. To use the rules, Calc builds the formula
address@hidden(x, [a, b, c, ...])} where @code{x} is the polynomial
+base variable and @code{a}, @code{b}, etc., are polynomial coefficients
+(which may be numbers or formulas). The constant term is written first,
+i.e., in the @code{a} position. When the rules complete, they should have
+changed the formula into the form @samp{thefactors(x, [f1, f2, f3, ...])}
+where each @code{fi} should be a factored term, e.g., @samp{x - ai}.
+Calc then multiplies these terms together to get the complete
+factored form of the polynomial. If the rules do not change the
address@hidden call to a @code{thefactors} call, @kbd{a f} leaves the
+polynomial alone on the assumption that it is unfactorable. (Note that
+the function names @code{thecoefs} and @code{thefactors} are used only
+as placeholders; there are no actual Calc functions by those names.)
+
address@hidden H a f
address@hidden factors
+The @kbd{H a f} address@hidden command also factors a polynomial,
+but it returns a list of factors instead of an expression which is the
+product of the factors. Each factor is represented by a sub-vector
+of the factor, and the power with which it appears. For example,
address@hidden + x^4 - 33 x^3 + 63 x^2} factors to @expr{(x + 7) x^2 (x - 3)^2}
+in @kbd{a f}, or to @expr{[ [x, 2], [x+7, 1], [x-3, 2] ]} in @kbd{H a f}.
+If there is an overall numeric factor, it always comes first in the list.
+The functions @code{factor} and @code{factors} allow a second argument
+when written in algebraic form; @samp{factor(x,v)} factors @expr{x} with
+respect to the specific variable @expr{v}. The default is to factor with
+respect to all the variables that appear in @expr{x}.
+
address@hidden a c
address@hidden calc-collect
address@hidden collect
+The @kbd{a c} (@code{calc-collect}) address@hidden command rearranges a
+formula as a
+polynomial in a given variable, ordered in decreasing powers of that
+variable. For example, given @expr{1 + 2 x + 3 y + 4 x y^2} on
+the stack, @kbd{a c x} would produce @expr{(2 + 4 y^2) x + (1 + 3 y)},
+and @kbd{a c y} would produce @expr{(4 x) y^2 + 3 y + (1 + 2 x)}.
+The polynomial will be expanded out using the distributive law as
+necessary: Collecting @expr{x} in @expr{(x - 1)^3} produces
address@hidden - 3 x^2 + 3 x - 1}. Terms not involving @expr{x} will
+not be expanded.
+
+The ``variable'' you specify at the prompt can actually be any
+expression: @kbd{a c ln(x+1)} will collect together all terms multiplied
+by @samp{ln(x+1)} or integer powers thereof. If @samp{x} also appears
+in the formula in a context other than @samp{ln(x+1)}, @kbd{a c} will
+treat those occurrences as unrelated to @samp{ln(x+1)}, i.e., as constants.
+
address@hidden a x
address@hidden calc-expand
address@hidden expand
+The @kbd{a x} (@code{calc-expand}) address@hidden command expands an
+expression by applying the distributive law everywhere. It applies to
+products, quotients, and powers involving sums. By default, it fully
+distributes all parts of the expression. With a numeric prefix argument,
+the distributive law is applied only the specified number of times, then
+the partially expanded expression is left on the stack.
+
+The @kbd{a x} and @kbd{j D} commands are somewhat redundant. Use
address@hidden x} if you want to expand all products of sums in your formula.
+Use @kbd{j D} if you want to expand a particular specified term of
+the formula. There is an exactly analogous correspondence between
address@hidden f} and @kbd{j M}. (The @kbd{j D} and @kbd{j M} commands
+also know many other kinds of expansions, such as
address@hidden(a + b) = exp(a) exp(b)}, which @kbd{a x} and @kbd{a f}
+do not do.)
+
+Calc's automatic simplifications will sometimes reverse a partial
+expansion. For example, the first step in expanding @expr{(x+1)^3} is
+to write @expr{(x+1) (x+1)^2}. If @kbd{a x} stops there and tries
+to put this formula onto the stack, though, Calc will automatically
+simplify it back to @expr{(x+1)^3} form. The solution is to turn
+simplification off first (@pxref{Simplification Modes}), or to run
address@hidden x} without a numeric prefix argument so that it expands all
+the way in one step.
+
address@hidden a a
address@hidden calc-apart
address@hidden apart
+The @kbd{a a} (@code{calc-apart}) address@hidden command expands a
+rational function by partial fractions. A rational function is the
+quotient of two polynomials; @code{apart} pulls this apart into a
+sum of rational functions with simple denominators. In algebraic
+notation, the @code{apart} function allows a second argument that
+specifies which variable to use as the ``base''; by default, Calc
+chooses the base variable automatically.
+
address@hidden a n
address@hidden calc-normalize-rat
address@hidden nrat
+The @kbd{a n} (@code{calc-normalize-rat}) address@hidden command
+attempts to arrange a formula into a quotient of two polynomials.
+For example, given @expr{1 + (a + b/c) / d}, the result would be
address@hidden(b + a c + c d) / c d}. The quotient is reduced, so that
address@hidden n} will simplify @expr{(x^2 + 2x + 1) / (x^2 - 1)} by dividing
+out the common factor @expr{x + 1}, yielding @expr{(x + 1) / (x - 1)}.
+
address@hidden a \
address@hidden calc-poly-div
address@hidden pdiv
+The @kbd{a \} (@code{calc-poly-div}) address@hidden command divides
+two polynomials @expr{u} and @expr{v}, yielding a new polynomial
address@hidden If several variables occur in the inputs, the inputs are
+considered multivariate polynomials. (Calc divides by the variable
+with the largest power in @expr{u} first, or, in the case of equal
+powers, chooses the variables in alphabetical order.) For example,
+dividing @expr{x^2 + 3 x + 2} by @expr{x + 2} yields @expr{x + 1}.
+The remainder from the division, if any, is reported at the bottom
+of the screen and is also placed in the Trail along with the quotient.
+
+Using @code{pdiv} in algebraic notation, you can specify the particular
+variable to be used as the base: @code{pdiv(@var{a},@var{b},@var{x})}.
+If @code{pdiv} is given only two arguments (as is always the case with
+the @kbd{a \} command), then it does a multivariate division as outlined
+above.
+
address@hidden a %
address@hidden calc-poly-rem
address@hidden prem
+The @kbd{a %} (@code{calc-poly-rem}) address@hidden command divides
+two polynomials and keeps the remainder @expr{r}. The quotient
address@hidden is discarded. For any formulas @expr{a} and @expr{b}, the
+results of @kbd{a \} and @kbd{a %} satisfy @expr{a = q b + r}.
+(This is analogous to plain @kbd{\} and @kbd{%}, which compute the
+integer quotient and remainder from dividing two numbers.)
+
address@hidden a /
address@hidden H a /
address@hidden calc-poly-div-rem
address@hidden pdivrem
address@hidden pdivide
+The @kbd{a /} (@code{calc-poly-div-rem}) address@hidden command
+divides two polynomials and reports both the quotient and the
+remainder as a vector @expr{[q, r]}. The @kbd{H a /} address@hidden
+command divides two polynomials and constructs the formula
address@hidden + r/b} on the stack. (Naturally if the remainder is zero,
+this will immediately simplify to @expr{q}.)
+
address@hidden a g
address@hidden calc-poly-gcd
address@hidden pgcd
+The @kbd{a g} (@code{calc-poly-gcd}) address@hidden command computes
+the greatest common divisor of two polynomials. (The GCD actually
+is unique only to within a constant multiplier; Calc attempts to
+choose a GCD which will be unsurprising.) For example, the @kbd{a n}
+command uses @kbd{a g} to take the GCD of the numerator and denominator
+of a quotient, then divides each by the result using @kbd{a \}. (The
+definition of GCD ensures that this division can take place without
+leaving a remainder.)
+
+While the polynomials used in operations like @kbd{a /} and @kbd{a g}
+often have integer coefficients, this is not required. Calc can also
+deal with polynomials over the rationals or floating-point reals.
+Polynomials with modulo-form coefficients are also useful in many
+applications; if you enter @samp{(x^2 + 3 x - 1) mod 5}, Calc
+automatically transforms this into a polynomial over the field of
+integers mod 5: @samp{(1 mod 5) x^2 + (3 mod 5) x + (4 mod 5)}.
+
+Congratulations and thanks go to Ove Ewerlid
+(@code{ewerlid@@mizar.DoCS.UU.SE}), who contributed many of the
+polynomial routines used in the above commands.
+
address@hidden Polynomials}, for several useful functions for
+extracting the individual coefficients of a polynomial.
+
address@hidden Calculus, Solving Equations, Polynomials, Algebra
address@hidden Calculus
+
address@hidden
+The following calculus commands do not automatically simplify their
+inputs or outputs using @code{calc-simplify}. You may find it helps
+to do this by hand by typing @kbd{a s} or @kbd{a e}. It may also help
+to use @kbd{a x} and/or @kbd{a c} to arrange a result in the most
+readable way.
+
address@hidden
+* Differentiation::
+* Integration::
+* Customizing the Integrator::
+* Numerical Integration::
+* Taylor Series::
address@hidden menu
+
address@hidden Differentiation, Integration, Calculus, Calculus
address@hidden Differentiation
+
address@hidden
address@hidden a d
address@hidden H a d
address@hidden calc-derivative
address@hidden deriv
address@hidden tderiv
+The @kbd{a d} (@code{calc-derivative}) address@hidden command computes
+the derivative of the expression on the top of the stack with respect to
+some variable, which it will prompt you to enter. Normally, variables
+in the formula other than the specified differentiation variable are
+considered constant, i.e., @samp{deriv(y,x)} is reduced to zero. With
+the Hyperbolic flag, the @code{tderiv} (total derivative) operation is used
+instead, in which derivatives of variables are not reduced to zero
+unless those variables are known to be ``constant,'' i.e., independent
+of any other variables. (The built-in special variables like @code{pi}
+are considered constant, as are variables that have been declared
address@hidden; @pxref{Declarations}.)
+
+With a numeric prefix argument @var{n}, this command computes the
address@hidden derivative.
+
+When working with trigonometric functions, it is best to switch to
+Radians mode first (with @address@hidden r}}). The derivative of @samp{sin(x)}
+in degrees is @samp{(pi/180) cos(x)}, probably not the expected
+answer!
+
+If you use the @code{deriv} function directly in an algebraic formula,
+you can write @samp{deriv(f,x,x0)} which represents the derivative
+of @expr{f} with respect to @expr{x}, evaluated at the point
address@hidden @math{x=x_0}.
address@hidden @expr{x=x0}.
+
+If the formula being differentiated contains functions which Calc does
+not know, the derivatives of those functions are produced by adding
+primes (apostrophe characters). For example, @samp{deriv(f(2x), x)}
+produces @samp{2 f'(2 x)}, where the function @code{f'} represents the
+derivative of @code{f}.
+
+For functions you have defined with the @kbd{Z F} command, Calc expands
+the functions according to their defining formulas unless you have
+also defined @code{f'} suitably. For example, suppose we define
address@hidden(x) = sin(x)/x} using @kbd{Z F}. If we then differentiate
+the formula @samp{sinc(2 x)}, the formula will be expanded to
address@hidden(2 x) / (2 x)} and differentiated. However, if we also
+define @samp{sinc'(x) = dsinc(x)}, say, then Calc will write the
+result as @samp{2 dsinc(2 x)}. @xref{Algebraic Definitions}.
+
+For multi-argument functions @samp{f(x,y,z)}, the derivative with respect
+to the first argument is written @samp{f'(x,y,z)}; derivatives with
+respect to the other arguments are @samp{f'2(x,y,z)} and @samp{f'3(x,y,z)}.
+Various higher-order derivatives can be formed in the obvious way, e.g.,
address@hidden'@var{}'(x)} (the second derivative of @code{f}) or
address@hidden'@var{}'2'3(x,y,z)} (@code{f} differentiated with respect to each
+argument once).
+
address@hidden Integration, Customizing the Integrator, Differentiation,
Calculus
address@hidden Integration
+
address@hidden
address@hidden a i
address@hidden calc-integral
address@hidden integ
+The @kbd{a i} (@code{calc-integral}) address@hidden command computes the
+indefinite integral of the expression on the top of the stack with
+respect to a prompted-for variable. The integrator is not guaranteed to
+work for all integrable functions, but it is able to integrate several
+large classes of formulas. In particular, any polynomial or rational
+function (a polynomial divided by a polynomial) is acceptable.
+(Rational functions don't have to be in explicit quotient form, however;
address@hidden @math{x/(1+x^{-2})}
address@hidden @expr{x/(1+x^-2)}
+is not strictly a quotient of polynomials, but it is equivalent to
address@hidden/(x^2+1)}, which is.) Also, square roots of terms involving
address@hidden and @expr{x^2} may appear in rational functions being
+integrated. Finally, rational functions involving trigonometric or
+hyperbolic functions can be integrated.
+
+With an argument (@kbd{C-u a i}), this command will compute the definite
+integral of the expression on top of the stack. In this case, the
+command will again prompt for an integration variable, then prompt for a
+lower limit and an upper limit.
+
address@hidden
+If you use the @code{integ} function directly in an algebraic formula,
+you can also write @samp{integ(f,x,v)} which expresses the resulting
+indefinite integral in terms of variable @code{v} instead of @code{x}.
+With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
+integral from @code{a} to @code{b}.
address@hidden ifnottex
address@hidden
+If you use the @code{integ} function directly in an algebraic formula,
+you can also write @samp{integ(f,x,v)} which expresses the resulting
+indefinite integral in terms of variable @code{v} instead of @code{x}.
+With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
+integral $\int_a^b f(x) \, dx$.
address@hidden tex
+
+Please note that the current implementation of Calc's integrator sometimes
+produces results that are significantly more complex than they need to
+be. For example, the integral Calc finds for
address@hidden @math{1/(x+\sqrt{x^2+1})}
address@hidden @expr{1/(x+sqrt(x^2+1))}
+is several times more complicated than the answer Mathematica
+returns for the same input, although the two forms are numerically
+equivalent. Also, any indefinite integral should be considered to have
+an arbitrary constant of integration added to it, although Calc does not
+write an explicit constant of integration in its result. For example,
+Calc's solution for
address@hidden @math{1/(1+\tan x)}
address@hidden @expr{1/(1+tan(x))}
+differs from the solution given in the @emph{CRC Math Tables} by a
+constant factor of
address@hidden @math{\pi i / 2}
address@hidden @expr{pi i / 2},
+due to a different choice of constant of integration.
+
+The Calculator remembers all the integrals it has done. If conditions
+change in a way that would invalidate the old integrals, say, a switch
+from Degrees to Radians mode, then they will be thrown out. If you
+suspect this is not happening when it should, use the
address@hidden command; @pxref{Caches}.
+
address@hidden IntegLimit
+Calc normally will pursue integration by substitution or integration by
+parts up to 3 nested times before abandoning an approach as fruitless.
+If the integrator is taking too long, you can lower this limit by storing
+a number (like 2) in the variable @code{IntegLimit}. (The @kbd{s I}
+command is a convenient way to edit @code{IntegLimit}.) If this variable
+has no stored value or does not contain a nonnegative integer, a limit
+of 3 is used. The lower this limit is, the greater the chance that Calc
+will be unable to integrate a function it could otherwise handle. Raising
+this limit allows the Calculator to solve more integrals, though the time
+it takes may grow exponentially. You can monitor the integrator's actions
+by creating an Emacs buffer called @code{*Trace*}. If such a buffer
+exists, the @kbd{a i} command will write a log of its actions there.
+
+If you want to manipulate integrals in a purely symbolic way, you can
+set the integration nesting limit to 0 to prevent all but fast
+table-lookup solutions of integrals. You might then wish to define
+rewrite rules for integration by parts, various kinds of substitutions,
+and so on. @xref{Rewrite Rules}.
+
address@hidden Customizing the Integrator, Numerical Integration, Integration,
Calculus
address@hidden Customizing the Integrator
+
address@hidden
address@hidden IntegRules
+Calc has two built-in rewrite rules called @code{IntegRules} and
address@hidden which you can edit to define new integration
+methods. @xref{Rewrite Rules}. At each step of the integration process,
+Calc wraps the current integrand in a call to the fictitious function
address@hidden(@var{expr},@var{var})}, where @var{expr} is the
+integrand and @var{var} is the integration variable. If your rules
+rewrite this to be a plain formula (not a call to @code{integtry}), then
+Calc will use this formula as the integral of @var{expr}. For example,
+the rule @samp{integtry(mysin(x),x) := -mycos(x)} would define a rule to
+integrate a function @code{mysin} that acts like the sine function.
+Then, putting @samp{4 mysin(2y+1)} on the stack and typing @kbd{a i y}
+will produce the integral @samp{-2 mycos(2y+1)}. Note that Calc has
+automatically made various transformations on the integral to allow it
+to use your rule; integral tables generally give rules for
address@hidden(a x + b)}, but you don't need to use this much generality
+in your @code{IntegRules}.
+
address@hidden Exponential integral Ei(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden Ei
+As a more serious example, the expression @samp{exp(x)/x} cannot be
+integrated in terms of the standard functions, so the ``exponential
+integral'' function
address@hidden @math{{\rm Ei}(x)}
address@hidden @expr{Ei(x)}
+was invented to describe it.
+We can get Calc to do this integral in terms of a made-up @code{Ei}
+function by adding the rule @samp{[integtry(exp(x)/x, x) := Ei(x)]}
+to @code{IntegRules}. Now entering @samp{exp(2x)/x} on the stack
+and typing @kbd{a i x} yields @samp{Ei(2 x)}. This new rule will
+work with Calc's various built-in integration methods (such as
+integration by substitution) to solve a variety of other problems
+involving @code{Ei}: For example, now Calc will also be able to
+integrate @samp{exp(exp(x))} and @samp{ln(ln(x))} (to get @samp{Ei(exp(x))}
+and @samp{x ln(ln(x)) - Ei(ln(x))}, respectively).
+
+Your rule may do further integration by calling @code{integ}. For
+example, @samp{integtry(twice(u),x) := twice(integ(u))} allows Calc
+to integrate @samp{twice(sin(x))} to get @samp{twice(-cos(x))}.
+Note that @code{integ} was called with only one argument. This notation
+is allowed only within @code{IntegRules}; it means ``integrate this
+with respect to the same integration variable.'' If Calc is unable
+to integrate @code{u}, the integration that invoked @code{IntegRules}
+also fails. Thus integrating @samp{twice(f(x))} fails, returning the
+unevaluated integral @samp{integ(twice(f(x)), x)}. It is still valid
+to call @code{integ} with two or more arguments, however; in this case,
+if @code{u} is not integrable, @code{twice} itself will still be
+integrated: If the above rule is changed to @samp{... := twice(integ(u,x))},
+then integrating @samp{twice(f(x))} will yield @samp{twice(integ(f(x),x))}.
+
+If a rule instead produces the formula @samp{integsubst(@var{sexpr},
address@hidden)}, either replacing the top-level @code{integtry} call or
+nested anywhere inside the expression, then Calc will apply the
+substitution @address@hidden = @var{sexpr}(@var{svar})} to try to
+integrate the original @var{expr}. For example, the rule
address@hidden(a) := integsubst(sqrt(x),x)} says that if Calc ever finds
+a square root in the integrand, it should attempt the substitution
address@hidden = sqrt(x)}. (This particular rule is unnecessary because
+Calc always tries ``obvious'' substitutions where @var{sexpr} actually
+appears in the integrand.) The variable @var{svar} may be the same
+as the @var{var} that appeared in the call to @code{integtry}, but
+it need not be.
+
+When integrating according to an @code{integsubst}, Calc uses the
+equation solver to find the inverse of @var{sexpr} (if the integrand
+refers to @var{var} anywhere except in subexpressions that exactly
+match @var{sexpr}). It uses the differentiator to find the derivative
+of @var{sexpr} and/or its inverse (it has two methods that use one
+derivative or the other). You can also specify these items by adding
+extra arguments to the @code{integsubst} your rules construct; the
+general form is @samp{integsubst(@var{sexpr}, @var{svar}, @var{sinv},
address@hidden)}, where @var{sinv} is the inverse of @var{sexpr} (still
+written as a function of @var{svar}), and @var{sprime} is the
+derivative of @var{sexpr} with respect to @var{svar}. If you don't
+specify these things, and Calc is not able to work them out on its
+own with the information it knows, then your substitution rule will
+work only in very specific, simple cases.
+
+Calc applies @code{IntegRules} as if by @kbd{C-u 1 a r IntegRules};
+in other words, Calc stops rewriting as soon as any rule in your rule
+set succeeds. (If it weren't for this, the @samp{integsubst(sqrt(x),x)}
+example above would keep on adding layers of @code{integsubst} calls
+forever!)
+
address@hidden IntegSimpRules
+Another set of rules, stored in @code{IntegSimpRules}, are applied
+every time the integrator uses @kbd{a s} to simplify an intermediate
+result. For example, putting the rule @samp{twice(x) := 2 x} into
address@hidden would tell Calc to convert the @code{twice}
+function into a form it knows whenever integration is attempted.
+
+One more way to influence the integrator is to define a function with
+the @kbd{Z F} command (@pxref{Algebraic Definitions}). Calc's
+integrator automatically expands such functions according to their
+defining formulas, even if you originally asked for the function to
+be left unevaluated for symbolic arguments. (Certain other Calc
+systems, such as the differentiator and the equation solver, also
+do this.)
+
address@hidden IntegAfterRules
+Sometimes Calc is able to find a solution to your integral, but it
+expresses the result in a way that is unnecessarily complicated. If
+this happens, you can either use @code{integsubst} as described
+above to try to hint at a more direct path to the desired result, or
+you can use @code{IntegAfterRules}. This is an extra rule set that
+runs after the main integrator returns its result; basically, Calc does
+an @kbd{a r IntegAfterRules} on the result before showing it to you.
+(It also does an @kbd{a s}, without @code{IntegSimpRules}, after that
+to further simplify the result.) For example, Calc's integrator
+sometimes produces expressions of the form @samp{ln(1+x) - ln(1-x)};
+the default @code{IntegAfterRules} rewrite this into the more readable
+form @samp{2 arctanh(x)}. Note that, unlike @code{IntegRules},
address@hidden and @code{IntegAfterRules} are applied any number
+of times until no further changes are possible. Rewriting by
address@hidden occurs only after the main integrator has
+finished, not at every step as for @code{IntegRules} and
address@hidden
+
address@hidden Numerical Integration, Taylor Series, Customizing the
Integrator, Calculus
address@hidden Numerical Integration
+
address@hidden
address@hidden a I
address@hidden calc-num-integral
address@hidden ninteg
+If you want a purely numerical answer to an integration problem, you can
+use the @kbd{a I} (@code{calc-num-integral}) address@hidden command. This
+command prompts for an integration variable, a lower limit, and an
+upper limit. Except for the integration variable, all other variables
+that appear in the integrand formula must have stored values. (A stored
+value, if any, for the integration variable itself is ignored.)
+
+Numerical integration works by evaluating your formula at many points in
+the specified interval. Calc uses an ``open Romberg'' method; this means
+that it does not evaluate the formula actually at the endpoints (so that
+it is safe to integrate @samp{sin(x)/x} from zero, for example). Also,
+the Romberg method works especially well when the function being
+integrated is fairly smooth. If the function is not smooth, Calc will
+have to evaluate it at quite a few points before it can accurately
+determine the value of the integral.
+
+Integration is much faster when the current precision is small. It is
+best to set the precision to the smallest acceptable number of digits
+before you use @kbd{a I}. If Calc appears to be taking too long, press
address@hidden to halt it and try a lower precision. If Calc still appears
+to need hundreds of evaluations, check to make sure your function is
+well-behaved in the specified interval.
+
+It is possible for the lower integration limit to be @samp{-inf} (minus
+infinity). Likewise, the upper limit may be plus infinity. Calc
+internally transforms the integral into an equivalent one with finite
+limits. However, integration to or across singularities is not supported:
+The integral of @samp{1/sqrt(x)} from 0 to 1 exists (it can be found
+by Calc's symbolic integrator, for example), but @kbd{a I} will fail
+because the integrand goes to infinity at one of the endpoints.
+
address@hidden Taylor Series, , Numerical Integration, Calculus
address@hidden Taylor Series
+
address@hidden
address@hidden a t
address@hidden calc-taylor
address@hidden taylor
+The @kbd{a t} (@code{calc-taylor}) address@hidden command computes a
+power series expansion or Taylor series of a function. You specify the
+variable and the desired number of terms. You may give an expression of
+the form @address@hidden = @var{a}} or @address@hidden - @var{a}} instead
+of just a variable to produce a Taylor expansion about the point @var{a}.
+You may specify the number of terms with a numeric prefix argument;
+otherwise the command will prompt you for the number of terms. Note that
+many series expansions have coefficients of zero for some terms, so you
+may appear to get fewer terms than you asked for.
+
+If the @kbd{a i} command is unable to find a symbolic integral for a
+function, you can get an approximation by integrating the function's
+Taylor series.
+
address@hidden Solving Equations, Numerical Solutions, Calculus, Algebra
address@hidden Solving Equations
+
address@hidden
address@hidden a S
address@hidden calc-solve-for
address@hidden solve
address@hidden Equations, solving
address@hidden Solving equations
+The @kbd{a S} (@code{calc-solve-for}) address@hidden command rearranges
+an equation to solve for a specific variable. An equation is an
+expression of the form @expr{L = R}. For example, the command @kbd{a S x}
+will rearrange @expr{y = 3x + 6} to the form, @expr{x = y/3 - 2}. If the
+input is not an equation, it is treated like an equation of the
+form @expr{X = 0}.
+
+This command also works for inequalities, as in @expr{y < 3x + 6}.
+Some inequalities cannot be solved where the analogous equation could
+be; for example, solving
address@hidden @math{a < b \, c}
address@hidden @expr{a < b c}
+for @expr{b} is impossible
+without knowing the sign of @expr{c}. In this case, @kbd{a S} will
+produce the result
address@hidden @math{b \mathbin{\hbox{\code{!=}}} a/c}
address@hidden @expr{b != a/c}
+(using the not-equal-to operator) to signify that the direction of the
+inequality is now unknown. The inequality
address@hidden @math{a \le b \, c}
address@hidden @expr{a <= b c}
+is not even partially solved. @xref{Declarations}, for a way to tell
+Calc that the signs of the variables in a formula are in fact known.
+
+Two useful commands for working with the result of @kbd{a S} are
address@hidden .} (@pxref{Logical Operations}), which converts @expr{x = y/3 -
2}
+to @expr{y/3 - 2}, and @kbd{s l} (@pxref{Let Command}) which evaluates
+another formula with @expr{x} set equal to @expr{y/3 - 2}.
+
address@hidden
+* Multiple Solutions::
+* Solving Systems of Equations::
+* Decomposing Polynomials::
address@hidden menu
+
address@hidden Multiple Solutions, Solving Systems of Equations, Solving
Equations, Solving Equations
address@hidden Multiple Solutions
+
address@hidden
address@hidden H a S
address@hidden fsolve
+Some equations have more than one solution. The Hyperbolic flag
+(@code{H a S}) address@hidden tells the solver to report the fully
+general family of solutions. It will invent variables @code{n1},
address@hidden, @dots{}, which represent independent arbitrary integers, and
address@hidden, @code{s2}, @dots{}, which represent independent arbitrary
+signs (either @mathit{+1} or @mathit{-1}). If you don't use the Hyperbolic
+flag, Calc will use zero in place of all arbitrary integers, and plus
+one in place of all arbitrary signs. Note that variables like @code{n1}
+and @code{s1} are not given any special interpretation in Calc except by
+the equation solver itself. As usual, you can use the @address@hidden l}}
+(@code{calc-let}) command to obtain solutions for various actual values
+of these variables.
+
+For example, @kbd{' x^2 = y @key{RET} H a S x @key{RET}} solves to
+get @samp{x = s1 sqrt(y)}, indicating that the two solutions to the
+equation are @samp{sqrt(y)} and @samp{-sqrt(y)}. Another way to
+think about it is that the square-root operation is really a
+two-valued function; since every Calc function must return a
+single result, @code{sqrt} chooses to return the positive result.
+Then @kbd{H a S} doctors this result using @code{s1} to indicate
+the full set of possible values of the mathematical square-root.
+
+There is a similar phenomenon going the other direction: Suppose
+we solve @samp{sqrt(y) = x} for @code{y}. Calc squares both sides
+to get @samp{y = x^2}. This is correct, except that it introduces
+some dubious solutions. Consider solving @samp{sqrt(y) = -3}:
+Calc will report @expr{y = 9} as a valid solution, which is true
+in the mathematical sense of square-root, but false (there is no
+solution) for the actual Calc positive-valued @code{sqrt}. This
+happens for both @kbd{a S} and @kbd{H a S}.
+
address@hidden @code{GenCount} variable
address@hidden GenCount
address@hidden
address@hidden
address@hidden ignore
address@hidden an
address@hidden
address@hidden
address@hidden ignore
address@hidden as
+If you store a positive integer in the Calc variable @code{GenCount},
+then Calc will generate formulas of the form @samp{as(@var{n})} for
+arbitrary signs, and @samp{an(@var{n})} for arbitrary integers,
+where @var{n} represents successive values taken by incrementing
address@hidden by one. While the normal arbitrary sign and
+integer symbols start over at @code{s1} and @code{n1} with each
+new Calc command, the @code{GenCount} approach will give each
+arbitrary value a name that is unique throughout the entire Calc
+session. Also, the arbitrary values are function calls instead
+of variables, which is advantageous in some cases. For example,
+you can make a rewrite rule that recognizes all arbitrary signs
+using a pattern like @samp{as(n)}. The @kbd{s l} command only works
+on variables, but you can use the @kbd{a b} (@code{calc-substitute})
+command to substitute actual values for function calls like @samp{as(3)}.
+
+The @kbd{s G} (@code{calc-edit-GenCount}) command is a convenient
+way to create or edit this variable. Press @kbd{C-c C-c} to finish.
+
+If you have not stored a value in @code{GenCount}, or if the value
+in that variable is not a positive integer, the regular
address@hidden/@code{n1} notation is used.
+
address@hidden I a S
address@hidden H I a S
address@hidden finv
address@hidden ffinv
+With the Inverse flag, @kbd{I a S} address@hidden treats the expression
+on top of the stack as a function of the specified variable and solves
+to find the inverse function, written in terms of the same variable.
+For example, @kbd{I a S x} inverts @expr{2x + 6} to @expr{x/2 - 3}.
+You can use both Inverse and Hyperbolic address@hidden to obtain a
+fully general inverse, as described above.
+
address@hidden a P
address@hidden calc-poly-roots
address@hidden roots
+Some equations, specifically polynomials, have a known, finite number
+of solutions. The @kbd{a P} (@code{calc-poly-roots}) address@hidden
+command uses @kbd{H a S} to solve an equation in general form, then, for
+all arbitrary-sign variables like @code{s1}, and all arbitrary-integer
+variables like @code{n1} for which @code{n1} only usefully varies over
+a finite range, it expands these variables out to all their possible
+values. The results are collected into a vector, which is returned.
+For example, @samp{roots(x^4 = 1, x)} returns the four solutions
address@hidden, -1, (0, 1), (0, -1)]}. Generally an @var{n}th degree
+polynomial will always have @var{n} roots on the complex plane.
+(If you have given a @code{real} declaration for the solution
+variable, then only the real-valued solutions, if any, will be
+reported; @pxref{Declarations}.)
+
+Note that because @kbd{a P} uses @kbd{H a S}, it is able to deliver
+symbolic solutions if the polynomial has symbolic coefficients. Also
+note that Calc's solver is not able to get exact symbolic solutions
+to all polynomials. Polynomials containing powers up to @expr{x^4}
+can always be solved exactly; polynomials of higher degree sometimes
+can be: @expr{x^6 + x^3 + 1} is converted to @expr{(x^3)^2 + (x^3) + 1},
+which can be solved for @expr{x^3} using the quadratic equation, and then
+for @expr{x} by taking cube roots. But in many cases, like
address@hidden + x + 1}, Calc does not know how to rewrite the polynomial
+into a form it can solve. The @kbd{a P} command can still deliver a
+list of numerical roots, however, provided that Symbolic mode (@kbd{m s})
+is not turned on. (If you work with Symbolic mode on, recall that the
address@hidden (@code{calc-eval-num}) key is a handy way to reevaluate the
+formula on the stack with Symbolic mode temporarily off.) Naturally,
address@hidden P} can only provide numerical roots if the polynomial
coefficients
+are all numbers (real or complex).
+
address@hidden Solving Systems of Equations, Decomposing Polynomials, Multiple
Solutions, Solving Equations
address@hidden Solving Systems of Equations
+
address@hidden
address@hidden Systems of equations, symbolic
+You can also use the commands described above to solve systems of
+simultaneous equations. Just create a vector of equations, then
+specify a vector of variables for which to solve. (You can omit
+the surrounding brackets when entering the vector of variables
+at the prompt.)
+
+For example, putting @samp{[x + y = a, x - y = b]} on the stack
+and typing @kbd{a S x,y @key{RET}} produces the vector of solutions
address@hidden = a - (a-b)/2, y = (a-b)/2]}. The result vector will
+have the same length as the variables vector, and the variables
+will be listed in the same order there. Note that the solutions
+are not always simplified as far as possible; the solution for
address@hidden here could be improved by an application of the @kbd{a n}
+command.
+
+Calc's algorithm works by trying to eliminate one variable at a
+time by solving one of the equations for that variable and then
+substituting into the other equations. Calc will try all the
+possibilities, but you can speed things up by noting that Calc
+first tries to eliminate the first variable with the first
+equation, then the second variable with the second equation,
+and so on. It also helps to put the simpler (e.g., more linear)
+equations toward the front of the list. Calc's algorithm will
+solve any system of linear equations, and also many kinds of
+nonlinear systems.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden elim
+Normally there will be as many variables as equations. If you
+give fewer variables than equations (an ``over-determined'' system
+of equations), Calc will find a partial solution. For example,
+typing @kbd{a S y @key{RET}} with the above system of equations
+would produce @samp{[y = a - x]}. There are now several ways to
+express this solution in terms of the original variables; Calc uses
+the first one that it finds. You can control the choice by adding
+variable specifiers of the form @samp{elim(@var{v})} to the
+variables list. This says that @var{v} should be eliminated from
+the equations; the variable will not appear at all in the solution.
+For example, typing @kbd{a S y,elim(x)} would yield
address@hidden = a - (b+a)/2]}.
+
+If the variables list contains only @code{elim} specifiers,
+Calc simply eliminates those variables from the equations
+and then returns the resulting set of equations. For example,
address@hidden S elim(x)} produces @samp{[a - 2 y = b]}. Every variable
+eliminated will reduce the number of equations in the system
+by one.
+
+Again, @kbd{a S} gives you one solution to the system of
+equations. If there are several solutions, you can use @kbd{H a S}
+to get a general family of solutions, or, if there is a finite
+number of solutions, you can use @kbd{a P} to get a list. (In
+the latter case, the result will take the form of a matrix where
+the rows are different solutions and the columns correspond to the
+variables you requested.)
+
+Another way to deal with certain kinds of overdetermined systems of
+equations is the @kbd{a F} command, which does least-squares fitting
+to satisfy the equations. @xref{Curve Fitting}.
+
address@hidden Decomposing Polynomials, , Solving Systems of Equations,
Solving Equations
address@hidden Decomposing Polynomials
+
address@hidden
address@hidden
address@hidden
address@hidden ignore
address@hidden poly
+The @code{poly} function takes a polynomial and a variable as
+arguments, and returns a vector of polynomial coefficients (constant
+coefficient first). For example, @samp{poly(x^3 + 2 x, x)} returns
address@hidden, 2, 0, 1]}. If the input is not a polynomial in @expr{x},
+the call to @code{poly} is left in symbolic form. If the input does
+not involve the variable @expr{x}, the input is returned in a list
+of length one, representing a polynomial with only a constant
+coefficient. The call @samp{poly(x, x)} returns the vector @expr{[0, 1]}.
+The last element of the returned vector is guaranteed to be nonzero;
+note that @samp{poly(0, x)} returns the empty vector @expr{[]}.
+Note also that @expr{x} may actually be any formula; for example,
address@hidden(sin(x)^2 - sin(x) + 3, sin(x))} returns @expr{[3, -1, 1]}.
+
address@hidden Coefficients of polynomial
address@hidden Degree of polynomial
+To get the @expr{x^k} coefficient of polynomial @expr{p}, use
address@hidden(p, x)_(k+1)}. To get the degree of polynomial @expr{p},
+use @samp{vlen(poly(p, x)) - 1}. For example, @samp{poly((x+1)^4, x)}
+returns @samp{[1, 4, 6, 4, 1]}, so @samp{poly((x+1)^4, x)_(2+1)}
+gives the @expr{x^2} coefficient of this polynomial, 6.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden gpoly
+One important feature of the solver is its ability to recognize
+formulas which are ``essentially'' polynomials. This ability is
+made available to the user through the @code{gpoly} function, which
+is used just like @code{poly}: @samp{gpoly(@var{expr}, @var{var})}.
+If @var{expr} is a polynomial in some term which includes @var{var}, then
+this function will return a vector @address@hidden, @var{c}, @var{a}]}
+where @var{x} is the term that depends on @var{var}, @var{c} is a
+vector of polynomial coefficients (like the one returned by @code{poly}),
+and @var{a} is a multiplier which is usually 1. Basically,
address@hidden@var{expr} = @var{a}*(@var{c}_1 + @var{c}_2 @var{x} +
address@hidden @var{x}^2 + ...)}. The last element of @var{c} is
+guaranteed to be non-zero, and @var{c} will not equal @samp{[1]}
+(i.e., the trivial decomposition @var{expr} = @var{x} is not
+considered a polynomial). One side effect is that @samp{gpoly(x, x)}
+and @samp{gpoly(6, x)}, both of which might be expected to recognize
+their arguments as polynomials, will not because the decomposition
+is considered trivial.
+
+For example, @samp{gpoly((x-2)^2, x)} returns @samp{[x, [4, -4, 1], 1]},
+since the expanded form of this polynomial is @expr{4 - 4 x + x^2}.
+
+The term @var{x} may itself be a polynomial in @var{var}. This is
+done to reduce the size of the @var{c} vector. For example,
address@hidden(x^4 + x^2 - 1, x)} returns @samp{[x^2, [-1, 1, 1], 1]},
+since a quadratic polynomial in @expr{x^2} is easier to solve than
+a quartic polynomial in @expr{x}.
+
+A few more examples of the kinds of polynomials @code{gpoly} can
+discover:
+
address@hidden
+sin(x) - 1 [sin(x), [-1, 1], 1]
+x + 1/x - 1 [x, [1, -1, 1], 1/x]
+x + 1/x [x^2, [1, 1], 1/x]
+x^3 + 2 x [x^2, [2, 1], x]
+x + x^2:3 + sqrt(x) [x^1:6, [1, 1, 0, 1], x^1:2]
+x^(2a) + 2 x^a + 5 [x^a, [5, 2, 1], 1]
+(exp(-x) + exp(x)) / 2 [e^(2 x), [0.5, 0.5], e^-x]
address@hidden smallexample
+
+The @code{poly} and @code{gpoly} functions accept a third integer argument
+which specifies the largest degree of polynomial that is acceptable.
+If this is @expr{n}, then only @var{c} vectors of length @expr{n+1}
+or less will be returned. Otherwise, the @code{poly} or @code{gpoly}
+call will remain in symbolic form. For example, the equation solver
+can handle quartics and smaller polynomials, so it calls
address@hidden(@var{expr}, @var{var}, 4)} to discover whether @var{expr}
+can be treated by its linear, quadratic, cubic, or quartic formulas.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden pdeg
+The @code{pdeg} function computes the degree of a polynomial;
address@hidden(p,x)} is the highest power of @code{x} that appears in
address@hidden This is the same as @samp{vlen(poly(p,x))-1}, but is
+much more efficient. If @code{p} is constant with respect to @code{x},
+then @samp{pdeg(p,x) = 0}. If @code{p} is not a polynomial in @code{x}
+(e.g., @samp{pdeg(2 cos(x), x)}, the function remains unevaluated.
+It is possible to omit the second argument @code{x}, in which case
address@hidden(p)} returns the highest total degree of any term of the
+polynomial, counting all variables that appear in @code{p}. Note
+that @code{pdeg(c) = pdeg(c,x) = 0} for any nonzero constant @code{c};
+the degree of the constant zero is considered to be @code{-inf}
+(minus infinity).
+
address@hidden
address@hidden
address@hidden ignore
address@hidden plead
+The @code{plead} function finds the leading term of a polynomial.
+Thus @samp{plead(p,x)} is equivalent to @samp{poly(p,x)_vlen(poly(p,x))},
+though again more efficient. In particular, @samp{plead((2x+1)^10, x)}
+returns 1024 without expanding out the list of coefficients. The
+value of @code{plead(p,x)} will be zero only if @expr{p = 0}.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden pcont
+The @code{pcont} function finds the @dfn{content} of a polynomial. This
+is the greatest common divisor of all the coefficients of the polynomial.
+With two arguments, @code{pcont(p,x)} effectively uses @samp{poly(p,x)}
+to get a list of coefficients, then uses @code{pgcd} (the polynomial
+GCD function) to combine these into an answer. For example,
address@hidden(4 x y^2 + 6 x^2 y, x)} is @samp{2 y}. The content is
+basically the ``biggest'' polynomial that can be divided into @code{p}
+exactly. The sign of the content is the same as the sign of the leading
+coefficient.
+
+With only one argument, @samp{pcont(p)} computes the numerical
+content of the polynomial, i.e., the @code{gcd} of the numerical
+coefficients of all the terms in the formula. Note that @code{gcd}
+is defined on rational numbers as well as integers; it computes
+the @code{gcd} of the numerators and the @code{lcm} of the
+denominators. Thus @samp{pcont(4:3 x y^2 + 6 x^2 y)} returns 2:3.
+Dividing the polynomial by this number will clear all the
+denominators, as well as dividing by any common content in the
+numerators. The numerical content of a polynomial is negative only
+if all the coefficients in the polynomial are negative.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden pprim
+The @code{pprim} function finds the @dfn{primitive part} of a
+polynomial, which is simply the polynomial divided (using @code{pdiv}
+if necessary) by its content. If the input polynomial has rational
+coefficients, the result will have integer coefficients in simplest
+terms.
+
address@hidden Numerical Solutions, Curve Fitting, Solving Equations, Algebra
address@hidden Numerical Solutions
+
address@hidden
+Not all equations can be solved symbolically. The commands in this
+section use numerical algorithms that can find a solution to a specific
+instance of an equation to any desired accuracy. Note that the
+numerical commands are slower than their algebraic cousins; it is a
+good idea to try @kbd{a S} before resorting to these commands.
+
+(@xref{Curve Fitting}, for some other, more specialized, operations
+on numerical data.)
+
address@hidden
+* Root Finding::
+* Minimization::
+* Numerical Systems of Equations::
address@hidden menu
+
address@hidden Root Finding, Minimization, Numerical Solutions, Numerical
Solutions
address@hidden Root Finding
+
address@hidden
address@hidden a R
address@hidden calc-find-root
address@hidden root
address@hidden Newton's method
address@hidden Roots of equations
address@hidden Numerical root-finding
+The @kbd{a R} (@code{calc-find-root}) address@hidden command finds a
+numerical solution (or @dfn{root}) of an equation. (This command treats
+inequalities the same as equations. If the input is any other kind
+of formula, it is interpreted as an equation of the form @expr{X = 0}.)
+
+The @kbd{a R} command requires an initial guess on the top of the
+stack, and a formula in the second-to-top position. It prompts for a
+solution variable, which must appear in the formula. All other variables
+that appear in the formula must have assigned values, i.e., when
+a value is assigned to the solution variable and the formula is
+evaluated with @kbd{=}, it should evaluate to a number. Any assigned
+value for the solution variable itself is ignored and unaffected by
+this command.
+
+When the command completes, the initial guess is replaced on the stack
+by a vector of two numbers: The value of the solution variable that
+solves the equation, and the difference between the lefthand and
+righthand sides of the equation at that value. Ordinarily, the second
+number will be zero or very nearly zero. (Note that Calc uses a
+slightly higher precision while finding the root, and thus the second
+number may be slightly different from the value you would compute from
+the equation yourself.)
+
+The @kbd{v h} (@code{calc-head}) command is a handy way to extract
+the first element of the result vector, discarding the error term.
+
+The initial guess can be a real number, in which case Calc searches
+for a real solution near that number, or a complex number, in which
+case Calc searches the whole complex plane near that number for a
+solution, or it can be an interval form which restricts the search
+to real numbers inside that interval.
+
+Calc tries to use @kbd{a d} to take the derivative of the equation.
+If this succeeds, it uses Newton's method. If the equation is not
+differentiable Calc uses a bisection method. (If Newton's method
+appears to be going astray, Calc switches over to bisection if it
+can, or otherwise gives up. In this case it may help to try again
+with a slightly different initial guess.) If the initial guess is a
+complex number, the function must be differentiable.
+
+If the formula (or the difference between the sides of an equation)
+is negative at one end of the interval you specify and positive at
+the other end, the root finder is guaranteed to find a root.
+Otherwise, Calc subdivides the interval into small parts looking for
+positive and negative values to bracket the root. When your guess is
+an interval, Calc will not look outside that interval for a root.
+
address@hidden H a R
address@hidden wroot
+The @kbd{H a R} address@hidden command is similar to @kbd{a R}, except
+that if the initial guess is an interval for which the function has
+the same sign at both ends, then rather than subdividing the interval
+Calc attempts to widen it to enclose a root. Use this mode if
+you are not sure if the function has a root in your interval.
+
+If the function is not differentiable, and you give a simple number
+instead of an interval as your initial guess, Calc uses this widening
+process even if you did not type the Hyperbolic flag. (If the function
address@hidden differentiable, Calc uses Newton's method which does not
+require a bounding interval in order to work.)
+
+If Calc leaves the @code{root} or @code{wroot} function in symbolic
+form on the stack, it will normally display an explanation for why
+no root was found. If you miss this explanation, press @kbd{w}
+(@code{calc-why}) to get it back.
+
address@hidden Minimization, Numerical Systems of Equations, Root Finding,
Numerical Solutions
address@hidden Minimization
+
address@hidden
address@hidden a N
address@hidden H a N
address@hidden a X
address@hidden H a X
address@hidden calc-find-minimum
address@hidden calc-find-maximum
address@hidden minimize
address@hidden maximize
address@hidden Minimization, numerical
+The @kbd{a N} (@code{calc-find-minimum}) address@hidden command
+finds a minimum value for a formula. It is very similar in operation
+to @kbd{a R} (@code{calc-find-root}): You give the formula and an initial
+guess on the stack, and are prompted for the name of a variable. The guess
+may be either a number near the desired minimum, or an interval enclosing
+the desired minimum. The function returns a vector containing the
+value of the variable which minimizes the formula's value, along
+with the minimum value itself.
+
+Note that this command looks for a @emph{local} minimum. Many functions
+have more than one minimum; some, like
address@hidden @math{x \sin x},
address@hidden @expr{x sin(x)},
+have infinitely many. In fact, there is no easy way to define the
+``global'' minimum of
address@hidden @math{x \sin x}
address@hidden @expr{x sin(x)}
+but Calc can still locate any particular local minimum
+for you. Calc basically goes downhill from the initial guess until it
+finds a point at which the function's value is greater both to the left
+and to the right. Calc does not use derivatives when minimizing a function.
+
+If your initial guess is an interval and it looks like the minimum
+occurs at one or the other endpoint of the interval, Calc will return
+that endpoint only if that endpoint is closed; thus, minimizing @expr{17 x}
+over @expr{[2..3]} will return @expr{[2, 38]}, but minimizing over
address@hidden(2..3]} would report no minimum found. In general, you should
+use closed intervals to find literally the minimum value in that
+range of @expr{x}, or open intervals to find the local minimum, if
+any, that happens to lie in that range.
+
+Most functions are smooth and flat near their minimum values. Because
+of this flatness, if the current precision is, say, 12 digits, the
+variable can only be determined meaningfully to about six digits. Thus
+you should set the precision to twice as many digits as you need in your
+answer.
+
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden wminimize
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden wmaximize
+The @kbd{H a N} address@hidden command, analogously to @kbd{H a R},
+expands the guess interval to enclose a minimum rather than requiring
+that the minimum lie inside the interval you supply.
+
+The @kbd{a X} (@code{calc-find-maximum}) address@hidden and
address@hidden a X} address@hidden commands effectively minimize the
+negative of the formula you supply.
+
+The formula must evaluate to a real number at all points inside the
+interval (or near the initial guess if the guess is a number). If
+the initial guess is a complex number the variable will be minimized
+over the complex numbers; if it is real or an interval it will
+be minimized over the reals.
+
address@hidden Numerical Systems of Equations, , Minimization, Numerical
Solutions
address@hidden Systems of Equations
+
address@hidden
address@hidden Systems of equations, numerical
+The @kbd{a R} command can also solve systems of equations. In this
+case, the equation should instead be a vector of equations, the
+guess should instead be a vector of numbers (intervals are not
+supported), and the variable should be a vector of variables. You
+can omit the brackets while entering the list of variables. Each
+equation must be differentiable by each variable for this mode to
+work. The result will be a vector of two vectors: The variable
+values that solved the system of equations, and the differences
+between the sides of the equations with those variable values.
+There must be the same number of equations as variables. Since
+only plain numbers are allowed as guesses, the Hyperbolic flag has
+no effect when solving a system of equations.
+
+It is also possible to minimize over many variables with @kbd{a N}
+(or maximize with @kbd{a X}). Once again the variable name should
+be replaced by a vector of variables, and the initial guess should
+be an equal-sized vector of initial guesses. But, unlike the case of
+multidimensional @kbd{a R}, the formula being minimized should
+still be a single formula, @emph{not} a vector. Beware that
+multidimensional minimization is currently @emph{very} slow.
+
address@hidden Curve Fitting, Summations, Numerical Solutions, Algebra
address@hidden Curve Fitting
+
address@hidden
+The @kbd{a F} command fits a set of data to a @dfn{model formula},
+such as @expr{y = m x + b} where @expr{m} and @expr{b} are parameters
+to be determined. For a typical set of measured data there will be
+no single @expr{m} and @expr{b} that exactly fit the data; in this
+case, Calc chooses values of the parameters that provide the closest
+possible fit. The model formula can be entered in various ways after
+the key sequence @kbd{a F} is pressed.
+
+If the letter @kbd{P} is pressed after @kbd{a F} but before the model
+description is entered, the data as well as the model formula will be
+plotted after the formula is determined. This will be indicated by a
+``P'' in the minibuffer after the help message.
+
address@hidden
+* Linear Fits::
+* Polynomial and Multilinear Fits::
+* Error Estimates for Fits::
+* Standard Nonlinear Models::
+* Curve Fitting Details::
+* Interpolation::
address@hidden menu
+
address@hidden Linear Fits, Polynomial and Multilinear Fits, Curve Fitting,
Curve Fitting
address@hidden Linear Fits
+
address@hidden
address@hidden a F
address@hidden calc-curve-fit
address@hidden fit
address@hidden Linear regression
address@hidden Least-squares fits
+The @kbd{a F} (@code{calc-curve-fit}) address@hidden command attempts
+to fit a set of data (@expr{x} and @expr{y} vectors of numbers) to a
+straight line, polynomial, or other function of @expr{x}. For the
+moment we will consider only the case of fitting to a line, and we
+will ignore the issue of whether or not the model was in fact a good
+fit for the data.
+
+In a standard linear least-squares fit, we have a set of @expr{(x,y)}
+data points that we wish to fit to the model @expr{y = m x + b}
+by adjusting the parameters @expr{m} and @expr{b} to make the @expr{y}
+values calculated from the formula be as close as possible to the actual
address@hidden values in the data set. (In a polynomial fit, the model is
+instead, say, @expr{y = a x^3 + b x^2 + c x + d}. In a multilinear fit,
+we have data points of the form @expr{(x_1,x_2,x_3,y)} and our model is
address@hidden = a x_1 + b x_2 + c x_3 + d}. These will be discussed later.)
+
+In the model formula, variables like @expr{x} and @expr{x_2} are called
+the @dfn{independent variables}, and @expr{y} is the @dfn{dependent
+variable}. Variables like @expr{m}, @expr{a}, and @expr{b} are called
+the @dfn{parameters} of the model.
+
+The @kbd{a F} command takes the data set to be fitted from the stack.
+By default, it expects the data in the form of a matrix. For example,
+for a linear or polynomial fit, this would be a
address@hidden @math{2\times N}
address@hidden 2xN
+matrix where the first row is a list of @expr{x} values and the second
+row has the corresponding @expr{y} values. For the multilinear fit
+shown above, the matrix would have four rows (@expr{x_1}, @expr{x_2},
address@hidden, and @expr{y}, respectively).
+
+If you happen to have an
address@hidden @math{N\times2}
address@hidden Nx2
+matrix instead of a
address@hidden @math{2\times N}
address@hidden 2xN
+matrix, just press @kbd{v t} first to transpose the matrix.
+
+After you type @kbd{a F}, Calc prompts you to select a model. For a
+linear fit, press the digit @kbd{1}.
+
+Calc then prompts for you to name the variables. By default it chooses
+high letters like @expr{x} and @expr{y} for independent variables and
+low letters like @expr{a} and @expr{b} for parameters. (The dependent
+variable doesn't need a name.) The two kinds of variables are separated
+by a semicolon. Since you generally care more about the names of the
+independent variables than of the parameters, Calc also allows you to
+name only those and let the parameters use default names.
+
+For example, suppose the data matrix
+
address@hidden
address@hidden
address@hidden
+[ [ 1, 2, 3, 4, 5 ]
+ [ 5, 7, 9, 11, 13 ] ]
address@hidden group
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\turnoffactive
+\beforedisplay
+$$ \pmatrix{ 1 & 2 & 3 & 4 & 5 \cr
+ 5 & 7 & 9 & 11 & 13 }
+$$
+\afterdisplay
address@hidden tex
+
address@hidden
+is on the stack and we wish to do a simple linear fit. Type
address@hidden F}, then @kbd{1} for the model, then @key{RET} to use
+the default names. The result will be the formula @expr{3. + 2. x}
+on the stack. Calc has created the model expression @kbd{a + b x},
+then found the optimal values of @expr{a} and @expr{b} to fit the
+data. (In this case, it was able to find an exact fit.) Calc then
+substituted those values for @expr{a} and @expr{b} in the model
+formula.
+
+The @kbd{a F} command puts two entries in the trail. One is, as
+always, a copy of the result that went to the stack; the other is
+a vector of the actual parameter values, written as equations:
address@hidden = 3, b = 2]}, in case you'd rather read them in a list
+than pick them out of the formula. (You can type @kbd{t y}
+to move this vector to the stack; see @ref{Trail Commands}.
+
+Specifying a different independent variable name will affect the
+resulting formula: @kbd{a F 1 k @key{RET}} produces @kbd{3 + 2 k}.
+Changing the parameter names (say, @kbd{a F 1 k;b,m @key{RET}}) will affect
+the equations that go into the trail.
+
address@hidden
+\bigskip
address@hidden tex
+
+To see what happens when the fit is not exact, we could change
+the number 13 in the data matrix to 14 and try the fit again.
+The result is:
+
address@hidden
+2.6 + 2.2 x
address@hidden example
+
+Evaluating this formula, say with @kbd{v x 5 @key{RET} @key{TAB} V M $
@key{RET}}, shows
+a reasonably close match to the y-values in the data.
+
address@hidden
+[4.8, 7., 9.2, 11.4, 13.6]
address@hidden example
+
+Since there is no line which passes through all the @var{n} data points,
+Calc has chosen a line that best approximates the data points using
+the method of least squares. The idea is to define the @dfn{chi-square}
+error measure
+
address@hidden
address@hidden
+chi^2 = sum((y_i - (a + b x_i))^2, i, 1, N)
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ \chi^2 = \sum_{i=1}^N (y_i - (a + b x_i))^2 $$
+\afterdisplay
address@hidden tex
+
address@hidden
+which is clearly zero if @expr{a + b x} exactly fits all data points,
+and increases as various @expr{a + b x_i} values fail to match the
+corresponding @expr{y_i} values. There are several reasons why the
+summand is squared, one of them being to ensure that
address@hidden @math{\chi^2 \ge 0}.
address@hidden @expr{chi^2 >= 0}.
+Least-squares fitting simply chooses the values of @expr{a} and @expr{b}
+for which the error
address@hidden @math{\chi^2}
address@hidden @expr{chi^2}
+is as small as possible.
+
+Other kinds of models do the same thing but with a different model
+formula in place of @expr{a + b x_i}.
+
address@hidden
+\bigskip
address@hidden tex
+
+A numeric prefix argument causes the @kbd{a F} command to take the
+data in some other form than one big matrix. A positive argument @var{n}
+will take @var{N} items from the stack, corresponding to the @var{n} rows
+of a data matrix. In the linear case, @var{n} must be 2 since there
+is always one independent variable and one dependent variable.
+
+A prefix of zero or plain @kbd{C-u} is a compromise; Calc takes two
+items from the stack, an @var{n}-row matrix of @expr{x} values, and a
+vector of @expr{y} values. If there is only one independent variable,
+the @expr{x} values can be either a one-row matrix or a plain vector,
+in which case the @kbd{C-u} prefix is the same as a @address@hidden 2}} prefix.
+
address@hidden Polynomial and Multilinear Fits, Error Estimates for Fits,
Linear Fits, Curve Fitting
address@hidden Polynomial and Multilinear Fits
+
address@hidden
+To fit the data to higher-order polynomials, just type one of the
+digits @kbd{2} through @kbd{9} when prompted for a model. For example,
+we could fit the original data matrix from the previous section
+(with 13, not 14) to a parabola instead of a line by typing
address@hidden F 2 @key{RET}}.
+
address@hidden
+2.00000000001 x - 1.5e-12 x^2 + 2.99999999999
address@hidden example
+
+Note that since the constant and linear terms are enough to fit the
+data exactly, it's no surprise that Calc chose a tiny contribution
+for @expr{x^2}. (The fact that it's not exactly zero is due only
+to roundoff error. Since our data are exact integers, we could get
+an exact answer by typing @kbd{m f} first to get Fraction mode.
+Then the @expr{x^2} term would vanish altogether. Usually, though,
+the data being fitted will be approximate floats so Fraction mode
+won't help.)
+
+Doing the @kbd{a F 2} fit on the data set with 14 instead of 13
+gives a much larger @expr{x^2} contribution, as Calc bends the
+line slightly to improve the fit.
+
address@hidden
+0.142857142855 x^2 + 1.34285714287 x + 3.59999999998
address@hidden example
+
+An important result from the theory of polynomial fitting is that it
+is always possible to fit @var{n} data points exactly using a polynomial
+of degree @address@hidden, sometimes called an @dfn{interpolating polynomial}.
+Using the modified (14) data matrix, a model number of 4 gives
+a polynomial that exactly matches all five data points:
+
address@hidden
+0.04167 x^4 - 0.4167 x^3 + 1.458 x^2 - 0.08333 x + 4.
address@hidden example
+
+The actual coefficients we get with a precision of 12, like
address@hidden, clearly suffer from loss of precision.
+It is a good idea to increase the working precision to several
+digits beyond what you need when you do a fitting operation.
+Or, if your data are exact, use Fraction mode to get exact
+results.
+
+You can type @kbd{i} instead of a digit at the model prompt to fit
+the data exactly to a polynomial. This just counts the number of
+columns of the data matrix to choose the degree of the polynomial
+automatically.
+
+Fitting data ``exactly'' to high-degree polynomials is not always
+a good idea, though. High-degree polynomials have a tendency to
+wiggle uncontrollably in between the fitting data points. Also,
+if the exact-fit polynomial is going to be used to interpolate or
+extrapolate the data, it is numerically better to use the @kbd{a p}
+command described below. @xref{Interpolation}.
+
address@hidden
+\bigskip
address@hidden tex
+
+Another generalization of the linear model is to assume the
address@hidden values are a sum of linear contributions from several
address@hidden values. This is a @dfn{multilinear} fit, and it is also
+selected by the @kbd{1} digit key. (Calc decides whether the fit
+is linear or multilinear by counting the rows in the data matrix.)
+
+Given the data matrix,
+
address@hidden
address@hidden
+[ [ 1, 2, 3, 4, 5 ]
+ [ 7, 2, 3, 5, 2 ]
+ [ 14.5, 15, 18.5, 22.5, 24 ] ]
address@hidden group
address@hidden example
+
address@hidden
+the command @kbd{a F 1 @key{RET}} will call the first row @expr{x} and the
+second row @expr{y}, and will fit the values in the third row to the
+model @expr{a + b x + c y}.
+
address@hidden
+8. + 3. x + 0.5 y
address@hidden example
+
+Calc can do multilinear fits with any number of independent variables
+(i.e., with any number of data rows).
+
address@hidden
+\bigskip
address@hidden tex
+
+Yet another variation is @dfn{homogeneous} linear models, in which
+the constant term is known to be zero. In the linear case, this
+means the model formula is simply @expr{a x}; in the multilinear
+case, the model might be @expr{a x + b y + c z}; and in the polynomial
+case, the model could be @expr{a x + b x^2 + c x^3}. You can get
+a homogeneous linear or multilinear model by pressing the letter
address@hidden followed by a regular model key, like @kbd{1} or @kbd{2}.
+This will be indicated by an ``h'' in the minibuffer after the help
+message.
+
+It is certainly possible to have other constrained linear models,
+like @expr{2.3 + a x} or @expr{a - 4 x}. While there is no single
+key to select models like these, a later section shows how to enter
+any desired model by hand. In the first case, for example, you
+would enter @kbd{a F ' 2.3 + a x}.
+
+Another class of models that will work but must be entered by hand
+are multinomial fits, e.g., @expr{a + b x + c y + d x^2 + e y^2 + f x y}.
+
address@hidden Error Estimates for Fits, Standard Nonlinear Models, Polynomial
and Multilinear Fits, Curve Fitting
address@hidden Error Estimates for Fits
+
address@hidden
address@hidden H a F
address@hidden efit
+With the Hyperbolic flag, @kbd{H a F} address@hidden performs the same
+fitting operation as @kbd{a F}, but reports the coefficients as error
+forms instead of plain numbers. Fitting our two data matrices (first
+with 13, then with 14) to a line with @kbd{H a F} gives the results,
+
address@hidden
+3. + 2. x
+2.6 +/- 0.382970843103 + 2.2 +/- 0.115470053838 x
address@hidden example
+
+In the first case the estimated errors are zero because the linear
+fit is perfect. In the second case, the errors are nonzero but
+moderately small, because the data are still very close to linear.
+
+It is also possible for the @emph{input} to a fitting operation to
+contain error forms. The data values must either all include errors
+or all be plain numbers. Error forms can go anywhere but generally
+go on the numbers in the last row of the data matrix. If the last
+row contains error forms
address@hidden address@hidden@w{ @tfn{+/-} address@hidden',
address@hidden address@hidden@w{ @tfn{+/-} address@hidden',
+then the
address@hidden @math{\chi^2}
address@hidden @expr{chi^2}
+statistic is now,
+
address@hidden
address@hidden
+chi^2 = sum(((y_i - (a + b x_i)) / sigma_i)^2, i, 1, N)
address@hidden example
address@hidden ifnottex
address@hidden
+\turnoffactive
+\beforedisplay
+$$ \chi^2 = \sum_{i=1}^N \left(y_i - (a + b x_i) \over \sigma_i\right)^2 $$
+\afterdisplay
address@hidden tex
+
address@hidden
+so that data points with larger error estimates contribute less to
+the fitting operation.
+
+If there are error forms on other rows of the data matrix, all the
+errors for a given data point are combined; the square root of the
+sum of the squares of the errors forms the
address@hidden @math{\sigma_i}
address@hidden @expr{sigma_i}
+used for the data point.
+
+Both @kbd{a F} and @kbd{H a F} can accept error forms in the input
+matrix, although if you are concerned about error analysis you will
+probably use @kbd{H a F} so that the output also contains error
+estimates.
+
+If the input contains error forms but all the
address@hidden @math{\sigma_i}
address@hidden @expr{sigma_i}
+values are the same, it is easy to see that the resulting fitted model
+will be the same as if the input did not have error forms at all
address@hidden (@math{\chi^2}
address@hidden (@expr{chi^2}
+is simply scaled uniformly by
address@hidden @math{1 / \sigma^2},
address@hidden @expr{1 / sigma^2},
+which doesn't affect where it has a minimum). But there @emph{will} be
+a difference in the estimated errors of the coefficients reported by
address@hidden a F}.
+
+Consult any text on statistical modeling of data for a discussion
+of where these error estimates come from and how they should be
+interpreted.
+
address@hidden
+\bigskip
address@hidden tex
+
address@hidden I a F
address@hidden xfit
+With the Inverse flag, @kbd{I a F} address@hidden produces even more
+information. The result is a vector of six items:
+
address@hidden
address@hidden
+The model formula with error forms for its coefficients or
+parameters. This is the result that @kbd{H a F} would have
+produced.
+
address@hidden
+A vector of ``raw'' parameter values for the model. These are the
+polynomial coefficients or other parameters as plain numbers, in the
+same order as the parameters appeared in the final prompt of the
address@hidden a F} command. For polynomials of degree @expr{d}, this vector
+will have length @expr{M = d+1} with the constant term first.
+
address@hidden
+The covariance matrix @expr{C} computed from the fit. This is
+an @address@hidden symmetric matrix; the diagonal elements
address@hidden @math{C_{jj}}
address@hidden @expr{C_j_j}
+are the variances
address@hidden @math{\sigma_j^2}
address@hidden @expr{sigma_j^2}
+of the parameters. The other elements are covariances
address@hidden @math{\sigma_{ij}^2}
address@hidden @expr{sigma_i_j^2}
+that describe the correlation between pairs of parameters. (A related
+set of numbers, the @dfn{linear correlation coefficients}
address@hidden @math{r_{ij}},
address@hidden @expr{r_i_j},
+are defined as
address@hidden @math{\sigma_{ij}^2 / \sigma_i \, \sigma_j}.)
address@hidden @expr{sigma_i_j^2 / sigma_i sigma_j}.)
+
address@hidden
+A vector of @expr{M} ``parameter filter'' functions whose
+meanings are described below. If no filters are necessary this
+will instead be an empty vector; this is always the case for the
+polynomial and multilinear fits described so far.
+
address@hidden
+The value of
address@hidden @math{\chi^2}
address@hidden @expr{chi^2}
+for the fit, calculated by the formulas shown above. This gives a
+measure of the quality of the fit; statisticians consider
address@hidden @math{\chi^2 \approx N - M}
address@hidden @expr{chi^2 = N - M}
+to indicate a moderately good fit (where again @expr{N} is the number of
+data points and @expr{M} is the number of parameters).
+
address@hidden
+A measure of goodness of fit expressed as a probability @expr{Q}.
+This is computed from the @code{utpc} probability distribution
+function using
address@hidden @math{\chi^2}
address@hidden @expr{chi^2}
+with @expr{N - M} degrees of freedom. A
+value of 0.5 implies a good fit; some texts recommend that often
address@hidden = 0.1} or even 0.001 can signify an acceptable fit. In
+particular,
address@hidden @math{\chi^2}
address@hidden @expr{chi^2}
+statistics assume the errors in your inputs
+follow a normal (Gaussian) distribution; if they don't, you may
+have to accept smaller values of @expr{Q}.
+
+The @expr{Q} value is computed only if the input included error
+estimates. Otherwise, Calc will report the symbol @code{nan}
+for @expr{Q}. The reason is that in this case the
address@hidden @math{\chi^2}
address@hidden @expr{chi^2}
+value has effectively been used to estimate the original errors
+in the input, and thus there is no redundant information left
+over to use for a confidence test.
address@hidden enumerate
+
address@hidden Standard Nonlinear Models, Curve Fitting Details, Error
Estimates for Fits, Curve Fitting
address@hidden Standard Nonlinear Models
+
address@hidden
+The @kbd{a F} command also accepts other kinds of models besides
+lines and polynomials. Some common models have quick single-key
+abbreviations; others must be entered by hand as algebraic formulas.
+
+Here is a complete list of the standard models recognized by @kbd{a F}:
+
address@hidden @kbd
address@hidden 1
+Linear or multilinear. @mathit{a + b x + c y + d z}.
address@hidden 2-9
+Polynomials. @mathit{a + b x + c x^2 + d x^3}.
address@hidden e
+Exponential. @mathit{a} @address@hidden(b x)} @address@hidden(c y)}.
address@hidden E
+Base-10 exponential. @mathit{a} @address@hidden(b x)} @address@hidden(c y)}.
address@hidden x
+Exponential (alternate notation). @address@hidden(a + b x + c y)}.
address@hidden X
+Base-10 exponential (alternate). @address@hidden(a + b x + c y)}.
address@hidden l
+Logarithmic. @mathit{a + b} @address@hidden(x) + c} @address@hidden(y)}.
address@hidden L
+Base-10 logarithmic. @mathit{a + b} @address@hidden(x) + c}
@address@hidden(y)}.
address@hidden ^
+General exponential. @mathit{a b^x c^y}.
address@hidden p
+Power law. @mathit{a x^b y^c}.
address@hidden q
+Quadratic. @mathit{a + b (x-c)^2 + d (x-e)^2}.
address@hidden g
+Gaussian.
address@hidden @math{{a \over b \sqrt{2 \pi}} \exp\left( -{1 \over 2} \left( x
- c \over b \right)^2 \right)}.
address@hidden @mathit{(a / b sqrt(2 pi)) exp(-0.5*((x-c)/b)^2)}.
address@hidden s
+Logistic @emph{s} curve.
address@hidden @math{a/(1+e^{b(x-c)})}.
address@hidden @mathit{a/(1 + exp(b (x - c)))}.
address@hidden b
+Logistic bell curve.
address@hidden @math{ae^{b(x-c)}/(1+e^{b(x-c)})^2}.
address@hidden @mathit{a exp(b (x - c))/(1 + exp(b (x - c)))^2}.
address@hidden o
+Hubbert linearization.
address@hidden @math{{y \over x} = a(1-x/b)}.
address@hidden @mathit{(y/x) = a (1 - x/b)}.
address@hidden table
+
+All of these models are used in the usual way; just press the appropriate
+letter at the model prompt, and choose variable names if you wish. The
+result will be a formula as shown in the above table, with the best-fit
+values of the parameters substituted. (You may find it easier to read
+the parameter values from the vector that is placed in the trail.)
+
+All models except Gaussian, logistics, Hubbert and polynomials can
+generalize as shown to any number of independent variables. Also, all
+the built-in models except for the logistic and Hubbert curves have an
+additive or multiplicative parameter shown as @expr{a} in the above table
+which can be replaced by zero or one, as appropriate, by typing @kbd{h}
+before the model key.
+
+Note that many of these models are essentially equivalent, but express
+the parameters slightly differently. For example, @expr{a b^x} and
+the other two exponential models are all algebraic rearrangements of
+each other. Also, the ``quadratic'' model is just a degree-2 polynomial
+with the parameters expressed differently. Use whichever form best
+matches the problem.
+
+The HP-28/48 calculators support four different models for curve
+fitting, called @code{LIN}, @code{LOG}, @code{EXP}, and @code{PWR}.
+These correspond to Calc models @samp{a + b x}, @samp{a + b ln(x)},
address@hidden exp(b x)}, and @samp{a x^b}, respectively. In each case,
address@hidden is what the HP-48 identifies as the ``intercept,'' and
address@hidden is what it calls the ``slope.''
+
address@hidden
+\bigskip
address@hidden tex
+
+If the model you want doesn't appear on this list, press @kbd{'}
+(the apostrophe key) at the model prompt to enter any algebraic
+formula, such as @kbd{m x - b}, as the model. (Not all models
+will work, though---see the next section for details.)
+
+The model can also be an equation like @expr{y = m x + b}.
+In this case, Calc thinks of all the rows of the data matrix on
+equal terms; this model effectively has two parameters
+(@expr{m} and @expr{b}) and two independent variables (@expr{x}
+and @expr{y}), with no ``dependent'' variables. Model equations
+do not need to take this @expr{y =} form. For example, the
+implicit line equation @expr{a x + b y = 1} works fine as a
+model.
+
+When you enter a model, Calc makes an alphabetical list of all
+the variables that appear in the model. These are used for the
+default parameters, independent variables, and dependent variable
+(in that order). If you enter a plain formula (not an equation),
+Calc assumes the dependent variable does not appear in the formula
+and thus does not need a name.
+
+For example, if the model formula has the variables @expr{a,mu,sigma,t,x},
+and the data matrix has three rows (meaning two independent variables),
+Calc will use @expr{a,mu,sigma} as the default parameters, and the
+data rows will be named @expr{t} and @expr{x}, respectively. If you
+enter an equation instead of a plain formula, Calc will use @expr{a,mu}
+as the parameters, and @expr{sigma,t,x} as the three independent
+variables.
+
+You can, of course, override these choices by entering something
+different at the prompt. If you leave some variables out of the list,
+those variables must have stored values and those stored values will
+be used as constants in the model. (Stored values for the parameters
+and independent variables are ignored by the @kbd{a F} command.)
+If you list only independent variables, all the remaining variables
+in the model formula will become parameters.
+
+If there are @kbd{$} signs in the model you type, they will stand
+for parameters and all other variables (in alphabetical order)
+will be independent. Use @kbd{$} for one parameter, @kbd{$$} for
+another, and so on. Thus @kbd{$ x + $$} is another way to describe
+a linear model.
+
+If you type a @kbd{$} instead of @kbd{'} at the model prompt itself,
+Calc will take the model formula from the stack. (The data must then
+appear at the second stack level.) The same conventions are used to
+choose which variables in the formula are independent by default and
+which are parameters.
+
+Models taken from the stack can also be expressed as vectors of
+two or three elements, @address@hidden, @var{vars}]} or
address@hidden@var{model}, @var{vars}, @var{params}]}. Each of @var{vars}
+and @var{params} may be either a variable or a vector of variables.
+(If @var{params} is omitted, all variables in @var{model} except
+those listed as @var{vars} are parameters.)
+
+When you enter a model manually with @kbd{'}, Calc puts a 3-vector
+describing the model in the trail so you can get it back if you wish.
+
address@hidden
+\bigskip
address@hidden tex
+
address@hidden Model1
address@hidden Model2
+Finally, you can store a model in one of the Calc variables
address@hidden or @code{Model2}, then use this model by typing
address@hidden F u} or @kbd{a F U} (respectively). The value stored in
+the variable can be any of the formats that @kbd{a F $} would
+accept for a model on the stack.
+
address@hidden
+\bigskip
address@hidden tex
+
+Calc uses the principal values of inverse functions like @code{ln}
+and @code{arcsin} when doing fits. For example, when you enter
+the model @samp{y = sin(a t + b)} Calc actually uses the easier
+form @samp{arcsin(y) = a t + b}. The @code{arcsin} function always
+returns results in the range from @mathit{-90} to 90 degrees (or the
+equivalent range in radians). Suppose you had data that you
+believed to represent roughly three oscillations of a sine wave,
+so that the argument of the sine might go from zero to
address@hidden @math{3\times360}
address@hidden @mathit{3*360}
+degrees.
+The above model would appear to be a good way to determine the
+true frequency and phase of the sine wave, but in practice it
+would fail utterly. The righthand side of the actual model
address@hidden(y) = a t + b} will grow smoothly with @expr{t}, but
+the lefthand side will bounce back and forth between @mathit{-90} and 90.
+No values of @expr{a} and @expr{b} can make the two sides match,
+even approximately.
+
+There is no good solution to this problem at present. You could
+restrict your data to small enough ranges so that the above problem
+doesn't occur (i.e., not straddling any peaks in the sine wave).
+Or, in this case, you could use a totally different method such as
+Fourier analysis, which is beyond the scope of the @kbd{a F} command.
+(Unfortunately, Calc does not currently have any facilities for
+taking Fourier and related transforms.)
+
address@hidden Curve Fitting Details, Interpolation, Standard Nonlinear Models,
Curve Fitting
address@hidden Curve Fitting Details
+
address@hidden
+Calc's internal least-squares fitter can only handle multilinear
+models. More precisely, it can handle any model of the form
address@hidden f(x,y,z) + b g(x,y,z) + c h(x,y,z)}, where @expr{a,b,c}
+are the parameters and @expr{x,y,z} are the independent variables
+(of course there can be any number of each, not just three).
+
+In a simple multilinear or polynomial fit, it is easy to see how
+to convert the model into this form. For example, if the model
+is @expr{a + b x + c x^2}, then @expr{f(x) = 1}, @expr{g(x) = x},
+and @expr{h(x) = x^2} are suitable functions.
+
+For most other models, Calc uses a variety of algebraic manipulations
+to try to put the problem into the form
+
address@hidden
+Y(x,y,z) = A(a,b,c) F(x,y,z) + B(a,b,c) G(x,y,z) + C(a,b,c) H(x,y,z)
address@hidden smallexample
+
address@hidden
+where @expr{Y,A,B,C,F,G,H} are arbitrary functions. It computes
address@hidden, @expr{F}, @expr{G}, and @expr{H} for all the data points,
+does a standard linear fit to find the values of @expr{A}, @expr{B},
+and @expr{C}, then uses the equation solver to solve for @expr{a,b,c}
+in terms of @expr{A,B,C}.
+
+A remarkable number of models can be cast into this general form.
+We'll look at two examples here to see how it works. The power-law
+model @expr{y = a x^b} with two independent variables and two parameters
+can be rewritten as follows:
+
address@hidden
+y = a x^b
+y = a exp(b ln(x))
+y = exp(ln(a) + b ln(x))
+ln(y) = ln(a) + b ln(x)
address@hidden example
+
address@hidden
+which matches the desired form with
address@hidden @math{Y = \ln(y)},
address@hidden @expr{Y = ln(y)},
address@hidden @math{A = \ln(a)},
address@hidden @expr{A = ln(a)},
address@hidden = 1}, @expr{B = b}, and
address@hidden @math{G = \ln(x)}.
address@hidden @expr{G = ln(x)}.
+Calc thus computes the logarithms of your @expr{y} and @expr{x} values,
+does a linear fit for @expr{A} and @expr{B}, then solves to get
address@hidden @math{a = \exp(A)}
address@hidden @expr{a = exp(A)}
+and @expr{b = B}.
+
+Another interesting example is the ``quadratic'' model, which can
+be handled by expanding according to the distributive law.
+
address@hidden
+y = a + b*(x - c)^2
+y = a + b c^2 - 2 b c x + b x^2
address@hidden example
+
address@hidden
+which matches with @expr{Y = y}, @expr{A = a + b c^2}, @expr{F = 1},
address@hidden = -2 b c}, @expr{G = x} (the @mathit{-2} factor could just as
easily
+have been put into @expr{G} instead of @expr{B}), @expr{C = b}, and
address@hidden = x^2}.
+
+The Gaussian model looks quite complicated, but a closer examination
+shows that it's actually similar to the quadratic model but with an
+exponential that can be brought to the top and moved into @expr{Y}.
+
+The logistic models cannot be put into general linear form. For these
+models, and the Hubbert linearization, Calc computes a rough
+approximation for the parameters, then uses the Levenberg-Marquardt
+iterative method to refine the approximations.
+
+Another model that cannot be put into general linear
+form is a Gaussian with a constant background added on, i.e.,
address@hidden + the regular Gaussian formula. If you have a model like
+this, your best bet is to replace enough of your parameters with
+constants to make the model linearizable, then adjust the constants
+manually by doing a series of fits. You can compare the fits by
+graphing them, by examining the goodness-of-fit measures returned by
address@hidden a F}, or by some other method suitable to your application.
+Note that some models can be linearized in several ways. The
address@hidden model can be linearized by setting @expr{d}
+(the background) to a constant, or by setting @expr{b} (the standard
+deviation) and @expr{c} (the mean) to constants.
+
+To fit a model with constants substituted for some parameters, just
+store suitable values in those parameter variables, then omit them
+from the list of parameters when you answer the variables prompt.
+
address@hidden
+\bigskip
address@hidden tex
+
+A last desperate step would be to use the general-purpose
address@hidden function rather than @code{fit}. After all, both
+functions solve the problem of minimizing an expression (the
address@hidden @math{\chi^2}
address@hidden @expr{chi^2}
+sum) by adjusting certain parameters in the expression. The @kbd{a F}
+command is able to use a vastly more efficient algorithm due to its
+special knowledge about linear chi-square sums, but the @kbd{a N}
+command can do the same thing by brute force.
+
+A compromise would be to pick out a few parameters without which the
+fit is linearizable, and use @code{minimize} on a call to @code{fit}
+which efficiently takes care of the rest of the parameters. The thing
+to be minimized would be the value of
address@hidden @math{\chi^2}
address@hidden @expr{chi^2}
+returned as the fifth result of the @code{xfit} function:
+
address@hidden
+minimize(xfit(gaus(a,b,c,d,x), x, [a,b,c], data)_5, d, guess)
address@hidden smallexample
+
address@hidden
+where @code{gaus} represents the Gaussian model with background,
address@hidden represents the data matrix, and @code{guess} represents
+the initial guess for @expr{d} that @code{minimize} requires.
+This operation will only be, shall we say, extraordinarily slow
+rather than astronomically slow (as would be the case if @code{minimize}
+were used by itself to solve the problem).
+
address@hidden
+\bigskip
address@hidden tex
+
+The @kbd{I a F} address@hidden command is somewhat trickier when
+nonlinear models are used. The second item in the result is the
+vector of ``raw'' parameters @expr{A}, @expr{B}, @expr{C}. The
+covariance matrix is written in terms of those raw parameters.
+The fifth item is a vector of @dfn{filter} expressions. This
+is the empty vector @samp{[]} if the raw parameters were the same
+as the requested parameters, i.e., if @expr{A = a}, @expr{B = b},
+and so on (which is always true if the model is already linear
+in the parameters as written, e.g., for polynomial fits). If the
+parameters had to be rearranged, the fifth item is instead a vector
+of one formula per parameter in the original model. The raw
+parameters are expressed in these ``filter'' formulas as
address@hidden(1)} for @expr{A}, @samp{fitdummy(2)} for @expr{B},
+and so on.
+
+When Calc needs to modify the model to return the result, it replaces
address@hidden(1)} in all the filters with the first item in the raw
+parameters list, and so on for the other raw parameters, then
+evaluates the resulting filter formulas to get the actual parameter
+values to be substituted into the original model. In the case of
address@hidden a F} and @kbd{I a F} where the parameters must be error forms,
+Calc uses the square roots of the diagonal entries of the covariance
+matrix as error values for the raw parameters, then lets Calc's
+standard error-form arithmetic take it from there.
+
+If you use @kbd{I a F} with a nonlinear model, be sure to remember
+that the covariance matrix is in terms of the raw parameters,
address@hidden the actual requested parameters. It's up to you to
+figure out how to interpret the covariances in the presence of
+nontrivial filter functions.
+
+Things are also complicated when the input contains error forms.
+Suppose there are three independent and dependent variables, @expr{x},
address@hidden, and @expr{z}, one or more of which are error forms in the
+data. Calc combines all the error values by taking the square root
+of the sum of the squares of the errors. It then changes @expr{x}
+and @expr{y} to be plain numbers, and makes @expr{z} into an error
+form with this combined error. The @expr{Y(x,y,z)} part of the
+linearized model is evaluated, and the result should be an error
+form. The error part of that result is used for
address@hidden @math{\sigma_i}
address@hidden @expr{sigma_i}
+for the data point. If for some reason @expr{Y(x,y,z)} does not return
+an error form, the combined error from @expr{z} is used directly for
address@hidden @math{\sigma_i}.
address@hidden @expr{sigma_i}.
+Finally, @expr{z} is also stripped of its error
+for use in computing @expr{F(x,y,z)}, @expr{G(x,y,z)} and so on;
+the righthand side of the linearized model is computed in regular
+arithmetic with no error forms.
+
+(While these rules may seem complicated, they are designed to do
+the most reasonable thing in the typical case that @expr{Y(x,y,z)}
+depends only on the dependent variable @expr{z}, and in fact is
+often simply equal to @expr{z}. For common cases like polynomials
+and multilinear models, the combined error is simply used as the
address@hidden @math{\sigma}
address@hidden @expr{sigma}
+for the data point with no further ado.)
+
address@hidden
+\bigskip
address@hidden tex
+
address@hidden FitRules
+It may be the case that the model you wish to use is linearizable,
+but Calc's built-in rules are unable to figure it out. Calc uses
+its algebraic rewrite mechanism to linearize a model. The rewrite
+rules are kept in the variable @code{FitRules}. You can edit this
+variable using the @kbd{s e FitRules} command; in fact, there is
+a special @kbd{s F} command just for editing @code{FitRules}.
address@hidden on Variables}.
+
address@hidden Rules}, for a discussion of rewrite rules.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden fitvar
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden fitparam
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden @null
address@hidden ignore
address@hidden fitmodel
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden @null
address@hidden ignore
address@hidden fitsystem
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden @null
address@hidden ignore
address@hidden fitdummy
+Calc uses @code{FitRules} as follows. First, it converts the model
+to an equation if necessary and encloses the model equation in a
+call to the function @code{fitmodel} (which is not actually a defined
+function in Calc; it is only used as a placeholder by the rewrite rules).
+Parameter variables are renamed to function calls @samp{fitparam(1)},
address@hidden(2)}, and so on, and independent variables are renamed
+to @samp{fitvar(1)}, @samp{fitvar(2)}, etc. The dependent variable
+is the highest-numbered @code{fitvar}. For example, the power law
+model @expr{a x^b} is converted to @expr{y = a x^b}, then to
+
address@hidden
address@hidden
+fitmodel(fitvar(2) = fitparam(1) fitvar(1)^fitparam(2))
address@hidden group
address@hidden smallexample
+
+Calc then applies the rewrites as if by @samp{C-u 0 a r FitRules}.
+(The zero prefix means that rewriting should continue until no further
+changes are possible.)
+
+When rewriting is complete, the @code{fitmodel} call should have
+been replaced by a @code{fitsystem} call that looks like this:
+
address@hidden
+fitsystem(@var{Y}, @var{FGH}, @var{abc})
address@hidden example
+
address@hidden
+where @var{Y} is a formula that describes the function @expr{Y(x,y,z)},
address@hidden is the vector of formulas @expr{[F(x,y,z), G(x,y,z), H(x,y,z)]},
+and @var{abc} is the vector of parameter filters which refer to the
+raw parameters as @samp{fitdummy(1)} for @expr{A}, @samp{fitdummy(2)}
+for @expr{B}, etc. While the number of raw parameters (the length of
+the @var{FGH} vector) is usually the same as the number of original
+parameters (the length of the @var{abc} vector), this is not required.
+
+The power law model eventually boils down to
+
address@hidden
address@hidden
+fitsystem(ln(fitvar(2)),
+ [1, ln(fitvar(1))],
+ [exp(fitdummy(1)), fitdummy(2)])
address@hidden group
address@hidden smallexample
+
+The actual implementation of @code{FitRules} is complicated; it
+proceeds in four phases. First, common rearrangements are done
+to try to bring linear terms together and to isolate functions like
address@hidden and @code{ln} either all the way ``out'' (so that they
+can be put into @var{Y}) or all the way ``in'' (so that they can
+be put into @var{abc} or @var{FGH}). In particular, all
+non-constant powers are converted to logs-and-exponentials form,
+and the distributive law is used to expand products of sums.
+Quotients are rewritten to use the @samp{fitinv} function, where
address@hidden(x)} represents @expr{1/x} while the @code{FitRules}
+are operating. (The use of @code{fitinv} makes recognition of
+linear-looking forms easier.) If you modify @code{FitRules}, you
+will probably only need to modify the rules for this phase.
+
+Phase two, whose rules can actually also apply during phases one
+and three, first rewrites @code{fitmodel} to a two-argument
+form @samp{fitmodel(@var{Y}, @var{model})}, where @var{Y} is
+initially zero and @var{model} has been changed from @expr{a=b}
+to @expr{a-b} form. It then tries to peel off invertible functions
+from the outside of @var{model} and put them into @var{Y} instead,
+calling the equation solver to invert the functions. Finally, when
+this is no longer possible, the @code{fitmodel} is changed to a
+four-argument @code{fitsystem}, where the fourth argument is
address@hidden and the @var{FGH} and @var{abc} vectors are initially
+empty. (The last vector is really @var{ABC}, corresponding to
+raw parameters, for now.)
+
+Phase three converts a sum of items in the @var{model} to a sum
+of @samp{fitpart(@var{a}, @var{b}, @var{c})} terms which represent
+terms @address@hidden@address@hidden of the sum, where @var{a}
+is all factors that do not involve any variables, @var{b} is all
+factors that involve only parameters, and @var{c} is the factors
+that involve only independent variables. (If this decomposition
+is not possible, the rule set will not complete and Calc will
+complain that the model is too complex.) Then @code{fitpart}s
+with equal @var{b} or @var{c} components are merged back together
+using the distributive law in order to minimize the number of
+raw parameters needed.
+
+Phase four moves the @code{fitpart} terms into the @var{FGH} and
address@hidden vectors. Also, some of the algebraic expansions that
+were done in phase 1 are undone now to make the formulas more
+computationally efficient. Finally, it calls the solver one more
+time to convert the @var{ABC} vector to an @var{abc} vector, and
+removes the fourth @var{model} argument (which by now will be zero)
+to obtain the three-argument @code{fitsystem} that the linear
+least-squares solver wants to see.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden hasfitparams
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden @null
address@hidden ignore
address@hidden hasfitvars
+Two functions which are useful in connection with @code{FitRules}
+are @samp{hasfitparams(x)} and @samp{hasfitvars(x)}, which check
+whether @expr{x} refers to any parameters or independent variables,
+respectively. Specifically, these functions return ``true'' if the
+argument contains any @code{fitparam} (or @code{fitvar}) function
+calls, and ``false'' otherwise. (Recall that ``true'' means a
+nonzero number, and ``false'' means zero. The actual nonzero number
+returned is the largest @var{n} from all the @samp{fitparam(@var{n})}s
+or @samp{fitvar(@var{n})}s, respectively, that appear in the formula.)
+
address@hidden
+\bigskip
address@hidden tex
+
+The @code{fit} function in algebraic notation normally takes four
+arguments, @samp{fit(@var{model}, @var{vars}, @var{params}, @var{data})},
+where @var{model} is the model formula as it would be typed after
address@hidden F '}, @var{vars} is the independent variable or a vector of
+independent variables, @var{params} likewise gives the parameter(s),
+and @var{data} is the data matrix. Note that the length of @var{vars}
+must be equal to the number of rows in @var{data} if @var{model} is
+an equation, or one less than the number of rows if @var{model} is
+a plain formula. (Actually, a name for the dependent variable is
+allowed but will be ignored in the plain-formula case.)
+
+If @var{params} is omitted, the parameters are all variables in
address@hidden except those that appear in @var{vars}. If @var{vars}
+is also omitted, Calc sorts all the variables that appear in
address@hidden alphabetically and uses the higher ones for @var{vars}
+and the lower ones for @var{params}.
+
+Alternatively, @samp{fit(@var{modelvec}, @var{data})} is allowed
+where @var{modelvec} is a 2- or 3-vector describing the model
+and variables, as discussed previously.
+
+If Calc is unable to do the fit, the @code{fit} function is left
+in symbolic form, ordinarily with an explanatory message. The
+message will be ``Model expression is too complex'' if the
+linearizer was unable to put the model into the required form.
+
+The @code{efit} (corresponding to @kbd{H a F}) and @code{xfit}
+(for @kbd{I a F}) functions are completely analogous.
+
address@hidden Interpolation, , Curve Fitting Details, Curve Fitting
address@hidden Polynomial Interpolation
+
address@hidden a p
address@hidden calc-poly-interp
address@hidden polint
+The @kbd{a p} (@code{calc-poly-interp}) address@hidden command does
+a polynomial interpolation at a particular @expr{x} value. It takes
+two arguments from the stack: A data matrix of the sort used by
address@hidden F}, and a single number which represents the desired @expr{x}
+value. Calc effectively does an exact polynomial fit as if by @kbd{a F i},
+then substitutes the @expr{x} value into the result in order to get an
+approximate @expr{y} value based on the fit. (Calc does not actually
+use @kbd{a F i}, however; it uses a direct method which is both more
+efficient and more numerically stable.)
+
+The result of @kbd{a p} is actually a vector of two values: The @expr{y}
+value approximation, and an error measure @expr{dy} that reflects Calc's
+estimation of the probable error of the approximation at that value of
address@hidden If the input @expr{x} is equal to any of the @expr{x} values
+in the data matrix, the output @expr{y} will be the corresponding @expr{y}
+value from the matrix, and the output @expr{dy} will be exactly zero.
+
+A prefix argument of 2 causes @kbd{a p} to take separate x- and
+y-vectors from the stack instead of one data matrix.
+
+If @expr{x} is a vector of numbers, @kbd{a p} will return a matrix of
+interpolated results for each of those @expr{x} values. (The matrix will
+have two columns, the @expr{y} values and the @expr{dy} values.)
+If @expr{x} is a formula instead of a number, the @code{polint} function
+remains in symbolic form; use the @kbd{a "} command to expand it out to
+a formula that describes the fit in symbolic terms.
+
+In all cases, the @kbd{a p} command leaves the data vectors or matrix
+on the stack. Only the @expr{x} value is replaced by the result.
+
address@hidden H a p
address@hidden ratint
+The @kbd{H a p} address@hidden command does a rational function
+interpolation. It is used exactly like @kbd{a p}, except that it
+uses as its model the quotient of two polynomials. If there are
address@hidden data points, the numerator and denominator polynomials will
+each have degree @expr{N/2} (if @expr{N} is odd, the denominator will
+have degree one higher than the numerator).
+
+Rational approximations have the advantage that they can accurately
+describe functions that have poles (points at which the function's value
+goes to infinity, so that the denominator polynomial of the approximation
+goes to zero). If @expr{x} corresponds to a pole of the fitted rational
+function, then the result will be a division by zero. If Infinite mode
+is enabled, the result will be @samp{[uinf, uinf]}.
+
+There is no way to get the actual coefficients of the rational function
+used by @kbd{H a p}. (The algorithm never generates these coefficients
+explicitly, and quotients of polynomials are beyond @address@hidden F}}'s
+capabilities to fit.)
+
address@hidden Summations, Logical Operations, Curve Fitting, Algebra
address@hidden Summations
+
address@hidden
address@hidden Summation of a series
address@hidden a +
address@hidden calc-summation
address@hidden sum
+The @kbd{a +} (@code{calc-summation}) address@hidden command computes
+the sum of a formula over a certain range of index values. The formula
+is taken from the top of the stack; the command prompts for the
+name of the summation index variable, the lower limit of the
+sum (any formula), and the upper limit of the sum. If you
+enter a blank line at any of these prompts, that prompt and
+any later ones are answered by reading additional elements from
+the stack. Thus, @kbd{' k^2 @key{RET} ' k @key{RET} 1 @key{RET} 5 @key{RET} a
+ @key{RET}}
+produces the result 55.
address@hidden
+\turnoffactive
+$$ \sum_{k=1}^5 k^2 = 55 $$
address@hidden tex
+
+The choice of index variable is arbitrary, but it's best not to
+use a variable with a stored value. In particular, while
address@hidden is often a favorite index variable, it should be avoided
+in Calc because @code{i} has the imaginary constant @expr{(0, 1)}
+as a value. If you pressed @kbd{=} on a sum over @code{i}, it would
+be changed to a nonsensical sum over the ``variable'' @expr{(0, 1)}!
+If you really want to use @code{i} as an index variable, use
address@hidden@kbd{s u i @key{RET}}} first to ``unstore'' this variable.
+(@xref{Storing Variables}.)
+
+A numeric prefix argument steps the index by that amount rather
+than by one. Thus @kbd{' a_k @key{RET} C-u -2 a + k @key{RET} 10 @key{RET} 0
@key{RET}}
+yields @samp{a_10 + a_8 + a_6 + a_4 + a_2 + a_0}. A prefix
+argument of plain @kbd{C-u} causes @kbd{a +} to prompt for the
+step value, in which case you can enter any formula or enter
+a blank line to take the step value from the stack. With the
address@hidden prefix, @kbd{a +} can take up to five arguments from
+the stack: The formula, the variable, the lower limit, the
+upper limit, and (at the top of the stack), the step value.
+
+Calc knows how to do certain sums in closed form. For example,
address@hidden(6 k^2, k, 1, n) = @w{2 n^3} + 3 n^2 + n}. In particular,
+this is possible if the formula being summed is polynomial or
+exponential in the index variable. Sums of logarithms are
+transformed into logarithms of products. Sums of trigonometric
+and hyperbolic functions are transformed to sums of exponentials
+and then done in closed form. Also, of course, sums in which the
+lower and upper limits are both numbers can always be evaluated
+just by grinding them out, although Calc will use closed forms
+whenever it can for the sake of efficiency.
+
+The notation for sums in algebraic formulas is
address@hidden(@var{expr}, @var{var}, @var{low}, @var{high}, @var{step})}.
+If @var{step} is omitted, it defaults to one. If @var{high} is
+omitted, @var{low} is actually the upper limit and the lower limit
+is one. If @var{low} is also omitted, the limits are @samp{-inf}
+and @samp{inf}, respectively.
+
+Infinite sums can sometimes be evaluated: @samp{sum(.5^k, k, 1, inf)}
+returns @expr{1}. This is done by evaluating the sum in closed
+form (to @samp{1. - 0.5^n} in this case), then evaluating this
+formula with @code{n} set to @code{inf}. Calc's usual rules
+for ``infinite'' arithmetic can find the answer from there. If
+infinite arithmetic yields a @samp{nan}, or if the sum cannot be
+solved in closed form, Calc leaves the @code{sum} function in
+symbolic form. @xref{Infinities}.
+
+As a special feature, if the limits are infinite (or omitted, as
+described above) but the formula includes vectors subscripted by
+expressions that involve the iteration variable, Calc narrows
+the limits to include only the range of integers which result in
+valid subscripts for the vector. For example, the sum
address@hidden(k [a,b,c,d,e,f,g]_(2k),k)} evaluates to @samp{b + 2 d + 3 f}.
+
+The limits of a sum do not need to be integers. For example,
address@hidden(a_k, k, 0, 2 n, n)} produces @samp{a_0 + a_n + a_(2 n)}.
+Calc computes the number of iterations using the formula
address@hidden + (@var{high} - @var{low}) / @var{step}}, which must,
+after simplification as if by @kbd{a s}, evaluate to an integer.
+
+If the number of iterations according to the above formula does
+not come out to an integer, the sum is invalid and will be left
+in symbolic form. However, closed forms are still supplied, and
+you are on your honor not to misuse the resulting formulas by
+substituting mismatched bounds into them. For example,
address@hidden(k, k, 1, 10, 2)} is invalid, but Calc will go ahead and
+evaluate the closed form solution for the limits 1 and 10 to get
+the rather dubious answer, 29.25.
+
+If the lower limit is greater than the upper limit (assuming a
+positive step size), the result is generally zero. However,
+Calc only guarantees a zero result when the upper limit is
+exactly one step less than the lower limit, i.e., if the number
+of iterations is @mathit{-1}. Thus @samp{sum(f(k), k, n, n-1)} is zero
+but the sum from @samp{n} to @samp{n-2} may report a nonzero value
+if Calc used a closed form solution.
+
+Calc's logical predicates like @expr{a < b} return 1 for ``true''
+and 0 for ``false.'' @xref{Logical Operations}. This can be
+used to advantage for building conditional sums. For example,
address@hidden(prime(k)*k^2, k, 1, 20)} is the sum of the squares of all
+prime numbers from 1 to 20; the @code{prime} predicate returns 1 if
+its argument is prime and 0 otherwise. You can read this expression
+as ``the sum of @expr{k^2}, where @expr{k} is prime.'' Indeed,
address@hidden(prime(k)*k^2, k)} would represent the sum of @emph{all} primes
+squared, since the limits default to plus and minus infinity, but
+there are no such sums that Calc's built-in rules can do in
+closed form.
+
+As another example, @samp{sum((k != k_0) * f(k), k, 1, n)} is the
+sum of @expr{f(k)} for all @expr{k} from 1 to @expr{n}, excluding
+one value @expr{k_0}. Slightly more tricky is the summand
address@hidden(k != k_0) / (k - k_0)}, which is an attempt to describe
+the sum of all @expr{1/(k-k_0)} except at @expr{k = k_0}, where
+this would be a division by zero. But at @expr{k = k_0}, this
+formula works out to the indeterminate form @expr{0 / 0}, which
+Calc will not assume is zero. Better would be to use
address@hidden(k != k_0) ? 1/(k-k_0) : 0}; the @samp{? :} operator does
+an ``if-then-else'' test: This expression says, ``if
address@hidden @math{k \ne k_0},
address@hidden @expr{k != k_0},
+then @expr{1/(k-k_0)}, else zero.'' Now the formula @expr{1/(k-k_0)}
+will not even be evaluated by Calc when @expr{k = k_0}.
+
address@hidden Alternating sums
address@hidden a -
address@hidden calc-alt-summation
address@hidden asum
+The @kbd{a -} (@code{calc-alt-summation}) address@hidden command
+computes an alternating sum. Successive terms of the sequence
+are given alternating signs, with the first term (corresponding
+to the lower index value) being positive. Alternating sums
+are converted to normal sums with an extra term of the form
address@hidden(-1)^(address@hidden)}. This formula is adjusted appropriately
+if the step value is other than one. For example, the Taylor
+series for the sine function is @samp{asum(x^k / k!, k, 1, inf, 2)}.
+(Calc cannot evaluate this infinite series, but it can approximate
+it if you replace @code{inf} with any particular odd number.)
+Calc converts this series to a regular sum with a step of one,
+namely @samp{sum((-1)^k x^(2k+1) / (2k+1)!, k, 0, inf)}.
+
address@hidden Product of a sequence
address@hidden a *
address@hidden calc-product
address@hidden prod
+The @kbd{a *} (@code{calc-product}) address@hidden command is
+the analogous way to take a product of many terms. Calc also knows
+some closed forms for products, such as @samp{prod(k, k, 1, n) = n!}.
+Conditional products can be written @samp{prod(k^prime(k), k, 1, n)}
+or @samp{prod(prime(k) ? k : 1, k, 1, n)}.
+
address@hidden a T
address@hidden calc-tabulate
address@hidden table
+The @kbd{a T} (@code{calc-tabulate}) address@hidden command
+evaluates a formula at a series of iterated index values, just
+like @code{sum} and @code{prod}, but its result is simply a
+vector of the results. For example, @samp{table(a_i, i, 1, 7, 2)}
+produces @samp{[a_1, a_3, a_5, a_7]}.
+
address@hidden Logical Operations, Rewrite Rules, Summations, Algebra
address@hidden Logical Operations
+
address@hidden
+The following commands and algebraic functions return true/false values,
+where 1 represents ``true'' and 0 represents ``false.'' In cases where
+a truth value is required (such as for the condition part of a rewrite
+rule, or as the condition for a @address@hidden [ Z ]}} control structure), any
+nonzero value is accepted to mean ``true.'' (Specifically, anything
+for which @code{dnonzero} returns 1 is ``true,'' and anything for
+which @code{dnonzero} returns 0 or cannot decide is assumed ``false.''
+Note that this means that @address@hidden [ Z ]}} will execute the ``then''
+portion if its condition is provably true, but it will execute the
+``else'' portion for any condition like @expr{a = b} that is not
+provably true, even if it might be true. Algebraic functions that
+have conditions as arguments, like @code{? :} and @code{&&}, remain
+unevaluated if the condition is neither provably true nor provably
+false. @xref{Declarations}.)
+
address@hidden a =
address@hidden calc-equal-to
address@hidden eq
address@hidden =
address@hidden ==
+The @kbd{a =} (@code{calc-equal-to}) command, or @samp{eq(a,b)} function
+(which can also be written @samp{a = b} or @samp{a == b} in an algebraic
+formula) is true if @expr{a} and @expr{b} are equal, either because they
+are identical expressions, or because they are numbers which are
+numerically equal. (Thus the integer 1 is considered equal to the float
+1.0.) If the equality of @expr{a} and @expr{b} cannot be determined,
+the comparison is left in symbolic form. Note that as a command, this
+operation pops two values from the stack and pushes back either a 1 or
+a 0, or a formula @samp{a = b} if the values' equality cannot be determined.
+
+Many Calc commands use @samp{=} formulas to represent @dfn{equations}.
+For example, the @kbd{a S} (@code{calc-solve-for}) command rearranges
+an equation to solve for a given variable. The @kbd{a M}
+(@code{calc-map-equation}) command can be used to apply any
+function to both sides of an equation; for example, @kbd{2 a M *}
+multiplies both sides of the equation by two. Note that just
address@hidden *} would not do the same thing; it would produce the formula
address@hidden (a = b)} which represents 2 if the equality is true or
+zero if not.
+
+The @code{eq} function with more than two arguments (e.g., @kbd{C-u 3 a =}
+or @samp{a = b = c}) tests if all of its arguments are equal. In
+algebraic notation, the @samp{=} operator is unusual in that it is
+neither left- nor right-associative: @samp{a = b = c} is not the
+same as @samp{(a = b) = c} or @samp{a = (b = c)} (which each compare
+one variable with the 1 or 0 that results from comparing two other
+variables).
+
address@hidden a #
address@hidden calc-not-equal-to
address@hidden neq
address@hidden !=
+The @kbd{a #} (@code{calc-not-equal-to}) command, or @samp{neq(a,b)} or
address@hidden != b} function, is true if @expr{a} and @expr{b} are not equal.
+This also works with more than two arguments; @samp{a != b != c != d}
+tests that all four of @expr{a}, @expr{b}, @expr{c}, and @expr{d} are
+distinct numbers.
+
address@hidden a <
address@hidden lt
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden a >
address@hidden
address@hidden @null
address@hidden ignore
address@hidden a [
address@hidden
address@hidden @null
address@hidden ignore
address@hidden a ]
address@hidden calc-less-than
address@hidden calc-greater-than
address@hidden calc-less-equal
address@hidden calc-greater-equal
address@hidden
address@hidden @null
address@hidden ignore
address@hidden gt
address@hidden
address@hidden @null
address@hidden ignore
address@hidden leq
address@hidden
address@hidden @null
address@hidden ignore
address@hidden geq
address@hidden
address@hidden @null
address@hidden ignore
address@hidden <
address@hidden
address@hidden @null
address@hidden ignore
address@hidden >
address@hidden
address@hidden @null
address@hidden ignore
address@hidden <=
address@hidden
address@hidden @null
address@hidden ignore
address@hidden >=
+The @kbd{a <} (@code{calc-less-than}) address@hidden(a,b)} or @samp{a < b}]
+operation is true if @expr{a} is less than @expr{b}. Similar functions
+are @kbd{a >} (@code{calc-greater-than}) address@hidden(a,b)} or @samp{a > b}],
address@hidden [} (@code{calc-less-equal}) address@hidden(a,b)} or @samp{a <=
b}], and
address@hidden ]} (@code{calc-greater-equal}) address@hidden(a,b)} or @samp{a
>= b}].
+
+While the inequality functions like @code{lt} do not accept more
+than two arguments, the syntax @address@hidden <= b < c}} is translated to an
+equivalent expression involving intervals: @samp{b in [a .. c)}.
+(See the description of @code{in} below.) All four combinations
+of @samp{<} and @samp{<=} are allowed, or any of the four combinations
+of @samp{>} and @samp{>=}. Four-argument constructions like
address@hidden < b < c < d}, and mixtures like @address@hidden < b = c}} that
+involve both equalities and inequalities, are not allowed.
+
address@hidden a .
address@hidden calc-remove-equal
address@hidden rmeq
+The @kbd{a .} (@code{calc-remove-equal}) address@hidden command extracts
+the righthand side of the equation or inequality on the top of the
+stack. It also works elementwise on vectors. For example, if
address@hidden = 2.34, y = z / 2]} is on the stack, then @kbd{a .} produces
address@hidden, z / 2]}. As a special case, if the righthand side is a
+variable and the lefthand side is a number (as in @samp{2.34 = x}), then
+Calc keeps the lefthand side instead. Finally, this command works with
+assignments @samp{x := 2.34} as well as equations, always taking the
+righthand side, and for @samp{=>} (evaluates-to) operators, always
+taking the lefthand side.
+
address@hidden a &
address@hidden calc-logical-and
address@hidden land
address@hidden &&
+The @kbd{a &} (@code{calc-logical-and}) address@hidden(a,b)} or @samp{a && b}]
+function is true if both of its arguments are true, i.e., are
+non-zero numbers. In this case, the result will be either @expr{a} or
address@hidden, chosen arbitrarily. If either argument is zero, the result is
+zero. Otherwise, the formula is left in symbolic form.
+
address@hidden a |
address@hidden calc-logical-or
address@hidden lor
address@hidden ||
+The @kbd{a |} (@code{calc-logical-or}) address@hidden(a,b)} or @samp{a || b}]
+function is true if either or both of its arguments are true (nonzero).
+The result is whichever argument was nonzero, choosing arbitrarily if both
+are nonzero. If both @expr{a} and @expr{b} are zero, the result is
+zero.
+
address@hidden a !
address@hidden calc-logical-not
address@hidden lnot
address@hidden !
+The @kbd{a !} (@code{calc-logical-not}) address@hidden(a)} or @samp{!@: a}]
+function is true if @expr{a} is false (zero), or false if @expr{a} is
+true (nonzero). It is left in symbolic form if @expr{a} is not a
+number.
+
address@hidden a :
address@hidden calc-logical-if
address@hidden if
address@hidden
address@hidden ? :
address@hidden ignore
address@hidden ?
address@hidden
address@hidden @null
address@hidden ignore
address@hidden :
address@hidden Arguments, not evaluated
+The @kbd{a :} (@code{calc-logical-if}) address@hidden(a,b,c)} or @samp{a ? b
:@: c}]
+function is equal to either @expr{b} or @expr{c} if @expr{a} is a nonzero
+number or zero, respectively. If @expr{a} is not a number, the test is
+left in symbolic form and neither @expr{b} nor @expr{c} is evaluated in
+any way. In algebraic formulas, this is one of the few Calc functions
+whose arguments are not automatically evaluated when the function itself
+is evaluated. The others are @code{lambda}, @code{quote}, and
address@hidden
+
+One minor surprise to watch out for is that the formula @samp{a?3:4}
+will not work because the @samp{3:4} is parsed as a fraction instead of
+as three separate symbols. Type something like @samp{a ? 3 : 4} or
address@hidden(3):4} instead.
+
+As a special case, if @expr{a} evaluates to a vector, then both @expr{b}
+and @expr{c} are evaluated; the result is a vector of the same length
+as @expr{a} whose elements are chosen from corresponding elements of
address@hidden and @expr{c} according to whether each element of @expr{a}
+is zero or nonzero. Each of @expr{b} and @expr{c} must be either a
+vector of the same length as @expr{a}, or a non-vector which is matched
+with all elements of @expr{a}.
+
address@hidden a @{
address@hidden calc-in-set
address@hidden in
+The @kbd{a @{} (@code{calc-in-set}) address@hidden(a,b)}] function is true if
+the number @expr{a} is in the set of numbers represented by @expr{b}.
+If @expr{b} is an interval form, @expr{a} must be one of the values
+encompassed by the interval. If @expr{b} is a vector, @expr{a} must be
+equal to one of the elements of the vector. (If any vector elements are
+intervals, @expr{a} must be in any of the intervals.) If @expr{b} is a
+plain number, @expr{a} must be numerically equal to @expr{b}.
address@hidden Operations}, for a group of commands that manipulate sets
+of this sort.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden typeof
+The @samp{typeof(a)} function produces an integer or variable which
+characterizes @expr{a}. If @expr{a} is a number, vector, or variable,
+the result will be one of the following numbers:
+
address@hidden
+ 1 Integer
+ 2 Fraction
+ 3 Floating-point number
+ 4 HMS form
+ 5 Rectangular complex number
+ 6 Polar complex number
+ 7 Error form
+ 8 Interval form
+ 9 Modulo form
+10 Date-only form
+11 Date/time form
+12 Infinity (inf, uinf, or nan)
+100 Variable
+101 Vector (but not a matrix)
+102 Matrix
address@hidden example
+
+Otherwise, @expr{a} is a formula, and the result is a variable which
+represents the name of the top-level function call.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden integer
address@hidden
address@hidden
address@hidden ignore
address@hidden real
address@hidden
address@hidden
address@hidden ignore
address@hidden constant
+The @samp{integer(a)} function returns true if @expr{a} is an integer.
+The @samp{real(a)} function
+is true if @expr{a} is a real number, either integer, fraction, or
+float. The @samp{constant(a)} function returns true if @expr{a} is
+any of the objects for which @code{typeof} would produce an integer
+code result except for variables, and provided that the components of
+an object like a vector or error form are themselves constant.
+Note that infinities do not satisfy any of these tests, nor do
+special constants like @code{pi} and @code{e}.
+
address@hidden, for a set of similar functions that recognize
+formulas as well as actual numbers. For example, @samp{dint(floor(x))}
+is true because @samp{floor(x)} is provably integer-valued, but
address@hidden(floor(x))} does not because @samp{floor(x)} is not
+literally an integer constant.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden refers
+The @samp{refers(a,b)} function is true if the variable (or sub-expression)
address@hidden appears in @expr{a}, or false otherwise. Unlike the other
+tests described here, this function returns a definite ``no'' answer
+even if its arguments are still in symbolic form. The only case where
address@hidden will be left unevaluated is if @expr{a} is a plain
+variable (different from @expr{b}).
+
address@hidden
address@hidden
address@hidden ignore
address@hidden negative
+The @samp{negative(a)} function returns true if @expr{a} ``looks'' negative,
+because it is a negative number, because it is of the form @expr{-x},
+or because it is a product or quotient with a term that looks negative.
+This is most useful in rewrite rules. Beware that @samp{negative(a)}
+evaluates to 1 or 0 for @emph{any} argument @expr{a}, so it can only
+be stored in a formula if the default simplifications are turned off
+first with @kbd{m O} (or if it appears in an unevaluated context such
+as a rewrite rule condition).
+
address@hidden
address@hidden
address@hidden ignore
address@hidden variable
+The @samp{variable(a)} function is true if @expr{a} is a variable,
+or false if not. If @expr{a} is a function call, this test is left
+in symbolic form. Built-in variables like @code{pi} and @code{inf}
+are considered variables like any others by this test.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden nonvar
+The @samp{nonvar(a)} function is true if @expr{a} is a non-variable.
+If its argument is a variable it is left unsimplified; it never
+actually returns zero. However, since Calc's condition-testing
+commands consider ``false'' anything not provably true, this is
+often good enough.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden lin
address@hidden
address@hidden
address@hidden ignore
address@hidden linnt
address@hidden
address@hidden
address@hidden ignore
address@hidden islin
address@hidden
address@hidden
address@hidden ignore
address@hidden islinnt
address@hidden Linearity testing
+The functions @code{lin}, @code{linnt}, @code{islin}, and @code{islinnt}
+check if an expression is ``linear,'' i.e., can be written in the form
address@hidden + b x} for some constants @expr{a} and @expr{b}, and some
+variable or subformula @expr{x}. The function @samp{islin(f,x)} checks
+if formula @expr{f} is linear in @expr{x}, returning 1 if so. For
+example, @samp{islin(x,x)}, @samp{islin(-x,x)}, @samp{islin(3,x)}, and
address@hidden(x y / 3 - 2, x)} all return 1. The @samp{lin(f,x)} function
+is similar, except that instead of returning 1 it returns the vector
address@hidden, b, x]}. For the above examples, this vector would be
address@hidden, 1, x]}, @expr{[0, -1, x]}, @expr{[3, 0, x]}, and
address@hidden, y/3, x]}, respectively. Both @code{lin} and @code{islin}
+generally remain unevaluated for expressions which are not linear,
+e.g., @samp{lin(2 x^2, x)} and @samp{lin(sin(x), x)}. The second
+argument can also be a formula; @samp{islin(2 + 3 sin(x), sin(x))}
+returns true.
+
+The @code{linnt} and @code{islinnt} functions perform a similar check,
+but require a ``non-trivial'' linear form, which means that the
address@hidden coefficient must be non-zero. For example, @samp{lin(2,x)}
+returns @expr{[2, 0, x]} and @samp{lin(y,x)} returns @expr{[y, 0, x]},
+but @samp{linnt(2,x)} and @samp{linnt(y,x)} are left unevaluated
+(in other words, these formulas are considered to be only ``trivially''
+linear in @expr{x}).
+
+All four linearity-testing functions allow you to omit the second
+argument, in which case the input may be linear in any non-constant
+formula. Here, the @expr{a=0}, @expr{b=1} case is also considered
+trivial, and only constant values for @expr{a} and @expr{b} are
+recognized. Thus, @samp{lin(2 x y)} returns @expr{[0, 2, x y]},
address@hidden(2 - x y)} returns @expr{[2, -1, x y]}, and @samp{lin(x y)}
+returns @expr{[0, 1, x y]}. The @code{linnt} function would allow the
+first two cases but not the third. Also, neither @code{lin} nor
address@hidden accept plain constants as linear in the one-argument
+case: @samp{islin(2,x)} is true, but @samp{islin(2)} is false.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden istrue
+The @samp{istrue(a)} function returns 1 if @expr{a} is a nonzero
+number or provably nonzero formula, or 0 if @expr{a} is anything else.
+Calls to @code{istrue} can only be manipulated if @kbd{m O} mode is
+used to make sure they are not evaluated prematurely. (Note that
+declarations are used when deciding whether a formula is true;
address@hidden returns 1 when @code{dnonzero} would return 1, and
+it returns 0 when @code{dnonzero} would return 0 or leave itself
+in symbolic form.)
+
address@hidden Rewrite Rules, , Logical Operations, Algebra
address@hidden Rewrite Rules
+
address@hidden
address@hidden Rewrite rules
address@hidden Transformations
address@hidden Pattern matching
address@hidden a r
address@hidden calc-rewrite
address@hidden rewrite
+The @kbd{a r} (@code{calc-rewrite}) address@hidden command makes
+substitutions in a formula according to a specified pattern or patterns
+known as @dfn{rewrite rules}. Whereas @kbd{a b} (@code{calc-substitute})
+matches literally, so that substituting @samp{sin(x)} with @samp{cos(x)}
+matches only the @code{sin} function applied to the variable @code{x},
+rewrite rules match general kinds of formulas; rewriting using the rule
address@hidden(x) := cos(x)} matches @code{sin} of any argument and replaces
+it with @code{cos} of that same argument. The only significance of the
+name @code{x} is that the same name is used on both sides of the rule.
+
+Rewrite rules rearrange formulas already in Calc's memory.
address@hidden Tables}, to read about @dfn{syntax rules}, which are
+similar to algebraic rewrite rules but operate when new algebraic
+entries are being parsed, converting strings of characters into
+Calc formulas.
+
address@hidden
+* Entering Rewrite Rules::
+* Basic Rewrite Rules::
+* Conditional Rewrite Rules::
+* Algebraic Properties of Rewrite Rules::
+* Other Features of Rewrite Rules::
+* Composing Patterns in Rewrite Rules::
+* Nested Formulas with Rewrite Rules::
+* Multi-Phase Rewrite Rules::
+* Selections with Rewrite Rules::
+* Matching Commands::
+* Automatic Rewrites::
+* Debugging Rewrites::
+* Examples of Rewrite Rules::
address@hidden menu
+
address@hidden Entering Rewrite Rules, Basic Rewrite Rules, Rewrite Rules,
Rewrite Rules
address@hidden Entering Rewrite Rules
+
address@hidden
+Rewrite rules normally use the ``assignment'' operator
address@hidden@var{old} := @var{new}}.
+This operator is equivalent to the function call @samp{assign(old, new)}.
+The @code{assign} function is undefined by itself in Calc, so an
+assignment formula such as a rewrite rule will be left alone by ordinary
+Calc commands. But certain commands, like the rewrite system, interpret
+assignments in special ways.
+
+For example, the rule @samp{sin(x)^2 := 1-cos(x)^2} says to replace
+every occurrence of the sine of something, squared, with one minus the
+square of the cosine of that same thing. All by itself as a formula
+on the stack it does nothing, but when given to the @kbd{a r} command
+it turns that command into a sine-squared-to-cosine-squared converter.
+
+To specify a set of rules to be applied all at once, make a vector of
+rules.
+
+When @kbd{a r} prompts you to enter the rewrite rules, you can answer
+in several ways:
+
address@hidden
address@hidden
+With a rule: @kbd{f(x) := g(x) @key{RET}}.
address@hidden
+With a vector of rules: @kbd{[f1(x) := g1(x), f2(x) := g2(x)] @key{RET}}.
+(You can omit the enclosing square brackets if you wish.)
address@hidden
+With the name of a variable that contains the rule or rules vector:
address@hidden @key{RET}}.
address@hidden
+With any formula except a rule, a vector, or a variable name; this
+will be interpreted as the @var{old} half of a rewrite rule,
+and you will be prompted a second time for the @var{new} half:
address@hidden(x) @key{RET} g(x) @key{RET}}.
address@hidden
+With a blank line, in which case the rule, rules vector, or variable
+will be taken from the top of the stack (and the formula to be
+rewritten will come from the second-to-top position).
address@hidden enumerate
+
+If you enter the rules directly (as opposed to using rules stored
+in a variable), those rules will be put into the Trail so that you
+can retrieve them later. @xref{Trail Commands}.
+
+It is most convenient to store rules you use often in a variable and
+invoke them by giving the variable name. The @kbd{s e}
+(@code{calc-edit-variable}) command is an easy way to create or edit a
+rule set stored in a variable. You may also wish to use @kbd{s p}
+(@code{calc-permanent-variable}) to save your rules permanently;
address@hidden on Variables}.
+
+Rewrite rules are compiled into a special internal form for faster
+matching. If you enter a rule set directly it must be recompiled
+every time. If you store the rules in a variable and refer to them
+through that variable, they will be compiled once and saved away
+along with the variable for later reference. This is another good
+reason to store your rules in a variable.
+
+Calc also accepts an obsolete notation for rules, as vectors
address@hidden@var{old}, @var{new}]}. But because it is easily confused with a
+vector of two rules, the use of this notation is no longer recommended.
+
address@hidden Basic Rewrite Rules, Conditional Rewrite Rules, Entering Rewrite
Rules, Rewrite Rules
address@hidden Basic Rewrite Rules
+
address@hidden
+To match a particular formula @expr{x} with a particular rewrite rule
address@hidden@var{old} := @var{new}}, Calc compares the structure of @expr{x}
with
+the structure of @var{old}. Variables that appear in @var{old} are
+treated as @dfn{meta-variables}; the corresponding positions in @expr{x}
+may contain any sub-formulas. For example, the pattern @samp{f(x,y)}
+would match the expression @samp{f(12, a+1)} with the meta-variable
address@hidden corresponding to 12 and with @samp{y} corresponding to
address@hidden However, this pattern would not match @samp{f(12)} or
address@hidden(12, a+1)}, since there is no assignment of the meta-variables
+that will make the pattern match these expressions. Notice that if
+the pattern is a single meta-variable, it will match any expression.
+
+If a given meta-variable appears more than once in @var{old}, the
+corresponding sub-formulas of @expr{x} must be identical. Thus
+the pattern @samp{f(x,x)} would match @samp{f(12, 12)} and
address@hidden(a+1, a+1)} but not @samp{f(12, a+1)} or @samp{f(a+b, b+a)}.
+(@xref{Conditional Rewrite Rules}, for a way to match the latter.)
+
+Things other than variables must match exactly between the pattern
+and the target formula. To match a particular variable exactly, use
+the pseudo-function @samp{quote(v)} in the pattern. For example, the
+pattern @samp{x+quote(y)} matches @samp{x+y}, @samp{2+y}, or
address@hidden(a)+y}.
+
+The special variable names @samp{e}, @samp{pi}, @samp{i}, @samp{phi},
address@hidden, @samp{inf}, @samp{uinf}, and @samp{nan} always match
+literally. Thus the pattern @samp{sin(d + e + f)} acts exactly like
address@hidden(d + quote(e) + f)}.
+
+If the @var{old} pattern is found to match a given formula, that
+formula is replaced by @var{new}, where any occurrences in @var{new}
+of meta-variables from the pattern are replaced with the sub-formulas
+that they matched. Thus, applying the rule @samp{f(x,y) := g(y+x,x)}
+to @samp{f(12, a+1)} would produce @samp{g(a+13, 12)}.
+
+The normal @kbd{a r} command applies rewrite rules over and over
+throughout the target formula until no further changes are possible
+(up to a limit of 100 times). Use @kbd{C-u 1 a r} to make only one
+change at a time.
+
address@hidden Conditional Rewrite Rules, Algebraic Properties of Rewrite
Rules, Basic Rewrite Rules, Rewrite Rules
address@hidden Conditional Rewrite Rules
+
address@hidden
+A rewrite rule can also be @dfn{conditional}, written in the form
address@hidden@var{old} := @var{new} :: @var{cond}}. (There is also the
obsolete
+form @address@hidden, @var{new}, @var{cond}]}.) If a @var{cond} part
+is present in the
+rule, this is an additional condition that must be satisfied before
+the rule is accepted. Once @var{old} has been successfully matched
+to the target expression, @var{cond} is evaluated (with all the
+meta-variables substituted for the values they matched) and simplified
+with @kbd{a s} (@code{calc-simplify}). If the result is a nonzero
+number or any other object known to be nonzero (@pxref{Declarations}),
+the rule is accepted. If the result is zero or if it is a symbolic
+formula that is not known to be nonzero, the rule is rejected.
address@hidden Operations}, for a number of functions that return
+1 or 0 according to the results of various tests.
+
+For example, the formula @samp{n > 0} simplifies to 1 or 0 if @expr{n}
+is replaced by a positive or nonpositive number, respectively (or if
address@hidden has been declared to be positive or nonpositive). Thus,
+the rule @samp{f(x,y) := g(y+x,x) :: x+y > 0} would apply to
address@hidden(0, 4)} but not to @samp{f(-3, 2)} or @samp{f(12, a+1)}
+(assuming no outstanding declarations for @expr{a}). In the case of
address@hidden(-3, 2)}, the condition can be shown not to be satisfied; in
+the case of @samp{f(12, a+1)}, the condition merely cannot be shown
+to be satisfied, but that is enough to reject the rule.
+
+While Calc will use declarations to reason about variables in the
+formula being rewritten, declarations do not apply to meta-variables.
+For example, the rule @samp{f(a) := g(a+1)} will match for any values
+of @samp{a}, such as complex numbers, vectors, or formulas, even if
address@hidden has been declared to be real or scalar. If you want the
+meta-variable @samp{a} to match only literal real numbers, use
address@hidden(a) := g(a+1) :: real(a)}. If you want @samp{a} to match only
+reals and formulas which are provably real, use @samp{dreal(a)} as
+the condition.
+
+The @samp{::} operator is a shorthand for the @code{condition}
+function; @address@hidden := @var{new} :: @var{cond}} is equivalent to
+the formula @samp{condition(assign(@var{old}, @var{new}), @var{cond})}.
+
+If you have several conditions, you can use @samp{... :: c1 :: c2 :: c3}
+or @samp{... :: c1 && c2 && c3}. The two are entirely equivalent.
+
+It is also possible to embed conditions inside the pattern:
address@hidden(x :: x>0, y) := g(y+x, x)}. This is purely a notational
+convenience, though; where a condition appears in a rule has no
+effect on when it is tested. The rewrite-rule compiler automatically
+decides when it is best to test each condition while a rule is being
+matched.
+
+Certain conditions are handled as special cases by the rewrite rule
+system and are tested very efficiently: Where @expr{x} is any
+meta-variable, these conditions are @samp{integer(x)}, @samp{real(x)},
address@hidden(x)}, @samp{negative(x)}, @samp{x >= y} where @expr{y}
+is either a constant or another meta-variable and @samp{>=} may be
+replaced by any of the six relational operators, and @samp{x % a = b}
+where @expr{a} and @expr{b} are constants. Other conditions, like
address@hidden >= y+1} or @samp{dreal(x)}, will be less efficient to check
+since Calc must bring the whole evaluator and simplifier into play.
+
+An interesting property of @samp{::} is that neither of its arguments
+will be touched by Calc's default simplifications. This is important
+because conditions often are expressions that cannot safely be
+evaluated early. For example, the @code{typeof} function never
+remains in symbolic form; entering @samp{typeof(a)} will put the
+number 100 (the type code for variables like @samp{a}) on the stack.
+But putting the condition @samp{... :: typeof(a) = 6} on the stack
+is safe since @samp{::} prevents the @code{typeof} from being
+evaluated until the condition is actually used by the rewrite system.
+
+Since @samp{::} protects its lefthand side, too, you can use a dummy
+condition to protect a rule that must itself not evaluate early.
+For example, it's not safe to put @samp{a(f,x) := apply(f, [x])} on
+the stack because it will immediately evaluate to @samp{a(f,x) := f(x)},
+where the meta-variable-ness of @code{f} on the righthand side has been
+lost. But @samp{a(f,x) := apply(f, [x]) :: 1} is safe, and of course
+the condition @samp{1} is always true (nonzero) so it has no effect on
+the functioning of the rule. (The rewrite compiler will ensure that
+it doesn't even impact the speed of matching the rule.)
+
address@hidden Algebraic Properties of Rewrite Rules, Other Features of Rewrite
Rules, Conditional Rewrite Rules, Rewrite Rules
address@hidden Algebraic Properties of Rewrite Rules
+
address@hidden
+The rewrite mechanism understands the algebraic properties of functions
+like @samp{+} and @samp{*}. In particular, pattern matching takes
+the associativity and commutativity of the following functions into
+account:
+
address@hidden
++ - * = != && || and or xor vint vunion vxor gcd lcm max min beta
address@hidden smallexample
+
+For example, the rewrite rule:
+
address@hidden
+a x + b x := (a + b) x
address@hidden example
+
address@hidden
+will match formulas of the form,
+
address@hidden
+a x + b x, x a + x b, a x + x b, x a + b x
address@hidden example
+
+Rewrites also understand the relationship between the @samp{+} and @samp{-}
+operators. The above rewrite rule will also match the formulas,
+
address@hidden
+a x - b x, x a - x b, a x - x b, x a - b x
address@hidden example
+
address@hidden
+by matching @samp{b} in the pattern to @samp{-b} from the formula.
+
+Applied to a sum of many terms like @samp{r + a x + s + b x + t}, this
+pattern will check all pairs of terms for possible matches. The rewrite
+will take whichever suitable pair it discovers first.
+
+In general, a pattern using an associative operator like @samp{a + b}
+will try @var{2 n} different ways to match a sum of @var{n} terms
+like @samp{x + y + z - w}. First, @samp{a} is matched against each
+of @samp{x}, @samp{y}, @samp{z}, and @samp{-w} in turn, with @samp{b}
+being matched to the remainders @samp{y + z - w}, @samp{x + z - w}, etc.
+If none of these succeed, then @samp{b} is matched against each of the
+four terms with @samp{a} matching the remainder. Half-and-half matches,
+like @samp{(x + y) + (z - w)}, are not tried.
+
+Note that @samp{*} is not commutative when applied to matrices, but
+rewrite rules pretend that it is. If you type @kbd{m v} to enable
+Matrix mode (@pxref{Matrix Mode}), rewrite rules will match @samp{*}
+literally, ignoring its usual commutativity property. (In the
+current implementation, the associativity also vanishes---it is as
+if the pattern had been enclosed in a @code{plain} marker; see below.)
+If you are applying rewrites to formulas with matrices, it's best to
+enable Matrix mode first to prevent algebraically incorrect rewrites
+from occurring.
+
+The pattern @samp{-x} will actually match any expression. For example,
+the rule
+
address@hidden
+f(-x) := -f(x)
address@hidden example
+
address@hidden
+will rewrite @samp{f(a)} to @samp{-f(-a)}. To avoid this, either use
+a @code{plain} marker as described below, or add a @samp{negative(x)}
+condition. The @code{negative} function is true if its argument
+``looks'' negative, for example, because it is a negative number or
+because it is a formula like @samp{-x}. The new rule using this
+condition is:
+
address@hidden
+f(x) := -f(-x) :: negative(x) @r{or, equivalently,}
+f(-x) := -f(x) :: negative(-x)
address@hidden example
+
+In the same way, the pattern @samp{x - y} will match the sum @samp{a + b}
+by matching @samp{y} to @samp{-b}.
+
+The pattern @samp{a b} will also match the formula @samp{x/y} if
address@hidden is a number. Thus the rule @samp{a x + @w{b x} := (a+b) x}
+will also convert @samp{a x + x / 2} to @samp{(a + 0.5) x} (or
address@hidden(a + 1:2) x}, depending on the current fraction mode).
+
+Calc will @emph{not} take other liberties with @samp{*}, @samp{/}, and
address@hidden For example, the pattern @samp{f(a b)} will not match
address@hidden(x^2)}, and @samp{f(a + b)} will not match @samp{f(2 x)}, even
+though conceivably these patterns could match with @samp{a = b = x}.
+Nor will @samp{f(a b)} match @samp{f(x / y)} if @samp{y} is not a
+constant, even though it could be considered to match with @samp{a = x}
+and @samp{b = 1/y}. The reasons are partly for efficiency, and partly
+because while few mathematical operations are substantively different
+for addition and subtraction, often it is preferable to treat the cases
+of multiplication, division, and integer powers separately.
+
+Even more subtle is the rule set
+
address@hidden
+[ f(a) + f(b) := f(a + b), -f(a) := f(-a) ]
address@hidden example
+
address@hidden
+attempting to match @samp{f(x) - f(y)}. You might think that Calc
+will view this subtraction as @samp{f(x) + (-f(y))} and then apply
+the above two rules in turn, but actually this will not work because
+Calc only does this when considering rules for @samp{+} (like the
+first rule in this set). So it will see first that @samp{f(x) + (-f(y))}
+does not match @samp{f(a) + f(b)} for any assignments of the
+meta-variables, and then it will see that @samp{f(x) - f(y)} does
+not match @samp{-f(a)} for any assignment of @samp{a}. Because Calc
+tries only one rule at a time, it will not be able to rewrite
address@hidden(x) - f(y)} with this rule set. An explicit @samp{f(a) - f(b)}
+rule will have to be added.
+
+Another thing patterns will @emph{not} do is break up complex numbers.
+The pattern @samp{myconj(a + @w{b i)} := a - b i} will work for formulas
+involving the special constant @samp{i} (such as @samp{3 - 4 i}), but
+it will not match actual complex numbers like @samp{(3, -4)}. A version
+of the above rule for complex numbers would be
+
address@hidden
+myconj(a) := re(a) - im(a) (0,1) :: im(a) != 0
address@hidden example
+
address@hidden
+(Because the @code{re} and @code{im} functions understand the properties
+of the special constant @samp{i}, this rule will also work for
address@hidden - 4 i}. In fact, this particular rule would probably be better
+without the @samp{im(a) != 0} condition, since if @samp{im(a) = 0} the
+righthand side of the rule will still give the correct answer for the
+conjugate of a real number.)
+
+It is also possible to specify optional arguments in patterns. The rule
+
address@hidden
+opt(a) x + opt(b) (x^opt(c) + opt(d)) := f(a, b, c, d)
address@hidden example
+
address@hidden
+will match the formula
+
address@hidden
+5 (x^2 - 4) + 3 x
address@hidden example
+
address@hidden
+in a fairly straightforward manner, but it will also match reduced
+formulas like
+
address@hidden
+x + x^2, 2(x + 1) - x, x + x
address@hidden example
+
address@hidden
+producing, respectively,
+
address@hidden
+f(1, 1, 2, 0), f(-1, 2, 1, 1), f(1, 1, 1, 0)
address@hidden example
+
+(The latter two formulas can be entered only if default simplifications
+have been turned off with @kbd{m O}.)
+
+The default value for a term of a sum is zero. The default value
+for a part of a product, for a power, or for the denominator of a
+quotient, is one. Also, @samp{-x} matches the pattern @samp{opt(a) b}
+with @samp{a = -1}.
+
+In particular, the distributive-law rule can be refined to
+
address@hidden
+opt(a) x + opt(b) x := (a + b) x
address@hidden example
+
address@hidden
+so that it will convert, e.g., @samp{a x - x}, to @samp{(a - 1) x}.
+
+The pattern @samp{opt(a) + opt(b) x} matches almost any formulas which
+are linear in @samp{x}. You can also use the @code{lin} and @code{islin}
+functions with rewrite conditions to test for this; @pxref{Logical
+Operations}. These functions are not as convenient to use in rewrite
+rules, but they recognize more kinds of formulas as linear:
address@hidden/z} is considered linear with @expr{b = 1/z} by @code{lin},
+but it will not match the above pattern because that pattern calls
+for a multiplication, not a division.
+
+As another example, the obvious rule to replace @samp{sin(x)^2 + cos(x)^2}
+by 1,
+
address@hidden
+sin(x)^2 + cos(x)^2 := 1
address@hidden example
+
address@hidden
+misses many cases because the sine and cosine may both be multiplied by
+an equal factor. Here's a more successful rule:
+
address@hidden
+opt(a) sin(x)^2 + opt(a) cos(x)^2 := a
address@hidden example
+
+Note that this rule will @emph{not} match @samp{sin(x)^2 + 6 cos(x)^2}
+because one @expr{a} would have ``matched'' 1 while the other matched 6.
+
+Calc automatically converts a rule like
+
address@hidden
+f(x-1, x) := g(x)
address@hidden example
+
address@hidden
+into the form
+
address@hidden
+f(temp, x) := g(x) :: temp = x-1
address@hidden example
+
address@hidden
+(where @code{temp} stands for a new, invented meta-variable that
+doesn't actually have a name). This modified rule will successfully
+match @samp{f(6, 7)}, binding @samp{temp} and @samp{x} to 6 and 7,
+respectively, then verifying that they differ by one even though
address@hidden does not superficially look like @samp{x-1}.
+
+However, Calc does not solve equations to interpret a rule. The
+following rule,
+
address@hidden
+f(x-1, x+1) := g(x)
address@hidden example
+
address@hidden
+will not work. That is, it will match @samp{f(a - 1 + b, a + 1 + b)}
+but not @samp{f(6, 8)}. Calc always interprets at least one occurrence
+of a variable by literal matching. If the variable appears ``isolated''
+then Calc is smart enough to use it for literal matching. But in this
+last example, Calc is forced to rewrite the rule to @samp{f(x-1, temp)
+:= g(x) :: temp = x+1} where the @samp{x-1} term must correspond to an
+actual ``something-minus-one'' in the target formula.
+
+A successful way to write this would be @samp{f(x, x+2) := g(x+1)}.
+You could make this resemble the original form more closely by using
address@hidden notation, which is described in the next section:
+
address@hidden
+f(xm1, x+1) := g(x) :: let(x := xm1+1)
address@hidden example
+
+Calc does this rewriting or ``conditionalizing'' for any sub-pattern
+which involves only the functions in the following list, operating
+only on constants and meta-variables which have already been matched
+elsewhere in the pattern. When matching a function call, Calc is
+careful to match arguments which are plain variables before arguments
+which are calls to any of the functions below, so that a pattern like
address@hidden(x-1, x)} can be conditionalized even though the isolated
address@hidden comes after the @samp{x-1}.
+
address@hidden
++ - * / \ % ^ abs sign round rounde roundu trunc floor ceil
+max min re im conj arg
address@hidden smallexample
+
+You can suppress all of the special treatments described in this
+section by surrounding a function call with a @code{plain} marker.
+This marker causes the function call which is its argument to be
+matched literally, without regard to commutativity, associativity,
+negation, or conditionalization. When you use @code{plain}, the
+``deep structure'' of the formula being matched can show through.
+For example,
+
address@hidden
+plain(a - a b) := f(a, b)
address@hidden example
+
address@hidden
+will match only literal subtractions. However, the @code{plain}
+marker does not affect its arguments' arguments. In this case,
+commutativity and associativity is still considered while matching
+the @address@hidden b}} sub-pattern, so the whole pattern will match
address@hidden - y x} as well as @samp{x - x y}. We could go still
+further and use
+
address@hidden
+plain(a - plain(a b)) := f(a, b)
address@hidden example
+
address@hidden
+which would do a completely strict match for the pattern.
+
+By contrast, the @code{quote} marker means that not only the
+function name but also the arguments must be literally the same.
+The above pattern will match @samp{x - x y} but
+
address@hidden
+quote(a - a b) := f(a, b)
address@hidden example
+
address@hidden
+will match only the single formula @samp{a - a b}. Also,
+
address@hidden
+quote(a - quote(a b)) := f(a, b)
address@hidden example
+
address@hidden
+will match only @samp{a - quote(a b)}---probably not the desired
+effect!
+
+A certain amount of algebra is also done when substituting the
+meta-variables on the righthand side of a rule. For example,
+in the rule
+
address@hidden
+a + f(b) := f(a + b)
address@hidden example
+
address@hidden
+matching @samp{f(x) - y} would produce @samp{f((-y) + x)} if
+taken literally, but the rewrite mechanism will simplify the
+righthand side to @samp{f(x - y)} automatically. (Of course,
+the default simplifications would do this anyway, so this
+special simplification is only noticeable if you have turned the
+default simplifications off.) This rewriting is done only when
+a meta-variable expands to a ``negative-looking'' expression.
+If this simplification is not desirable, you can use a @code{plain}
+marker on the righthand side:
+
address@hidden
+a + f(b) := f(plain(a + b))
address@hidden example
+
address@hidden
+In this example, we are still allowing the pattern-matcher to
+use all the algebra it can muster, but the righthand side will
+always simplify to a literal addition like @samp{f((-y) + x)}.
+
address@hidden Other Features of Rewrite Rules, Composing Patterns in Rewrite
Rules, Algebraic Properties of Rewrite Rules, Rewrite Rules
address@hidden Other Features of Rewrite Rules
+
address@hidden
+Certain ``function names'' serve as markers in rewrite rules.
+Here is a complete list of these markers. First are listed the
+markers that work inside a pattern; then come the markers that
+work in the righthand side of a rule.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden import
+One kind of marker, @samp{import(x)}, takes the place of a whole
+rule. Here @expr{x} is the name of a variable containing another
+rule set; those rules are ``spliced into'' the rule set that
+imports them. For example, if @samp{[f(a+b) := f(a) + f(b),
+f(a b) := a f(b) :: real(a)]} is stored in variable @samp{linearF},
+then the rule set @samp{[f(0) := 0, import(linearF)]} will apply
+all three rules. It is possible to modify the imported rules
+slightly: @samp{import(x, v1, x1, v2, x2, @dots{})} imports
+the rule set @expr{x} with all occurrences of
address@hidden @math{v_1},
address@hidden @expr{v1},
+as either a variable name or a function name, replaced with
address@hidden @math{x_1}
address@hidden @expr{x1}
+and so on. (If
address@hidden @math{v_1}
address@hidden @expr{v1}
+is used as a function name, then
address@hidden @math{x_1}
address@hidden @expr{x1}
+must be either a function name itself or a @address@hidden< >}} nameless
+function; @pxref{Specifying Operators}.) For example, @samp{[g(0) := 0,
+import(linearF, f, g)]} applies the linearity rules to the function
address@hidden instead of @samp{f}. Imports can be nested, but the
+import-with-renaming feature may fail to rename sub-imports properly.
+
+The special functions allowed in patterns are:
+
address@hidden @samp
address@hidden quote(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden quote
+This pattern matches exactly @expr{x}; variable names in @expr{x} are
+not interpreted as meta-variables. The only flexibility is that
+numbers are compared for numeric equality, so that the pattern
address@hidden(quote(12))} will match both @samp{f(12)} and @samp{f(12.0)}.
+(Numbers are always treated this way by the rewrite mechanism:
+The rule @samp{f(x,x) := g(x)} will match @samp{f(12, 12.0)}.
+The rewrite may produce either @samp{g(12)} or @samp{g(12.0)}
+as a result in this case.)
+
address@hidden plain(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden plain
+Here @expr{x} must be a function call @samp{f(x1,x2,@dots{})}. This
+pattern matches a call to function @expr{f} with the specified
+argument patterns. No special knowledge of the properties of the
+function @expr{f} is used in this case; @samp{+} is not commutative or
+associative. Unlike @code{quote}, the arguments @samp{x1,x2,@dots{}}
+are treated as patterns. If you wish them to be treated ``plainly''
+as well, you must enclose them with more @code{plain} markers:
address@hidden(plain(@w{-a}) + plain(b c))}.
+
address@hidden opt(x,def)
address@hidden
address@hidden
address@hidden ignore
address@hidden opt
+Here @expr{x} must be a variable name. This must appear as an
+argument to a function or an element of a vector; it specifies that
+the argument or element is optional.
+As an argument to @samp{+}, @samp{-}, @samp{*}, @samp{&&}, or @samp{||},
+or as the second argument to @samp{/} or @samp{^}, the value @var{def}
+may be omitted. The pattern @samp{x + opt(y)} matches a sum by
+binding one summand to @expr{x} and the other to @expr{y}, and it
+matches anything else by binding the whole expression to @expr{x} and
+zero to @expr{y}. The other operators above work similarly.
+
+For general miscellaneous functions, the default value @code{def}
+must be specified. Optional arguments are dropped starting with
+the rightmost one during matching. For example, the pattern
address@hidden(opt(a,0), b, opt(c,b))} will match @samp{f(b)}, @samp{f(a,b)},
+or @samp{f(a,b,c)}. Default values of zero and @expr{b} are
+supplied in this example for the omitted arguments. Note that
+the literal variable @expr{b} will be the default in the latter
+case, @emph{not} the value that matched the meta-variable @expr{b}.
+In other words, the default @var{def} is effectively quoted.
+
address@hidden condition(x,c)
address@hidden
address@hidden
address@hidden ignore
address@hidden condition
address@hidden ::
+This matches the pattern @expr{x}, with the attached condition
address@hidden It is the same as @samp{x :: c}.
+
address@hidden pand(x,y)
address@hidden
address@hidden
address@hidden ignore
address@hidden pand
address@hidden &&&
+This matches anything that matches both pattern @expr{x} and
+pattern @expr{y}. It is the same as @samp{x &&& y}.
address@hidden Patterns in Rewrite Rules}.
+
address@hidden por(x,y)
address@hidden
address@hidden
address@hidden ignore
address@hidden por
address@hidden |||
+This matches anything that matches either pattern @expr{x} or
+pattern @expr{y}. It is the same as @address@hidden ||| y}}.
+
address@hidden pnot(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden pnot
address@hidden !!!
+This matches anything that does not match pattern @expr{x}.
+It is the same as @samp{!!! x}.
+
address@hidden cons(h,t)
address@hidden
address@hidden cons
address@hidden ignore
address@hidden cons (rewrites)
+This matches any vector of one or more elements. The first
+element is matched to @expr{h}; a vector of the remaining
+elements is matched to @expr{t}. Note that vectors of fixed
+length can also be matched as actual vectors: The rule
address@hidden(a,cons(b,[])) := cons(a+b,[])} is equivalent
+to the rule @samp{[a,b] := [a+b]}.
+
address@hidden rcons(t,h)
address@hidden
address@hidden rcons
address@hidden ignore
address@hidden rcons (rewrites)
+This is like @code{cons}, except that the @emph{last} element
+is matched to @expr{h}, with the remaining elements matched
+to @expr{t}.
+
address@hidden apply(f,args)
address@hidden
address@hidden apply
address@hidden ignore
address@hidden apply (rewrites)
+This matches any function call. The name of the function, in
+the form of a variable, is matched to @expr{f}. The arguments
+of the function, as a vector of zero or more objects, are
+matched to @samp{args}. Constants, variables, and vectors
+do @emph{not} match an @code{apply} pattern. For example,
address@hidden(f,x)} matches any function call, @samp{apply(quote(f),x)}
+matches any call to the function @samp{f}, @samp{apply(f,[a,b])}
+matches any function call with exactly two arguments, and
address@hidden(quote(f), cons(a,cons(b,x)))} matches any call
+to the function @samp{f} with two or more arguments. Another
+way to implement the latter, if the rest of the rule does not
+need to refer to the first two arguments of @samp{f} by name,
+would be @samp{apply(quote(f), x :: vlen(x) >= 2)}.
+Here's a more interesting sample use of @code{apply}:
+
address@hidden
+apply(f,[x+n]) := n + apply(f,[x])
+ :: in(f, [floor,ceil,round,trunc]) :: integer(n)
address@hidden example
+
+Note, however, that this will be slower to match than a rule
+set with four separate rules. The reason is that Calc sorts
+the rules of a rule set according to top-level function name;
+if the top-level function is @code{apply}, Calc must try the
+rule for every single formula and sub-formula. If the top-level
+function in the pattern is, say, @code{floor}, then Calc invokes
+the rule only for sub-formulas which are calls to @code{floor}.
+
+Formulas normally written with operators like @code{+} are still
+considered function calls: @code{apply(f,x)} matches @samp{a+b}
+with @samp{f = add}, @samp{x = [a,b]}.
+
+You must use @code{apply} for meta-variables with function names
+on both sides of a rewrite rule: @samp{apply(f, [x]) := f(x+1)}
+is @emph{not} correct, because it rewrites @samp{spam(6)} into
address@hidden(7)}. The righthand side should be @samp{apply(f, [x+1])}.
+Also note that you will have to use No-Simplify mode (@kbd{m O})
+when entering this rule so that the @code{apply} isn't
+evaluated immediately to get the new rule @samp{f(x) := f(x+1)}.
+Or, use @kbd{s e} to enter the rule without going through the stack,
+or enter the rule as @samp{apply(f, [x]) := apply(f, [x+1]) @w{:: 1}}.
address@hidden Rewrite Rules}.
+
address@hidden select(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden select
+This is used for applying rules to formulas with selections;
address@hidden with Rewrite Rules}.
address@hidden table
+
+Special functions for the righthand sides of rules are:
+
address@hidden @samp
address@hidden quote(x)
+The notation @samp{quote(x)} is changed to @samp{x} when the
+righthand side is used. As far as the rewrite rule is concerned,
address@hidden is invisible. However, @code{quote} has the special
+property in Calc that its argument is not evaluated. Thus,
+while it will not work to put the rule @samp{t(a) := typeof(a)}
+on the stack because @samp{typeof(a)} is evaluated immediately
+to produce @samp{t(a) := 100}, you can use @code{quote} to
+protect the righthand side: @samp{t(a) := quote(typeof(a))}.
+(@xref{Conditional Rewrite Rules}, for another trick for
+protecting rules from evaluation.)
+
address@hidden plain(x)
+Special properties of and simplifications for the function call
address@hidden are not used. One interesting case where @code{plain}
+is useful is the rule, @samp{q(x) := quote(x)}, trying to expand a
+shorthand notation for the @code{quote} function. This rule will
+not work as shown; instead of replacing @samp{q(foo)} with
address@hidden(foo)}, it will replace it with @samp{foo}! The correct
+rule would be @samp{q(x) := plain(quote(x))}.
+
address@hidden cons(h,t)
+Where @expr{t} is a vector, this is converted into an expanded
+vector during rewrite processing. Note that @code{cons} is a regular
+Calc function which normally does this anyway; the only way @code{cons}
+is treated specially by rewrites is that @code{cons} on the righthand
+side of a rule will be evaluated even if default simplifications
+have been turned off.
+
address@hidden rcons(t,h)
+Analogous to @code{cons} except putting @expr{h} at the @emph{end} of
+the vector @expr{t}.
+
address@hidden apply(f,args)
+Where @expr{f} is a variable and @var{args} is a vector, this
+is converted to a function call. Once again, note that @code{apply}
+is also a regular Calc function.
+
address@hidden eval(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden eval
+The formula @expr{x} is handled in the usual way, then the
+default simplifications are applied to it even if they have
+been turned off normally. This allows you to treat any function
+similarly to the way @code{cons} and @code{apply} are always
+treated. However, there is a slight difference: @samp{cons(2+3, [])}
+with default simplifications off will be converted to @samp{[2+3]},
+whereas @samp{eval(cons(2+3, []))} will be converted to @samp{[5]}.
+
address@hidden evalsimp(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden evalsimp
+The formula @expr{x} has meta-variables substituted in the usual
+way, then algebraically simplified as if by the @kbd{a s} command.
+
address@hidden evalextsimp(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden evalextsimp
+The formula @expr{x} has meta-variables substituted in the normal
+way, then ``extendedly'' simplified as if by the @kbd{a e} command.
+
address@hidden select(x)
address@hidden with Rewrite Rules}.
address@hidden table
+
+There are also some special functions you can use in conditions.
+
address@hidden @samp
address@hidden let(v := x)
address@hidden
address@hidden
address@hidden ignore
address@hidden let
+The expression @expr{x} is evaluated with meta-variables substituted.
+The @kbd{a s} command's simplifications are @emph{not} applied by
+default, but @expr{x} can include calls to @code{evalsimp} or
address@hidden as described above to invoke higher levels
+of simplification. The
+result of @expr{x} is then bound to the meta-variable @expr{v}. As
+usual, if this meta-variable has already been matched to something
+else the two values must be equal; if the meta-variable is new then
+it is bound to the result of the expression. This variable can then
+appear in later conditions, and on the righthand side of the rule.
+In fact, @expr{v} may be any pattern in which case the result of
+evaluating @expr{x} is matched to that pattern, binding any
+meta-variables that appear in that pattern. Note that @code{let}
+can only appear by itself as a condition, or as one term of an
address@hidden&&} which is a whole condition: It cannot be inside
+an @samp{||} term or otherwise buried.
+
+The alternate, equivalent form @samp{let(v, x)} is also recognized.
+Note that the use of @samp{:=} by @code{let}, while still being
+assignment-like in character, is unrelated to the use of @samp{:=}
+in the main part of a rewrite rule.
+
+As an example, @samp{f(a) := g(ia) :: let(ia := 1/a) :: constant(ia)}
+replaces @samp{f(a)} with @samp{g} of the inverse of @samp{a}, if
+that inverse exists and is constant. For example, if @samp{a} is a
+singular matrix the operation @samp{1/a} is left unsimplified and
address@hidden(ia)} fails, but if @samp{a} is an invertible matrix
+then the rule succeeds. Without @code{let} there would be no way
+to express this rule that didn't have to invert the matrix twice.
+Note that, because the meta-variable @samp{ia} is otherwise unbound
+in this rule, the @code{let} condition itself always ``succeeds''
+because no matter what @samp{1/a} evaluates to, it can successfully
+be bound to @code{ia}.
+
+Here's another example, for integrating cosines of linear
+terms: @samp{myint(cos(y),x) := sin(y)/b :: let([a,b,x] := lin(y,x))}.
+The @code{lin} function returns a 3-vector if its argument is linear,
+or leaves itself unevaluated if not. But an unevaluated @code{lin}
+call will not match the 3-vector on the lefthand side of the @code{let},
+so this @code{let} both verifies that @code{y} is linear, and binds
+the coefficients @code{a} and @code{b} for use elsewhere in the rule.
+(It would have been possible to use @samp{sin(a x + b)/b} for the
+righthand side instead, but using @samp{sin(y)/b} avoids gratuitous
+rearrangement of the argument of the sine.)
+
address@hidden
address@hidden
address@hidden ignore
address@hidden ierf
+Similarly, here is a rule that implements an address@hidden
+function. It uses @code{root} to search for a solution. If
address@hidden succeeds, it will return a vector of two numbers
+where the first number is the desired solution. If no solution
+is found, @code{root} remains in symbolic form. So we use
address@hidden to check that the result was indeed a vector.
+
address@hidden
+ierf(x) := y :: let([y,z] := root(erf(a) = x, a, .5))
address@hidden example
+
address@hidden matches(v,p)
+The meta-variable @var{v}, which must already have been matched
+to something elsewhere in the rule, is compared against pattern
address@hidden Since @code{matches} is a standard Calc function, it
+can appear anywhere in a condition. But if it appears alone or
+as a term of a top-level @samp{&&}, then you get the special
+extra feature that meta-variables which are bound to things
+inside @var{p} can be used elsewhere in the surrounding rewrite
+rule.
+
+The only real difference between @samp{let(p := v)} and
address@hidden(v, p)} is that the former evaluates @samp{v} using
+the default simplifications, while the latter does not.
+
address@hidden remember
address@hidden remember
+This is actually a variable, not a function. If @code{remember}
+appears as a condition in a rule, then when that rule succeeds
+the original expression and rewritten expression are added to the
+front of the rule set that contained the rule. If the rule set
+was not stored in a variable, @code{remember} is ignored. The
+lefthand side is enclosed in @code{quote} in the added rule if it
+contains any variables.
+
+For example, the rule @samp{f(n) := n f(n-1) :: remember} applied
+to @samp{f(7)} will add the rule @samp{f(7) := 7 f(6)} to the front
+of the rule set. The rule set @code{EvalRules} works slightly
+differently: There, the evaluation of @samp{f(6)} will complete before
+the result is added to the rule set, in this case as @samp{f(7) := 5040}.
+Thus @code{remember} is most useful inside @code{EvalRules}.
+
+It is up to you to ensure that the optimization performed by
address@hidden is safe. For example, the rule @samp{foo(n) := n
+:: evalv(eatfoo) > 0 :: remember} is a bad idea (@code{evalv} is
+the function equivalent of the @kbd{=} command); if the variable
address@hidden ever contains 1, rules like @samp{foo(7) := 7} will
+be added to the rule set and will continue to operate even if
address@hidden is later changed to 0.
+
address@hidden remember(c)
address@hidden
address@hidden
address@hidden ignore
address@hidden remember
+Remember the match as described above, but only if condition @expr{c}
+is true. For example, @samp{remember(n % 4 = 0)} in the above factorial
+rule remembers only every fourth result. Note that @samp{remember(1)}
+is equivalent to @samp{remember}, and @samp{remember(0)} has no effect.
address@hidden table
+
address@hidden Composing Patterns in Rewrite Rules, Nested Formulas with
Rewrite Rules, Other Features of Rewrite Rules, Rewrite Rules
address@hidden Composing Patterns in Rewrite Rules
+
address@hidden
+There are three operators, @samp{&&&}, @samp{|||}, and @samp{!!!},
+that combine rewrite patterns to make larger patterns. The
+combinations are ``and,'' ``or,'' and ``not,'' respectively, and
+these operators are the pattern equivalents of @samp{&&}, @samp{||}
+and @samp{!} (which operate on zero-or-nonzero logical values).
+
+Note that @samp{&&&}, @samp{|||}, and @samp{!!!} are left in symbolic
+form by all regular Calc features; they have special meaning only in
+the context of rewrite rule patterns.
+
+The pattern @address@hidden &&& @var{p2}} matches anything that
+matches both @var{p1} and @var{p2}. One especially useful case is
+when one of @var{p1} or @var{p2} is a meta-variable. For example,
+here is a rule that operates on error forms:
+
address@hidden
+f(x &&& a +/- b, x) := g(x)
address@hidden example
+
+This does the same thing, but is arguably simpler than, the rule
+
address@hidden
+f(a +/- b, a +/- b) := g(a +/- b)
address@hidden example
+
address@hidden
address@hidden
address@hidden ignore
address@hidden ends
+Here's another interesting example:
+
address@hidden
+ends(cons(a, x) &&& rcons(y, b)) := [a, b]
address@hidden example
+
address@hidden
+which effectively clips out the middle of a vector leaving just
+the first and last elements. This rule will change a one-element
+vector @samp{[a]} to @samp{[a, a]}. The similar rule
+
address@hidden
+ends(cons(a, rcons(y, b))) := [a, b]
address@hidden example
+
address@hidden
+would do the same thing except that it would fail to match a
+one-element vector.
+
address@hidden
+\bigskip
address@hidden tex
+
+The pattern @address@hidden ||| @var{p2}} matches anything that
+matches either @var{p1} or @var{p2}. Calc first tries matching
+against @var{p1}; if that fails, it goes on to try @var{p2}.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden curve
+A simple example of @samp{|||} is
+
address@hidden
+curve(inf ||| -inf) := 0
address@hidden example
+
address@hidden
+which converts both @samp{curve(inf)} and @samp{curve(-inf)} to zero.
+
+Here is a larger example:
+
address@hidden
+log(a, b) ||| (ln(a) :: let(b := e)) := mylog(a, b)
address@hidden example
+
+This matches both generalized and natural logarithms in a single rule.
+Note that the @samp{::} term must be enclosed in parentheses because
+that operator has lower precedence than @samp{|||} or @samp{:=}.
+
+(In practice this rule would probably include a third alternative,
+omitted here for brevity, to take care of @code{log10}.)
+
+While Calc generally treats interior conditions exactly the same as
+conditions on the outside of a rule, it does guarantee that if all the
+variables in the condition are special names like @code{e}, or already
+bound in the pattern to which the condition is attached (say, if
address@hidden had appeared in this condition), then Calc will process this
+condition right after matching the pattern to the left of the @samp{::}.
+Thus, we know that @samp{b} will be bound to @samp{e} only if the
address@hidden branch of the @samp{|||} was taken.
+
+Note that this rule was careful to bind the same set of meta-variables
+on both sides of the @samp{|||}. Calc does not check this, but if
+you bind a certain meta-variable only in one branch and then use that
+meta-variable elsewhere in the rule, results are unpredictable:
+
address@hidden
+f(a,b) ||| g(b) := h(a,b)
address@hidden example
+
+Here if the pattern matches @samp{g(17)}, Calc makes no promises about
+the value that will be substituted for @samp{a} on the righthand side.
+
address@hidden
+\bigskip
address@hidden tex
+
+The pattern @samp{!!! @var{pat}} matches anything that does not
+match @var{pat}. Any meta-variables that are bound while matching
address@hidden remain unbound outside of @var{pat}.
+
+For example,
+
address@hidden
+f(x &&& !!! a +/- b, !!![]) := g(x)
address@hidden example
+
address@hidden
+converts @code{f} whose first argument is anything @emph{except} an
+error form, and whose second argument is not the empty vector, into
+a similar call to @code{g} (but without the second argument).
+
+If we know that the second argument will be a vector (empty or not),
+then an equivalent rule would be:
+
address@hidden
+f(x, y) := g(x) :: typeof(x) != 7 :: vlen(y) > 0
address@hidden example
+
address@hidden
+where of course 7 is the @code{typeof} code for error forms.
+Another final condition, that works for any kind of @samp{y},
+would be @samp{!istrue(y == [])}. (The @code{istrue} function
+returns an explicit 0 if its argument was left in symbolic form;
+plain @samp{!(y == [])} or @samp{y != []} would not work to replace
address@hidden since these would be left unsimplified, and thus cause
+the rule to fail, if @samp{y} was something like a variable name.)
+
+It is possible for a @samp{!!!} to refer to meta-variables bound
+elsewhere in the pattern. For example,
+
address@hidden
+f(a, !!!a) := g(a)
address@hidden example
+
address@hidden
+matches any call to @code{f} with different arguments, changing
+this to @code{g} with only the first argument.
+
+If a function call is to be matched and one of the argument patterns
+contains a @samp{!!!} somewhere inside it, that argument will be
+matched last. Thus
+
address@hidden
+f(!!!a, a) := g(a)
address@hidden example
+
address@hidden
+will be careful to bind @samp{a} to the second argument of @code{f}
+before testing the first argument. If Calc had tried to match the
+first argument of @code{f} first, the results would have been
+disastrous: since @code{a} was unbound so far, the pattern @samp{a}
+would have matched anything at all, and the pattern @samp{!!!a}
+therefore would @emph{not} have matched anything at all!
+
address@hidden Nested Formulas with Rewrite Rules, Multi-Phase Rewrite Rules,
Composing Patterns in Rewrite Rules, Rewrite Rules
address@hidden Nested Formulas with Rewrite Rules
+
address@hidden
+When @kbd{a r} (@code{calc-rewrite}) is used, it takes an expression from
+the top of the stack and attempts to match any of the specified rules
+to any part of the expression, starting with the whole expression
+and then, if that fails, trying deeper and deeper sub-expressions.
+For each part of the expression, the rules are tried in the order
+they appear in the rules vector. The first rule to match the first
+sub-expression wins; it replaces the matched sub-expression according
+to the @var{new} part of the rule.
+
+Often, the rule set will match and change the formula several times.
+The top-level formula is first matched and substituted repeatedly until
+it no longer matches the pattern; then, sub-formulas are tried, and
+so on. Once every part of the formula has gotten its chance, the
+rewrite mechanism starts over again with the top-level formula
+(in case a substitution of one of its arguments has caused it again
+to match). This continues until no further matches can be made
+anywhere in the formula.
+
+It is possible for a rule set to get into an infinite loop. The
+most obvious case, replacing a formula with itself, is not a problem
+because a rule is not considered to ``succeed'' unless the righthand
+side actually comes out to something different than the original
+formula or sub-formula that was matched. But if you accidentally
+had both @samp{ln(a b) := ln(a) + ln(b)} and the reverse
address@hidden(a) + ln(b) := ln(a b)} in your rule set, Calc would
+run forever switching a formula back and forth between the two
+forms.
+
+To avoid disaster, Calc normally stops after 100 changes have been
+made to the formula. This will be enough for most multiple rewrites,
+but it will keep an endless loop of rewrites from locking up the
+computer forever. (On most systems, you can also type @kbd{C-g} to
+halt any Emacs command prematurely.)
+
+To change this limit, give a positive numeric prefix argument.
+In particular, @kbd{M-1 a r} applies only one rewrite at a time,
+useful when you are first testing your rule (or just if repeated
+rewriting is not what is called for by your application).
+
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden iterations
+You can also put a ``function call'' @samp{iterations(@var{n})}
+in place of a rule anywhere in your rules vector (but usually at
+the top). Then, @var{n} will be used instead of 100 as the default
+number of iterations for this rule set. You can use
address@hidden(inf)} if you want no iteration limit by default.
+A prefix argument will override the @code{iterations} limit in the
+rule set.
+
address@hidden
+[ iterations(1),
+ f(x) := f(x+1) ]
address@hidden example
+
+More precisely, the limit controls the number of ``iterations,''
+where each iteration is a successful matching of a rule pattern whose
+righthand side, after substituting meta-variables and applying the
+default simplifications, is different from the original sub-formula
+that was matched.
+
+A prefix argument of zero sets the limit to infinity. Use with caution!
+
+Given a negative numeric prefix argument, @kbd{a r} will match and
+substitute the top-level expression up to that many times, but
+will not attempt to match the rules to any sub-expressions.
+
+In a formula, @code{rewrite(@var{expr}, @var{rules}, @var{n})}
+does a rewriting operation. Here @var{expr} is the expression
+being rewritten, @var{rules} is the rule, vector of rules, or
+variable containing the rules, and @var{n} is the optional
+iteration limit, which may be a positive integer, a negative
+integer, or @samp{inf} or @samp{-inf}. If @var{n} is omitted
+the @code{iterations} value from the rule set is used; if both
+are omitted, 100 is used.
+
address@hidden Multi-Phase Rewrite Rules, Selections with Rewrite Rules, Nested
Formulas with Rewrite Rules, Rewrite Rules
address@hidden Multi-Phase Rewrite Rules
+
address@hidden
+It is possible to separate a rewrite rule set into several @dfn{phases}.
+During each phase, certain rules will be enabled while certain others
+will be disabled. A @dfn{phase schedule} controls the order in which
+phases occur during the rewriting process.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden phase
address@hidden all
+If a call to the marker function @code{phase} appears in the rules
+vector in place of a rule, all rules following that point will be
+members of the phase(s) identified in the arguments to @code{phase}.
+Phases are given integer numbers. The markers @samp{phase()} and
address@hidden(all)} both mean the following rules belong to all phases;
+this is the default at the start of the rule set.
+
+If you do not explicitly schedule the phases, Calc sorts all phase
+numbers that appear in the rule set and executes the phases in
+ascending order. For example, the rule set
+
address@hidden
address@hidden
+[ f0(x) := g0(x),
+ phase(1),
+ f1(x) := g1(x),
+ phase(2),
+ f2(x) := g2(x),
+ phase(3),
+ f3(x) := g3(x),
+ phase(1,2),
+ f4(x) := g4(x) ]
address@hidden group
address@hidden example
+
address@hidden
+has three phases, 1 through 3. Phase 1 consists of the @code{f0},
address@hidden, and @code{f4} rules (in that order). Phase 2 consists of
address@hidden, @code{f2}, and @code{f4}. Phase 3 consists of @code{f0}
+and @code{f3}.
+
+When Calc rewrites a formula using this rule set, it first rewrites
+the formula using only the phase 1 rules until no further changes are
+possible. Then it switches to the phase 2 rule set and continues
+until no further changes occur, then finally rewrites with phase 3.
+When no more phase 3 rules apply, rewriting finishes. (This is
+assuming @kbd{a r} with a large enough prefix argument to allow the
+rewriting to run to completion; the sequence just described stops
+early if the number of iterations specified in the prefix argument,
+100 by default, is reached.)
+
+During each phase, Calc descends through the nested levels of the
+formula as described previously. (@xref{Nested Formulas with Rewrite
+Rules}.) Rewriting starts at the top of the formula, then works its
+way down to the parts, then goes back to the top and works down again.
+The phase 2 rules do not begin until no phase 1 rules apply anywhere
+in the formula.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden schedule
+A @code{schedule} marker appearing in the rule set (anywhere, but
+conventionally at the top) changes the default schedule of phases.
+In the simplest case, @code{schedule} has a sequence of phase numbers
+for arguments; each phase number is invoked in turn until the
+arguments to @code{schedule} are exhausted. Thus adding
address@hidden(3,2,1)} at the top of the above rule set would
+reverse the order of the phases; @samp{schedule(1,2,3)} would have
+no effect since this is the default schedule; and @samp{schedule(1,2,1,3)}
+would give phase 1 a second chance after phase 2 has completed, before
+moving on to phase 3.
+
+Any argument to @code{schedule} can instead be a vector of phase
+numbers (or even of sub-vectors). Then the sub-sequence of phases
+described by the vector are tried repeatedly until no change occurs
+in any phase in the sequence. For example, @samp{schedule([1, 2], 3)}
+tries phase 1, then phase 2, then, if either phase made any changes
+to the formula, repeats these two phases until they can make no
+further progress. Finally, it goes on to phase 3 for finishing
+touches.
+
+Also, items in @code{schedule} can be variable names as well as
+numbers. A variable name is interpreted as the name of a function
+to call on the whole formula. For example, @samp{schedule(1, simplify)}
+says to apply the phase-1 rules (presumably, all of them), then to
+call @code{simplify} which is the function name equivalent of @kbd{a s}.
+Likewise, @samp{schedule([1, simplify])} says to alternate between
+phase 1 and @kbd{a s} until no further changes occur.
+
+Phases can be used purely to improve efficiency; if it is known that
+a certain group of rules will apply only at the beginning of rewriting,
+and a certain other group will apply only at the end, then rewriting
+will be faster if these groups are identified as separate phases.
+Once the phase 1 rules are done, Calc can put them aside and no longer
+spend any time on them while it works on phase 2.
+
+There are also some problems that can only be solved with several
+rewrite phases. For a real-world example of a multi-phase rule set,
+examine the set @code{FitRules}, which is used by the curve-fitting
+command to convert a model expression to linear form.
address@hidden Fitting Details}. This set is divided into four phases.
+The first phase rewrites certain kinds of expressions to be more
+easily linearizable, but less computationally efficient. After the
+linear components have been picked out, the final phase includes the
+opposite rewrites to put each component back into an efficient form.
+If both sets of rules were included in one big phase, Calc could get
+into an infinite loop going back and forth between the two forms.
+
+Elsewhere in @code{FitRules}, the components are first isolated,
+then recombined where possible to reduce the complexity of the linear
+fit, then finally packaged one component at a time into vectors.
+If the packaging rules were allowed to begin before the recombining
+rules were finished, some components might be put away into vectors
+before they had a chance to recombine. By putting these rules in
+two separate phases, this problem is neatly avoided.
+
address@hidden Selections with Rewrite Rules, Matching Commands, Multi-Phase
Rewrite Rules, Rewrite Rules
address@hidden Selections with Rewrite Rules
+
address@hidden
+If a sub-formula of the current formula is selected (as by @kbd{j s};
address@hidden Subformulas}), the @kbd{a r} (@code{calc-rewrite})
+command applies only to that sub-formula. Together with a negative
+prefix argument, you can use this fact to apply a rewrite to one
+specific part of a formula without affecting any other parts.
+
address@hidden j r
address@hidden calc-rewrite-selection
+The @kbd{j r} (@code{calc-rewrite-selection}) command allows more
+sophisticated operations on selections. This command prompts for
+the rules in the same way as @kbd{a r}, but it then applies those
+rules to the whole formula in question even though a sub-formula
+of it has been selected. However, the selected sub-formula will
+first have been surrounded by a @samp{select( )} function call.
+(Calc's evaluator does not understand the function name @code{select};
+this is only a tag used by the @kbd{j r} command.)
+
+For example, suppose the formula on the stack is @samp{2 (a + b)^2}
+and the sub-formula @samp{a + b} is selected. This formula will
+be rewritten to @samp{2 select(a + b)^2} and then the rewrite
+rules will be applied in the usual way. The rewrite rules can
+include references to @code{select} to tell where in the pattern
+the selected sub-formula should appear.
+
+If there is still exactly one @samp{select( )} function call in
+the formula after rewriting is done, it indicates which part of
+the formula should be selected afterwards. Otherwise, the
+formula will be unselected.
+
+You can make @kbd{j r} act much like @kbd{a r} by enclosing both parts
+of the rewrite rule with @samp{select()}. However, @kbd{j r}
+allows you to use the current selection in more flexible ways.
+Suppose you wished to make a rule which removed the exponent from
+the selected term; the rule @samp{select(a)^x := select(a)} would
+work. In the above example, it would rewrite @samp{2 select(a + b)^2}
+to @samp{2 select(a + b)}. This would then be returned to the
+stack as @samp{2 (a + b)} with the @samp{a + b} selected.
+
+The @kbd{j r} command uses one iteration by default, unlike
address@hidden r} which defaults to 100 iterations. A numeric prefix
+argument affects @kbd{j r} in the same way as @kbd{a r}.
address@hidden Formulas with Rewrite Rules}.
+
+As with other selection commands, @kbd{j r} operates on the stack
+entry that contains the cursor. (If the cursor is on the top-of-stack
address@hidden marker, it works as if the cursor were on the formula
+at stack level 1.)
+
+If you don't specify a set of rules, the rules are taken from the
+top of the stack, just as with @kbd{a r}. In this case, the
+cursor must indicate stack entry 2 or above as the formula to be
+rewritten (otherwise the same formula would be used as both the
+target and the rewrite rules).
+
+If the indicated formula has no selection, the cursor position within
+the formula temporarily selects a sub-formula for the purposes of this
+command. If the cursor is not on any sub-formula (e.g., it is in
+the line-number area to the left of the formula), the @samp{select( )}
+markers are ignored by the rewrite mechanism and the rules are allowed
+to apply anywhere in the formula.
+
+As a special feature, the normal @kbd{a r} command also ignores
address@hidden( )} calls in rewrite rules. For example, if you used the
+above rule @samp{select(a)^x := select(a)} with @kbd{a r}, it would apply
+the rule as if it were @samp{a^x := a}. Thus, you can write general
+purpose rules with @samp{select( )} hints inside them so that they
+will ``do the right thing'' in both @kbd{a r} and @kbd{j r},
+both with and without selections.
+
address@hidden Matching Commands, Automatic Rewrites, Selections with Rewrite
Rules, Rewrite Rules
address@hidden Matching Commands
+
address@hidden
address@hidden a m
address@hidden calc-match
address@hidden match
+The @kbd{a m} (@code{calc-match}) address@hidden function takes a
+vector of formulas and a rewrite-rule-style pattern, and produces
+a vector of all formulas which match the pattern. The command
+prompts you to enter the pattern; as for @kbd{a r}, you can enter
+a single pattern (i.e., a formula with meta-variables), or a
+vector of patterns, or a variable which contains patterns, or
+you can give a blank response in which case the patterns are taken
+from the top of the stack. The pattern set will be compiled once
+and saved if it is stored in a variable. If there are several
+patterns in the set, vector elements are kept if they match any
+of the patterns.
+
+For example, @samp{match(a+b, [x, x+y, x-y, 7, x+y+z])}
+will return @samp{[x+y, x-y, x+y+z]}.
+
+The @code{import} mechanism is not available for pattern sets.
+
+The @kbd{a m} command can also be used to extract all vector elements
+which satisfy any condition: The pattern @samp{x :: x>0} will select
+all the positive vector elements.
+
address@hidden I a m
address@hidden matchnot
+With the Inverse flag address@hidden, this command extracts all
+vector elements which do @emph{not} match the given pattern.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden matches
+There is also a function @samp{matches(@var{x}, @var{p})} which
+evaluates to 1 if expression @var{x} matches pattern @var{p}, or
+to 0 otherwise. This is sometimes useful for including into the
+conditional clauses of other rewrite rules.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden vmatches
+The function @code{vmatches} is just like @code{matches}, except
+that if the match succeeds it returns a vector of assignments to
+the meta-variables instead of the number 1. For example,
address@hidden(f(1,2), f(a,b))} returns @samp{[a := 1, b := 2]}.
+If the match fails, the function returns the number 0.
+
address@hidden Automatic Rewrites, Debugging Rewrites, Matching Commands,
Rewrite Rules
address@hidden Automatic Rewrites
+
address@hidden
address@hidden @code{EvalRules} variable
address@hidden EvalRules
+It is possible to get Calc to apply a set of rewrite rules on all
+results, effectively adding to the built-in set of default
+simplifications. To do this, simply store your rule set in the
+variable @code{EvalRules}. There is a convenient @kbd{s E} command
+for editing @code{EvalRules}; @pxref{Operations on Variables}.
+
+For example, suppose you want @samp{sin(a + b)} to be expanded out
+to @samp{sin(b) cos(a) + cos(b) sin(a)} wherever it appears, and
+similarly for @samp{cos(a + b)}. The corresponding rewrite rule
+set would be,
+
address@hidden
address@hidden
+[ sin(a + b) := cos(a) sin(b) + sin(a) cos(b),
+ cos(a + b) := cos(a) cos(b) - sin(a) sin(b) ]
address@hidden group
address@hidden smallexample
+
+To apply these manually, you could put them in a variable called
address@hidden and then use @kbd{a r trigexp} every time you wanted
+to expand trig functions. But if instead you store them in the
+variable @code{EvalRules}, they will automatically be applied to all
+sines and cosines of sums. Then, with @samp{2 x} and @samp{45} on
+the stack, typing @kbd{+ S} will (assuming Degrees mode) result in
address@hidden sin(2 x) + 0.7071 cos(2 x)} automatically.
+
+As each level of a formula is evaluated, the rules from
address@hidden are applied before the default simplifications.
+Rewriting continues until no further @code{EvalRules} apply.
+Note that this is different from the usual order of application of
+rewrite rules: @code{EvalRules} works from the bottom up, simplifying
+the arguments to a function before the function itself, while @kbd{a r}
+applies rules from the top down.
+
+Because the @code{EvalRules} are tried first, you can use them to
+override the normal behavior of any built-in Calc function.
+
+It is important not to write a rule that will get into an infinite
+loop. For example, the rule set @samp{[f(0) := 1, f(n) := n f(n-1)]}
+appears to be a good definition of a factorial function, but it is
+unsafe. Imagine what happens if @samp{f(2.5)} is simplified. Calc
+will continue to subtract 1 from this argument forever without reaching
+zero. A safer second rule would be @samp{f(n) := n f(n-1) :: n>0}.
+Another dangerous rule is @samp{g(x, y) := g(y, x)}. Rewriting
address@hidden(2, 4)}, this would bounce back and forth between that and
address@hidden(4, 2)} forever. If an infinite loop in @code{EvalRules}
+occurs, Emacs will eventually stop with a ``Computation got stuck
+or ran too long'' message.
+
+Another subtle difference between @code{EvalRules} and regular rewrites
+concerns rules that rewrite a formula into an identical formula. For
+example, @samp{f(n) := f(floor(n))} ``fails to match'' when @expr{n} is
+already an integer. But in @code{EvalRules} this case is detected only
+if the righthand side literally becomes the original formula before any
+further simplification. This means that @samp{f(n) := f(floor(n))} will
+get into an infinite loop if it occurs in @code{EvalRules}. Calc will
+replace @samp{f(6)} with @samp{f(floor(6))}, which is different from
address@hidden(6)}, so it will consider the rule to have matched and will
+continue simplifying that formula; first the argument is simplified
+to get @samp{f(6)}, then the rule matches again to get @samp{f(floor(6))}
+again, ad infinitum. A much safer rule would check its argument first,
+say, with @samp{f(n) := f(floor(n)) :: !dint(n)}.
+
+(What really happens is that the rewrite mechanism substitutes the
+meta-variables in the righthand side of a rule, compares to see if the
+result is the same as the original formula and fails if so, then uses
+the default simplifications to simplify the result and compares again
+(and again fails if the formula has simplified back to its original
+form). The only special wrinkle for the @code{EvalRules} is that the
+same rules will come back into play when the default simplifications
+are used. What Calc wants to do is build @samp{f(floor(6))}, see that
+this is different from the original formula, simplify to @samp{f(6)},
+see that this is the same as the original formula, and thus halt the
+rewriting. But while simplifying, @samp{f(6)} will again trigger
+the same @code{EvalRules} rule and Calc will get into a loop inside
+the rewrite mechanism itself.)
+
+The @code{phase}, @code{schedule}, and @code{iterations} markers do
+not work in @code{EvalRules}. If the rule set is divided into phases,
+only the phase 1 rules are applied, and the schedule is ignored.
+The rules are always repeated as many times as possible.
+
+The @code{EvalRules} are applied to all function calls in a formula,
+but not to numbers (and other number-like objects like error forms),
+nor to vectors or individual variable names. (Though they will apply
+to @emph{components} of vectors and error forms when appropriate.) You
+might try to make a variable @code{phihat} which automatically expands
+to its definition without the need to press @kbd{=} by writing the
+rule @samp{quote(phihat) := (1-sqrt(5))/2}, but unfortunately this rule
+will not work as part of @code{EvalRules}.
+
+Finally, another limitation is that Calc sometimes calls its built-in
+functions directly rather than going through the default simplifications.
+When it does this, @code{EvalRules} will not be able to override those
+functions. For example, when you take the absolute value of the complex
+number @expr{(2, 3)}, Calc computes @samp{sqrt(2*2 + 3*3)} by calling
+the multiplication, addition, and square root functions directly rather
+than applying the default simplifications to this formula. So an
address@hidden rule that (perversely) rewrites @samp{sqrt(13) := 6}
+would not apply. (However, if you put Calc into Symbolic mode so that
address@hidden(13)} will be left in symbolic form by the built-in square
+root function, your rule will be able to apply. But if the complex
+number were @expr{(3,4)}, so that @samp{sqrt(25)} must be calculated,
+then Symbolic mode will not help because @samp{sqrt(25)} can be
+evaluated exactly to 5.)
+
+One subtle restriction that normally only manifests itself with
address@hidden is that while a given rewrite rule is in the process
+of being checked, that same rule cannot be recursively applied. Calc
+effectively removes the rule from its rule set while checking the rule,
+then puts it back once the match succeeds or fails. (The technical
+reason for this is that compiled pattern programs are not reentrant.)
+For example, consider the rule @samp{foo(x) := x :: foo(x/2) > 0}
+attempting to match @samp{foo(8)}. This rule will be inactive while
+the condition @samp{foo(4) > 0} is checked, even though it might be
+an integral part of evaluating that condition. Note that this is not
+a problem for the more usual recursive type of rule, such as
address@hidden(x) := foo(x/2)}, because there the rule has succeeded and
+been reactivated by the time the righthand side is evaluated.
+
+If @code{EvalRules} has no stored value (its default state), or if
+anything but a vector is stored in it, then it is ignored.
+
+Even though Calc's rewrite mechanism is designed to compare rewrite
+rules to formulas as quickly as possible, storing rules in
address@hidden may make Calc run substantially slower. This is
+particularly true of rules where the top-level call is a commonly used
+function, or is not fixed. The rule @samp{f(n) := n f(n-1) :: n>0} will
+only activate the rewrite mechanism for calls to the function @code{f},
+but @samp{lg(n) + lg(m) := lg(n m)} will check every @samp{+} operator.
+
address@hidden
+apply(f, [a*b]) := apply(f, [a]) + apply(f, [b]) :: in(f, [ln, log10])
address@hidden smallexample
+
address@hidden
+may seem more ``efficient'' than two separate rules for @code{ln} and
address@hidden, but actually it is vastly less efficient because rules
+with @code{apply} as the top-level pattern must be tested against
address@hidden function call that is simplified.
+
address@hidden @code{AlgSimpRules} variable
address@hidden AlgSimpRules
+Suppose you want @samp{sin(a + b)} to be expanded out not all the time,
+but only when @kbd{a s} is used to simplify the formula. The variable
address@hidden holds rules for this purpose. The @kbd{a s} command
+will apply @code{EvalRules} and @code{AlgSimpRules} to the formula, as
+well as all of its built-in simplifications.
+
+Most of the special limitations for @code{EvalRules} don't apply to
address@hidden Calc simply does an @kbd{a r AlgSimpRules}
+command with an infinite repeat count as the first step of @kbd{a s}.
+It then applies its own built-in simplifications throughout the
+formula, and then repeats these two steps (along with applying the
+default simplifications) until no further changes are possible.
+
address@hidden @code{ExtSimpRules} variable
address@hidden @code{UnitSimpRules} variable
address@hidden ExtSimpRules
address@hidden UnitSimpRules
+There are also @code{ExtSimpRules} and @code{UnitSimpRules} variables
+that are used by @kbd{a e} and @kbd{u s}, respectively; these commands
+also apply @code{EvalRules} and @code{AlgSimpRules}. The variable
address@hidden contains simplification rules that are used
+only during integration by @kbd{a i}.
+
address@hidden Debugging Rewrites, Examples of Rewrite Rules, Automatic
Rewrites, Rewrite Rules
address@hidden Debugging Rewrites
+
address@hidden
+If a buffer named @samp{*Trace*} exists, the rewrite mechanism will
+record some useful information there as it operates. The original
+formula is written there, as is the result of each successful rewrite,
+and the final result of the rewriting. All phase changes are also
+noted.
+
+Calc always appends to @samp{*Trace*}. You must empty this buffer
+yourself periodically if it is in danger of growing unwieldy.
+
+Note that the rewriting mechanism is substantially slower when the
address@hidden buffer exists, even if the buffer is not visible on
+the screen. Once you are done, you will probably want to kill this
+buffer (with @kbd{C-x k *Trace* @key{RET}}). If you leave it in
+existence and forget about it, all your future rewrite commands will
+be needlessly slow.
+
address@hidden Examples of Rewrite Rules, , Debugging Rewrites, Rewrite Rules
address@hidden Examples of Rewrite Rules
+
address@hidden
+Returning to the example of substituting the pattern
address@hidden(x)^2 + cos(x)^2} with 1, we saw that the rule
address@hidden(a) sin(x)^2 + opt(a) cos(x)^2 := a} does a good job of
+finding suitable cases. Another solution would be to use the rule
address@hidden(x)^2 := 1 - sin(x)^2}, followed by algebraic simplification
+if necessary. This rule will be the most effective way to do the job,
+but at the expense of making some changes that you might not desire.
+
+Another algebraic rewrite rule is @samp{exp(x+y) := exp(x) exp(y)}.
+To make this work with the @address@hidden r}} command so that it can be
+easily targeted to a particular exponential in a large formula,
+you might wish to write the rule as @samp{select(exp(x+y)) :=
+select(exp(x) exp(y))}. The @samp{select} markers will be
+ignored by the regular @kbd{a r} command
+(@pxref{Selections with Rewrite Rules}).
+
+A surprisingly useful rewrite rule is @samp{a/(b-c) := a*(b+c)/(b^2-c^2)}.
+This will simplify the formula whenever @expr{b} and/or @expr{c} can
+be made simpler by squaring. For example, applying this rule to
address@hidden / (sqrt(2) + 3)} yields @samp{6:7 - 2:7 sqrt(2)} (assuming
+Symbolic mode has been enabled to keep the square root from being
+evaluated to a floating-point approximation). This rule is also
+useful when working with symbolic complex numbers, e.g.,
address@hidden(a + b i) / (c + d i)}.
+
+As another example, we could define our own ``triangular numbers'' function
+with the rules @samp{[tri(0) := 0, tri(n) := n + tri(n-1) :: n>0]}. Enter
+this vector and store it in a variable: @address@hidden t} trirules}. Now,
given
+a suitable formula like @samp{tri(5)} on the stack, type @samp{a r trirules}
+to apply these rules repeatedly. After six applications, @kbd{a r} will
+stop with 15 on the stack. Once these rules are debugged, it would probably
+be most useful to add them to @code{EvalRules} so that Calc will evaluate
+the new @code{tri} function automatically. We could then use @kbd{Z K} on
+the keyboard macro @kbd{' tri($) @key{RET}} to make a command that applies
address@hidden to the value on the top of the stack. @xref{Programming}.
+
address@hidden Quaternions
+The following rule set, contributed by
address@hidden Fran\c cois
address@hidden Francois
+Pinard, implements @dfn{quaternions}, a generalization of the concept of
+complex numbers. Quaternions have four components, and are here
+represented by function calls @samp{quat(@var{w}, address@hidden, @var{y},
address@hidden)} with ``real part'' @var{w} and the three ``imaginary'' parts
+collected into a vector. Various arithmetical operations on quaternions
+are supported. To use these rules, either add them to @code{EvalRules},
+or create a command based on @kbd{a r} for simplifying quaternion
+formulas. A convenient way to enter quaternions would be a command
+defined by a keyboard macro containing: @kbd{' quat($$$$, [$$$, $$, $])
address@hidden
+
address@hidden
+[ quat(w, x, y, z) := quat(w, [x, y, z]),
+ quat(w, [0, 0, 0]) := w,
+ abs(quat(w, v)) := hypot(w, v),
+ -quat(w, v) := quat(-w, -v),
+ r + quat(w, v) := quat(r + w, v) :: real(r),
+ r - quat(w, v) := quat(r - w, -v) :: real(r),
+ quat(w1, v1) + quat(w2, v2) := quat(w1 + w2, v1 + v2),
+ r * quat(w, v) := quat(r * w, r * v) :: real(r),
+ plain(quat(w1, v1) * quat(w2, v2))
+ := quat(w1 * w2 - v1 * v2, w1 * v2 + w2 * v1 + cross(v1, v2)),
+ quat(w1, v1) / r := quat(w1 / r, v1 / r) :: real(r),
+ z / quat(w, v) := z * quatinv(quat(w, v)),
+ quatinv(quat(w, v)) := quat(w, -v) / (w^2 + v^2),
+ quatsqr(quat(w, v)) := quat(w^2 - v^2, 2 * w * v),
+ quat(w, v)^k := quatsqr(quat(w, v)^(k / 2))
+ :: integer(k) :: k > 0 :: k % 2 = 0,
+ quat(w, v)^k := quatsqr(quat(w, v)^((k - 1) / 2)) * quat(w, v)
+ :: integer(k) :: k > 2,
+ quat(w, v)^-k := quatinv(quat(w, v)^k) :: integer(k) :: k > 0 ]
address@hidden smallexample
+
+Quaternions, like matrices, have non-commutative multiplication.
+In other words, @expr{q1 * q2 = q2 * q1} is not necessarily true if
address@hidden and @expr{q2} are @code{quat} forms. The @samp{quat*quat}
+rule above uses @code{plain} to prevent Calc from rearranging the
+product. It may also be wise to add the line @samp{[quat(), matrix]}
+to the @code{Decls} matrix, to ensure that Calc's other algebraic
+operations will not rearrange a quaternion product. @xref{Declarations}.
+
+These rules also accept a four-argument @code{quat} form, converting
+it to the preferred form in the first rule. If you would rather see
+results in the four-argument form, just append the two items
address@hidden(2), quat(w, [x, y, z]) := quat(w, x, y, z)} to the end
+of the rule set. (But remember that multi-phase rule sets don't work
+in @code{EvalRules}.)
+
address@hidden Units, Store and Recall, Algebra, Top
address@hidden Operating on Units
+
address@hidden
+One special interpretation of algebraic formulas is as numbers with units.
+For example, the formula @samp{5 m / s^2} can be read ``five meters
+per second squared.'' The commands in this chapter help you
+manipulate units expressions in this form. Units-related commands
+begin with the @kbd{u} prefix key.
+
address@hidden
+* Basic Operations on Units::
+* The Units Table::
+* Predefined Units::
+* User-Defined Units::
address@hidden menu
+
address@hidden Basic Operations on Units, The Units Table, Units, Units
address@hidden Basic Operations on Units
+
address@hidden
+A @dfn{units expression} is a formula which is basically a number
+multiplied and/or divided by one or more @dfn{unit names}, which may
+optionally be raised to integer powers. Actually, the value part need not
+be a number; any product or quotient involving unit names is a units
+expression. Many of the units commands will also accept any formula,
+where the command applies to all units expressions which appear in the
+formula.
+
+A unit name is a variable whose name appears in the @dfn{unit table},
+or a variable whose name is a prefix character like @samp{k} (for ``kilo'')
+or @samp{u} (for ``micro'') followed by a name in the unit table.
+A substantial table of built-in units is provided with Calc;
address@hidden Units}. You can also define your own unit names;
address@hidden Units}.
+
+Note that if the value part of a units expression is exactly @samp{1},
+it will be removed by the Calculator's automatic algebra routines: The
+formula @samp{1 mm} is ``simplified'' to @samp{mm}. This is only a
+display anomaly, however; @samp{mm} will work just fine as a
+representation of one millimeter.
+
+You may find that Algebraic mode (@pxref{Algebraic Entry}) makes working
+with units expressions easier. Otherwise, you will have to remember
+to hit the apostrophe key every time you wish to enter units.
+
address@hidden u s
address@hidden calc-simplify-units
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden usimplify
+The @kbd{u s} (@code{calc-simplify-units}) address@hidden command
+simplifies a units
+expression. It uses @kbd{a s} (@code{calc-simplify}) to simplify the
+expression first as a regular algebraic formula; it then looks for
+features that can be further simplified by converting one object's units
+to be compatible with another's. For example, @samp{5 m + 23 mm} will
+simplify to @samp{5.023 m}. When different but compatible units are
+added, the righthand term's units are converted to match those of the
+lefthand term. @xref{Simplification Modes}, for a way to have this done
+automatically at all times.
+
+Units simplification also handles quotients of two units with the same
+dimensionality, as in @address@hidden in s/L cm}} to @samp{5.08 s/L};
fractional
+powers of unit expressions, as in @samp{sqrt(9 mm^2)} to @samp{3 mm} and
address@hidden(9 acre)} to a quantity in meters; and @code{floor},
address@hidden, @code{round}, @code{rounde}, @code{roundu}, @code{trunc},
address@hidden, @code{frac}, @code{abs}, and @code{clean}
+applied to units expressions, in which case
+the operation in question is applied only to the numeric part of the
+expression. Finally, trigonometric functions of quantities with units
+of angle are evaluated, regardless of the current angular mode.
+
address@hidden u c
address@hidden calc-convert-units
+The @kbd{u c} (@code{calc-convert-units}) command converts a units
+expression to new, compatible units. For example, given the units
+expression @samp{55 mph}, typing @kbd{u c m/s @key{RET}} produces
address@hidden m/s}. If you have previously converted a units expression
+with the same type of units (in this case, distance over time), you will
+be offered the previous choice of new units as a default. Continuing
+the above example, entering the units expression @samp{100 km/hr} and
+typing @kbd{u c @key{RET}} (without specifying new units) produces
address@hidden m/s}.
+
+While many of Calc's conversion factors are exact, some are necessarily
+approximate. If Calc is in fraction mode (@pxref{Fraction Mode}), then
+unit conversions will try to give exact, rational conversions, but it
+isn't always possible. Given @samp{55 mph} in fraction mode, typing
address@hidden c m/s @key{RET}} produces @samp{15367:625 m/s}, for example,
+while typing @kbd{u c au/yr @key{RET}} produces
address@hidden au/yr}.
+
+If the units you request are inconsistent with the original units, the
+number will be converted into your units times whatever ``remainder''
+units are left over. For example, converting @samp{55 mph} into acres
+produces @samp{6.08e-3 acre / m s}. (Recall that multiplication binds
+more strongly than division in Calc formulas, so the units here are
+acres per meter-second.) Remainder units are expressed in terms of
+``fundamental'' units like @samp{m} and @samp{s}, regardless of the
+input units.
+
+One special exception is that if you specify a single unit name, and
+a compatible unit appears somewhere in the units expression, then
+that compatible unit will be converted to the new unit and the
+remaining units in the expression will be left alone. For example,
+given the input @samp{980 cm/s^2}, the command @kbd{u c ms} will
+change the @samp{s} to @samp{ms} to get @samp{9.8e-4 cm/ms^2}.
+The ``remainder unit'' @samp{cm} is left alone rather than being
+changed to the base unit @samp{m}.
+
+You can use explicit unit conversion instead of the @kbd{u s} command
+to gain more control over the units of the result of an expression.
+For example, given @samp{5 m + 23 mm}, you can type @kbd{u c m} or
address@hidden c mm} to express the result in either meters or millimeters.
+(For that matter, you could type @kbd{u c fath} to express the result
+in fathoms, if you preferred!)
+
+In place of a specific set of units, you can also enter one of the
+units system names @code{si}, @code{mks} (equivalent), or @code{cgs}.
+For example, @kbd{u c si @key{RET}} converts the expression into
+International System of Units (SI) base units. Also, @kbd{u c base}
+converts to Calc's base units, which are the same as @code{si} units
+except that @code{base} uses @samp{g} as the fundamental unit of mass
+whereas @code{si} uses @samp{kg}.
+
address@hidden Composite units
+The @kbd{u c} command also accepts @dfn{composite units}, which
+are expressed as the sum of several compatible unit names. For
+example, converting @samp{30.5 in} to units @samp{mi+ft+in} (miles,
+feet, and inches) produces @samp{2 ft + 6.5 in}. Calc first
+sorts the unit names into order of decreasing relative size.
+It then accounts for as much of the input quantity as it can
+using an integer number times the largest unit, then moves on
+to the next smaller unit, and so on. Only the smallest unit
+may have a non-integer amount attached in the result. A few
+standard unit names exist for common combinations, such as
address@hidden for @samp{mi+ft+in}, and @code{tpo} for @samp{ton+lb+oz}.
+Composite units are expanded as if by @kbd{a x}, so that
address@hidden(ft+in)/hr} is first converted to @samp{ft/hr+in/hr}.
+
+If the value on the stack does not contain any units, @kbd{u c} will
+prompt first for the old units which this value should be considered
+to have, then for the new units. Assuming the old and new units you
+give are consistent with each other, the result also will not contain
+any units. For example, @address@hidden c} cm @key{RET} in @key{RET}}
converts the number
+2 on the stack to 5.08.
+
address@hidden u b
address@hidden calc-base-units
+The @kbd{u b} (@code{calc-base-units}) command is shorthand for
address@hidden c base}; it converts the units expression on the top of the
+stack into @code{base} units. If @kbd{u s} does not simplify a
+units expression as far as you would like, try @kbd{u b}.
+
+The @kbd{u c} and @kbd{u b} commands treat temperature units (like
address@hidden and @samp{K}) as relative temperatures. For example,
address@hidden c} converts @samp{10 degC} to @samp{18 degF}: A change of 10
+degrees Celsius corresponds to a change of 18 degrees Fahrenheit.
+
address@hidden u t
address@hidden calc-convert-temperature
address@hidden Temperature conversion
+The @kbd{u t} (@code{calc-convert-temperature}) command converts
+absolute temperatures. The value on the stack must be a simple units
+expression with units of temperature only. This command would convert
address@hidden degC} to @samp{50 degF}, the equivalent temperature on the
+Fahrenheit scale.
+
address@hidden u r
address@hidden calc-remove-units
address@hidden u x
address@hidden calc-extract-units
+The @kbd{u r} (@code{calc-remove-units}) command removes units from the
+formula at the top of the stack. The @kbd{u x}
+(@code{calc-extract-units}) command extracts only the units portion of a
+formula. These commands essentially replace every term of the formula
+that does or doesn't (respectively) look like a unit name by the
+constant 1, then resimplify the formula.
+
address@hidden u a
address@hidden calc-autorange-units
+The @kbd{u a} (@code{calc-autorange-units}) command turns on and off a
+mode in which unit prefixes like @code{k} (``kilo'') are automatically
+applied to keep the numeric part of a units expression in a reasonable
+range. This mode affects @kbd{u s} and all units conversion commands
+except @kbd{u b}. For example, with autoranging on, @samp{12345 Hz}
+will be simplified to @samp{12.345 kHz}. Autoranging is useful for
+some kinds of units (like @code{Hz} and @code{m}), but is probably
+undesirable for non-metric units like @code{ft} and @code{tbsp}.
+(Composite units are more appropriate for those; see above.)
+
+Autoranging always applies the prefix to the leftmost unit name.
+Calc chooses the largest prefix that causes the number to be greater
+than or equal to 1.0. Thus an increasing sequence of adjusted times
+would be @samp{1 ms, 10 ms, 100 ms, 1 s, 10 s, 100 s, 1 ks}.
+Generally the rule of thumb is that the number will be adjusted
+to be in the interval @samp{[1 .. 1000)}, although there are several
+exceptions to this rule. First, if the unit has a power then this
+is not possible; @samp{0.1 s^2} simplifies to @samp{100000 ms^2}.
+Second, the ``centi-'' prefix is allowed to form @code{cm} (centimeters),
+but will not apply to other units. The ``deci-,'' ``deka-,'' and
+``hecto-'' prefixes are never used. Thus the allowable interval is
address@hidden .. 10)} for millimeters and @samp{[1 .. 100)} for centimeters.
+Finally, a prefix will not be added to a unit if the resulting name
+is also the actual name of another unit; @samp{1e-15 t} would normally
+be considered a ``femto-ton,'' but it is written as @samp{1000 at}
+(1000 atto-tons) instead because @code{ft} would be confused with feet.
+
address@hidden The Units Table, Predefined Units, Basic Operations on Units,
Units
address@hidden The Units Table
+
address@hidden
address@hidden u v
address@hidden calc-enter-units-table
+The @kbd{u v} (@code{calc-enter-units-table}) command displays the units table
+in another buffer called @code{*Units Table*}. Each entry in this table
+gives the unit name as it would appear in an expression, the definition
+of the unit in terms of simpler units, and a full name or description of
+the unit. Fundamental units are defined as themselves; these are the
+units produced by the @kbd{u b} command. The fundamental units are
+meters, seconds, grams, kelvins, amperes, candelas, moles, radians,
+and steradians.
+
+The Units Table buffer also displays the Unit Prefix Table. Note that
+two prefixes, ``kilo'' and ``hecto,'' accept either upper- or lower-case
+prefix letters. @samp{Meg} is also accepted as a synonym for the @samp{M}
+prefix. Whenever a unit name can be interpreted as either a built-in name
+or a prefix followed by another built-in name, the former interpretation
+wins. For example, @samp{2 pt} means two pints, not two pico-tons.
+
+The Units Table buffer, once created, is not rebuilt unless you define
+new units. To force the buffer to be rebuilt, give any numeric prefix
+argument to @kbd{u v}.
+
address@hidden u V
address@hidden calc-view-units-table
+The @kbd{u V} (@code{calc-view-units-table}) command is like @kbd{u v} except
+that the cursor is not moved into the Units Table buffer. You can
+type @kbd{u V} again to remove the Units Table from the display. To
+return from the Units Table buffer after a @kbd{u v}, type @kbd{C-x * c}
+again or use the regular Emacs @address@hidden o}} (@code{other-window})
+command. You can also kill the buffer with @kbd{C-x k} if you wish;
+the actual units table is safely stored inside the Calculator.
+
address@hidden u g
address@hidden calc-get-unit-definition
+The @kbd{u g} (@code{calc-get-unit-definition}) command retrieves a unit's
+defining expression and pushes it onto the Calculator stack. For example,
address@hidden g in} will produce the expression @samp{2.54 cm}. This is the
+same definition for the unit that would appear in the Units Table buffer.
+Note that this command works only for actual unit names; @kbd{u g km}
+will report that no such unit exists, for example, because @code{km} is
+really the unit @code{m} with a @code{k} (``kilo'') prefix. To see a
+definition of a unit in terms of base units, it is easier to push the
+unit name on the stack and then reduce it to base units with @kbd{u b}.
+
address@hidden u e
address@hidden calc-explain-units
+The @kbd{u e} (@code{calc-explain-units}) command displays an English
+description of the units of the expression on the stack. For example,
+for the expression @samp{62 km^2 g / s^2 mol K}, the description is
+``Square-Kilometer Gram per (Second-squared Mole Degree-Kelvin).'' This
+command uses the English descriptions that appear in the righthand
+column of the Units Table.
+
address@hidden Predefined Units, User-Defined Units, The Units Table, Units
address@hidden Predefined Units
+
address@hidden
+Since the exact definitions of many kinds of units have evolved over the
+years, and since certain countries sometimes have local differences in
+their definitions, it is a good idea to examine Calc's definition of a
+unit before depending on its exact value. For example, there are three
+different units for gallons, corresponding to the US (@code{gal}),
+Canadian (@code{galC}), and British (@code{galUK}) definitions. Also,
+note that @code{oz} is a standard ounce of mass, @code{ozt} is a Troy
+ounce, and @code{ozfl} is a fluid ounce.
+
+The temperature units corresponding to degrees Kelvin and Centigrade
+(Celsius) are the same in this table, since most units commands treat
+temperatures as being relative. The @code{calc-convert-temperature}
+command has special rules for handling the different absolute magnitudes
+of the various temperature scales.
+
+The unit of volume ``liters'' can be referred to by either the lower-case
address@hidden or the upper-case @code{L}.
+
+The unit @code{A} stands for Amperes; the name @code{Ang} is used
address@hidden
+for \AA ngstroms.
address@hidden tex
address@hidden
+for Angstroms.
address@hidden ifnottex
+
+The unit @code{pt} stands for pints; the name @code{point} stands for
+a typographical point, defined by @samp{72 point = 1 in}. This is
+slightly different than the point defined by the American Typefounder's
+Association in 1886, but the point used by Calc has become standard
+largely due to its use by the PostScript page description language.
+There is also @code{texpt}, which stands for a printer's point as
+defined by the @TeX{} typesetting system: @samp{72.27 texpt = 1 in}.
+Other units used by @TeX{} are available; they are @code{texpc} (a pica),
address@hidden (a ``big point'', equal to a standard point which is larger
+than the point used by @TeX{}), @code{texdd} (a Didot point),
address@hidden (a Cicero) and @code{texsp} (a scaled @TeX{} point,
+all dimensions representable in @TeX{} are multiples of this value).
+
+The unit @code{e} stands for the elementary (electron) unit of charge;
+because algebra command could mistake this for the special constant
address@hidden, Calc provides the alternate unit name @code{ech} which is
+preferable to @code{e}.
+
+The name @code{g} stands for one gram of mass; there is also @code{gf},
+one gram of force. (Likewise for @kbd{lb}, pounds, and @kbd{lbf}.)
+Meanwhile, one address@hidden'' of acceleration is denoted @code{ga}.
+
+The unit @code{ton} is a U.S. ton of @samp{2000 lb}, and @code{t} is
+a metric ton of @samp{1000 kg}.
+
+The names @code{s} (or @code{sec}) and @code{min} refer to units of
+time; @code{arcsec} and @code{arcmin} are units of angle.
+
+Some ``units'' are really physical constants; for example, @code{c}
+represents the speed of light, and @code{h} represents Planck's
+constant. You can use these just like other units: converting
address@hidden c} to @samp{m/s} expresses one-half the speed of light in
+meters per second. You can also use this merely as a handy reference;
+the @kbd{u g} command gets the definition of one of these constants
+in its normal terms, and @kbd{u b} expresses the definition in base
+units.
+
+Two units, @code{pi} and @code{alpha} (the fine structure constant,
+approximately @mathit{1/137}) are dimensionless. The units simplification
+commands simply treat these names as equivalent to their corresponding
+values. However you can, for example, use @kbd{u c} to convert a pure
+number into multiples of the fine structure constant, or @kbd{u b} to
+convert this back into a pure number. (When @kbd{u c} prompts for the
+``old units,'' just enter a blank line to signify that the value
+really is unitless.)
+
address@hidden Describe angular units, luminosity vs. steradians problem.
+
address@hidden User-Defined Units, , Predefined Units, Units
address@hidden User-Defined Units
+
address@hidden
+Calc provides ways to get quick access to your selected ``favorite''
+units, as well as ways to define your own new units.
+
address@hidden u 0-9
address@hidden calc-quick-units
address@hidden Units
address@hidden @code{Units} variable
address@hidden Quick units
+To select your favorite units, store a vector of unit names or
+expressions in the Calc variable @code{Units}. The @kbd{u 1}
+through @kbd{u 9} commands (@code{calc-quick-units}) provide access
+to these units. If the value on the top of the stack is a plain
+number (with no units attached), then @kbd{u 1} gives it the
+specified units. (Basically, it multiplies the number by the
+first item in the @code{Units} vector.) If the number on the
+stack @emph{does} have units, then @kbd{u 1} converts that number
+to the new units. For example, suppose the vector @samp{[in, ft]}
+is stored in @code{Units}. Then @kbd{30 u 1} will create the
+expression @samp{30 in}, and @kbd{u 2} will convert that expression
+to @samp{2.5 ft}.
+
+The @kbd{u 0} command accesses the tenth element of @code{Units}.
+Only ten quick units may be defined at a time. If the @code{Units}
+variable has no stored value (the default), or if its value is not
+a vector, then the quick-units commands will not function. The
address@hidden U} command is a convenient way to edit the @code{Units}
+variable; @pxref{Operations on Variables}.
+
address@hidden u d
address@hidden calc-define-unit
address@hidden User-defined units
+The @kbd{u d} (@code{calc-define-unit}) command records the units
+expression on the top of the stack as the definition for a new,
+user-defined unit. For example, putting @samp{16.5 ft} on the stack and
+typing @kbd{u d rod} defines the new unit @samp{rod} to be equivalent to
+16.5 feet. The unit conversion and simplification commands will now
+treat @code{rod} just like any other unit of length. You will also be
+prompted for an optional English description of the unit, which will
+appear in the Units Table.
+
address@hidden u u
address@hidden calc-undefine-unit
+The @kbd{u u} (@code{calc-undefine-unit}) command removes a user-defined
+unit. It is not possible to remove one of the predefined units,
+however.
+
+If you define a unit with an existing unit name, your new definition
+will replace the original definition of that unit. If the unit was a
+predefined unit, the old definition will not be replaced, only
+``shadowed.'' The built-in definition will reappear if you later use
address@hidden u} to remove the shadowing definition.
+
+To create a new fundamental unit, use either 1 or the unit name itself
+as the defining expression. Otherwise the expression can involve any
+other units that you like (except for composite units like @samp{mfi}).
+You can create a new composite unit with a sum of other units as the
+defining expression. The next unit operation like @kbd{u c} or @kbd{u v}
+will rebuild the internal unit table incorporating your modifications.
+Note that erroneous definitions (such as two units defined in terms of
+each other) will not be detected until the unit table is next rebuilt;
address@hidden v} is a convenient way to force this to happen.
+
+Temperature units are treated specially inside the Calculator; it is not
+possible to create user-defined temperature units.
+
address@hidden u p
address@hidden calc-permanent-units
address@hidden Calc init file, user-defined units
+The @kbd{u p} (@code{calc-permanent-units}) command stores the user-defined
+units in your Calc init file (the file given by the variable
address@hidden, typically @file{~/.calc.el}), so that the
+units will still be available in subsequent Emacs sessions. If there
+was already a set of user-defined units in your Calc init file, it
+is replaced by the new set. (@xref{General Mode Commands}, for a way to
+tell Calc to use a different file for the Calc init file.)
+
address@hidden Store and Recall, Graphics, Units, Top
address@hidden Storing and Recalling
+
address@hidden
+Calculator variables are really just Lisp variables that contain numbers
+or formulas in a form that Calc can understand. The commands in this
+section allow you to manipulate variables conveniently. Commands related
+to variables use the @kbd{s} prefix key.
+
address@hidden
+* Storing Variables::
+* Recalling Variables::
+* Operations on Variables::
+* Let Command::
+* Evaluates-To Operator::
address@hidden menu
+
address@hidden Storing Variables, Recalling Variables, Store and Recall, Store
and Recall
address@hidden Storing Variables
+
address@hidden
address@hidden s s
address@hidden calc-store
address@hidden Storing variables
address@hidden Quick variables
address@hidden q0
address@hidden q9
+The @kbd{s s} (@code{calc-store}) command stores the value at the top of
+the stack into a specified variable. It prompts you to enter the
+name of the variable. If you press a single digit, the value is stored
+immediately in one of the ``quick'' variables @code{q0} through
address@hidden Or you can enter any variable name.
+
address@hidden s t
address@hidden calc-store-into
+The @kbd{s s} command leaves the stored value on the stack. There is
+also an @kbd{s t} (@code{calc-store-into}) command, which removes a
+value from the stack and stores it in a variable.
+
+If the top of stack value is an equation @samp{a = 7} or assignment
address@hidden := 7} with a variable on the lefthand side, then Calc will
+assign that variable with that value by default, i.e., if you type
address@hidden s @key{RET}} or @kbd{s t @key{RET}}. In this example, the
+value 7 would be stored in the variable @samp{a}. (If you do type
+a variable name at the prompt, the top-of-stack value is stored in
+its entirety, even if it is an equation: @samp{s s b @key{RET}}
+with @samp{a := 7} on the stack stores @samp{a := 7} in @code{b}.)
+
+In fact, the top of stack value can be a vector of equations or
+assignments with different variables on their lefthand sides; the
+default will be to store all the variables with their corresponding
+righthand sides simultaneously.
+
+It is also possible to type an equation or assignment directly at
+the prompt for the @kbd{s s} or @kbd{s t} command: @kbd{s s foo = 7}.
+In this case the expression to the right of the @kbd{=} or @kbd{:=}
+symbol is evaluated as if by the @kbd{=} command, and that value is
+stored in the variable. No value is taken from the stack; @kbd{s s}
+and @kbd{s t} are equivalent when used in this way.
+
address@hidden s 0-9
address@hidden t 0-9
+The prefix keys @kbd{s} and @kbd{t} may be followed immediately by a
+digit; @kbd{s 9} is equivalent to @kbd{s s 9}, and @kbd{t 9} is
+equivalent to @kbd{s t 9}. (The @kbd{t} prefix is otherwise used
+for trail and time/date commands.)
+
address@hidden s +
address@hidden s -
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden s *
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s /
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s ^
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s |
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s n
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s &
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s [
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s ]
address@hidden calc-store-plus
address@hidden calc-store-minus
address@hidden calc-store-times
address@hidden calc-store-div
address@hidden calc-store-power
address@hidden calc-store-concat
address@hidden calc-store-neg
address@hidden calc-store-inv
address@hidden calc-store-decr
address@hidden calc-store-incr
+There are also several ``arithmetic store'' commands. For example,
address@hidden +} removes a value from the stack and adds it to the specified
+variable. The other arithmetic stores are @kbd{s -}, @kbd{s *}, @kbd{s /},
address@hidden ^}, and @address@hidden |}} (vector concatenation), plus @kbd{s
n} and
address@hidden &} which negate or invert the value in a variable, and
@address@hidden [}}
+and @kbd{s ]} which decrease or increase a variable by one.
+
+All the arithmetic stores accept the Inverse prefix to reverse the
+order of the operands. If @expr{v} represents the contents of the
+variable, and @expr{a} is the value drawn from the stack, then regular
address@hidden@kbd{s -}} assigns
address@hidden @math{v \coloneq v - a},
address@hidden @expr{v := v - a},
+but @kbd{I s -} assigns
address@hidden @math{v \coloneq a - v}.
address@hidden @expr{v := a - v}.
+While @kbd{I s *} might seem pointless, it is
+useful if matrix multiplication is involved. Actually, all the
+arithmetic stores use formulas designed to behave usefully both
+forwards and backwards:
+
address@hidden
address@hidden
+s + v := v + a v := a + v
+s - v := v - a v := a - v
+s * v := v * a v := a * v
+s / v := v / a v := a / v
+s ^ v := v ^ a v := a ^ v
+s | v := v | a v := a | v
+s n v := v / (-1) v := (-1) / v
+s & v := v ^ (-1) v := (-1) ^ v
+s [ v := v - 1 v := 1 - v
+s ] v := v - (-1) v := (-1) - v
address@hidden group
address@hidden example
+
+In the last four cases, a numeric prefix argument will be used in
+place of the number one. (For example, @kbd{M-2 s ]} increases
+a variable by 2, and @kbd{M-2 I s ]} replaces a variable by
+minus-two minus the variable.
+
+The first six arithmetic stores can also be typed @kbd{s t +}, @kbd{s t -},
+etc. The commands @kbd{s s +}, @kbd{s s -}, and so on are analogous
+arithmetic stores that don't remove the value @expr{a} from the stack.
+
+All arithmetic stores report the new value of the variable in the
+Trail for your information. They signal an error if the variable
+previously had no stored value. If default simplifications have been
+turned off, the arithmetic stores temporarily turn them on for numeric
+arguments only (i.e., they temporarily do an @kbd{m N} command).
address@hidden Modes}. Large vectors put in the trail by
+these commands always use abbreviated (@kbd{t .}) mode.
+
address@hidden s m
address@hidden calc-store-map
+The @kbd{s m} command is a general way to adjust a variable's value
+using any Calc function. It is a ``mapping'' command analogous to
address@hidden M}, @kbd{V R}, etc. @xref{Reducing and Mapping}, to see
+how to specify a function for a mapping command. Basically,
+all you do is type the Calc command key that would invoke that
+function normally. For example, @kbd{s m n} applies the @kbd{n}
+key to negate the contents of the variable, so @kbd{s m n} is
+equivalent to @kbd{s n}. Also, @kbd{s m Q} takes the square root
+of the value stored in a variable, @kbd{s m v v} uses @kbd{v v} to
+reverse the vector stored in the variable, and @kbd{s m H I S}
+takes the hyperbolic arcsine of the variable contents.
+
+If the mapping function takes two or more arguments, the additional
+arguments are taken from the stack; the old value of the variable
+is provided as the first argument. Thus @kbd{s m -} with @expr{a}
+on the stack computes @expr{v - a}, just like @kbd{s -}. With the
+Inverse prefix, the variable's original value becomes the @emph{last}
+argument instead of the first. Thus @kbd{I s m -} is also
+equivalent to @kbd{I s -}.
+
address@hidden s x
address@hidden calc-store-exchange
+The @kbd{s x} (@code{calc-store-exchange}) command exchanges the value
+of a variable with the value on the top of the stack. Naturally, the
+variable must already have a stored value for this to work.
+
+You can type an equation or assignment at the @kbd{s x} prompt. The
+command @kbd{s x a=6} takes no values from the stack; instead, it
+pushes the old value of @samp{a} on the stack and stores @samp{a = 6}.
+
address@hidden s u
address@hidden calc-unstore
address@hidden Void variables
address@hidden Un-storing variables
+Until you store something in them, most variables are ``void,'' that is,
+they contain no value at all. If they appear in an algebraic formula
+they will be left alone even if you press @kbd{=} (@code{calc-evaluate}).
+The @kbd{s u} (@code{calc-unstore}) command returns a variable to the
+void state.
+
address@hidden s c
address@hidden calc-copy-variable
+The @kbd{s c} (@code{calc-copy-variable}) command copies the stored
+value of one variable to another. One way it differs from a simple
address@hidden r} followed by an @kbd{s t} (aside from saving keystrokes) is
+that the value never goes on the stack and thus is never rounded,
+evaluated, or simplified in any way; it is not even rounded down to the
+current precision.
+
+The only variables with predefined values are the ``special constants''
address@hidden, @code{e}, @code{i}, @code{phi}, and @code{gamma}. You are free
+to unstore these variables or to store new values into them if you like,
+although some of the algebraic-manipulation functions may assume these
+variables represent their standard values. Calc displays a warning if
+you change the value of one of these variables, or of one of the other
+special variables @code{inf}, @code{uinf}, and @code{nan} (which are
+normally void).
+
+Note that @code{pi} doesn't actually have 3.14159265359 stored in it,
+but rather a special magic value that evaluates to @cpi{} at the current
+precision. Likewise @code{e}, @code{i}, and @code{phi} evaluate
+according to the current precision or polar mode. If you recall a value
+from @code{pi} and store it back, this magic property will be lost. The
+magic property is preserved, however, when a variable is copied with
address@hidden c}.
+
address@hidden s k
address@hidden calc-copy-special-constant
+If one of the ``special constants'' is redefined (or undefined) so that
+it no longer has its magic property, the property can be restored with
address@hidden k} (@code{calc-copy-special-constant}). This command will prompt
+for a special constant and a variable to store it in, and so a special
+constant can be stored in any variable. Here, the special constant that
+you enter doesn't depend on the value of the corresponding variable;
address@hidden will represent address@hidden regardless of what is currently
+stored in the Calc variable @code{pi}. If one of the other special
+variables, @code{inf}, @code{uinf} or @code{nan}, is given a value, its
+original behavior can be restored by voiding it with @kbd{s u}.
+
address@hidden Recalling Variables, Operations on Variables, Storing Variables,
Store and Recall
address@hidden Recalling Variables
+
address@hidden
address@hidden s r
address@hidden calc-recall
address@hidden Recalling variables
+The most straightforward way to extract the stored value from a variable
+is to use the @kbd{s r} (@code{calc-recall}) command. This command prompts
+for a variable name (similarly to @code{calc-store}), looks up the value
+of the specified variable, and pushes that value onto the stack. It is
+an error to try to recall a void variable.
+
+It is also possible to recall the value from a variable by evaluating a
+formula containing that variable. For example, @kbd{' a @key{RET} =} is
+the same as @kbd{s r a @key{RET}} except that if the variable is void, the
+former will simply leave the formula @samp{a} on the stack whereas the
+latter will produce an error message.
+
address@hidden r 0-9
+The @kbd{r} prefix may be followed by a digit, so that @kbd{r 9} is
+equivalent to @kbd{s r 9}. (The @kbd{r} prefix is otherwise unused
+in the current version of Calc.)
+
address@hidden Operations on Variables, Let Command, Recalling Variables, Store
and Recall
address@hidden Other Operations on Variables
+
address@hidden
address@hidden s e
address@hidden calc-edit-variable
+The @kbd{s e} (@code{calc-edit-variable}) command edits the stored
+value of a variable without ever putting that value on the stack
+or simplifying or evaluating the value. It prompts for the name of
+the variable to edit. If the variable has no stored value, the
+editing buffer will start out empty. If the editing buffer is
+empty when you press @kbd{C-c C-c} to finish, the variable will
+be made void. @xref{Editing Stack Entries}, for a general
+description of editing.
+
+The @kbd{s e} command is especially useful for creating and editing
+rewrite rules which are stored in variables. Sometimes these rules
+contain formulas which must not be evaluated until the rules are
+actually used. (For example, they may refer to @samp{deriv(x,y)},
+where @code{x} will someday become some expression involving @code{y};
+if you let Calc evaluate the rule while you are defining it, Calc will
+replace @samp{deriv(x,y)} with 0 because the formula @code{x} does
+not itself refer to @code{y}.) By contrast, recalling the variable,
+editing with @kbd{`}, and storing will evaluate the variable's value
+as a side effect of putting the value on the stack.
+
address@hidden s A
address@hidden s D
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden s E
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s F
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s G
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s H
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s I
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s L
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s P
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s R
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s T
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s U
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s X
address@hidden calc-store-AlgSimpRules
address@hidden calc-store-Decls
address@hidden calc-store-EvalRules
address@hidden calc-store-FitRules
address@hidden calc-store-GenCount
address@hidden calc-store-Holidays
address@hidden calc-store-IntegLimit
address@hidden calc-store-LineStyles
address@hidden calc-store-PointStyles
address@hidden calc-store-PlotRejects
address@hidden calc-store-TimeZone
address@hidden calc-store-Units
address@hidden calc-store-ExtSimpRules
+There are several special-purpose variable-editing commands that
+use the @kbd{s} prefix followed by a shifted letter:
+
address@hidden @kbd
address@hidden s A
+Edit @code{AlgSimpRules}. @xref{Algebraic Simplifications}.
address@hidden s D
+Edit @code{Decls}. @xref{Declarations}.
address@hidden s E
+Edit @code{EvalRules}. @xref{Default Simplifications}.
address@hidden s F
+Edit @code{FitRules}. @xref{Curve Fitting}.
address@hidden s G
+Edit @code{GenCount}. @xref{Solving Equations}.
address@hidden s H
+Edit @code{Holidays}. @xref{Business Days}.
address@hidden s I
+Edit @code{IntegLimit}. @xref{Calculus}.
address@hidden s L
+Edit @code{LineStyles}. @xref{Graphics}.
address@hidden s P
+Edit @code{PointStyles}. @xref{Graphics}.
address@hidden s R
+Edit @code{PlotRejects}. @xref{Graphics}.
address@hidden s T
+Edit @code{TimeZone}. @xref{Time Zones}.
address@hidden s U
+Edit @code{Units}. @xref{User-Defined Units}.
address@hidden s X
+Edit @code{ExtSimpRules}. @xref{Unsafe Simplifications}.
address@hidden table
+
+These commands are just versions of @kbd{s e} that use fixed variable
+names rather than prompting for the variable name.
+
address@hidden s p
address@hidden calc-permanent-variable
address@hidden Storing variables
address@hidden Permanent variables
address@hidden Calc init file, variables
+The @kbd{s p} (@code{calc-permanent-variable}) command saves a
+variable's value permanently in your Calc init file (the file given by
+the variable @code{calc-settings-file}, typically @file{~/.calc.el}), so
+that its value will still be available in future Emacs sessions. You
+can re-execute @address@hidden p}} later on to update the saved value, but the
+only way to remove a saved variable is to edit your calc init file
+by hand. (@xref{General Mode Commands}, for a way to tell Calc to
+use a different file for the Calc init file.)
+
+If you do not specify the name of a variable to save (i.e.,
address@hidden p @key{RET}}), all Calc variables with defined values
+are saved except for the special constants @code{pi}, @code{e},
address@hidden, @code{phi}, and @code{gamma}; the variables @code{TimeZone}
+and @code{PlotRejects};
address@hidden, @code{DistribRules}, and other built-in rewrite
+rules; and @address@hidden variables generated
+by the graphics commands. (You can still save these variables by
+explicitly naming them in an @kbd{s p} command.)
+
address@hidden s i
address@hidden calc-insert-variables
+The @kbd{s i} (@code{calc-insert-variables}) command writes
+the values of all Calc variables into a specified buffer.
+The variables are written with the prefix @code{var-} in the form of
+Lisp @code{setq} commands
+which store the values in string form. You can place these commands
+in your Calc init file (or @file{.emacs}) if you wish, though in this case it
+would be easier to use @kbd{s p @key{RET}}. (Note that @kbd{s i}
+omits the same set of variables as @address@hidden p @key{RET}}}; the
difference
+is that @kbd{s i} will store the variables in any buffer, and it also
+stores in a more human-readable format.)
+
address@hidden Let Command, Evaluates-To Operator, Operations on Variables,
Store and Recall
address@hidden The Let Command
+
address@hidden
address@hidden s l
address@hidden calc-let
address@hidden Variables, temporary assignment
address@hidden Temporary assignment to variables
+If you have an expression like @samp{a+b^2} on the stack and you wish to
+compute its value where @expr{b=3}, you can simply store 3 in @expr{b} and
+then press @kbd{=} to reevaluate the formula. This has the side-effect
+of leaving the stored value of 3 in @expr{b} for future operations.
+
+The @kbd{s l} (@code{calc-let}) command evaluates a formula under a
address@hidden assignment of a variable. It stores the value on the
+top of the stack into the specified variable, then evaluates the
+second-to-top stack entry, then restores the original value (or lack of one)
+in the variable. Thus after @kbd{'@w{ }a+b^2 @key{RET} 3 s l b @key{RET}},
+the stack will contain the formula @samp{a + 9}. The subsequent command
address@hidden@w{5 s l a} @key{RET}} will replace this formula with the number
14.
+The variables @samp{a} and @samp{b} are not permanently affected in any way
+by these commands.
+
+The value on the top of the stack may be an equation or assignment, or
+a vector of equations or assignments, in which case the default will be
+analogous to the case of @kbd{s t @key{RET}}. @xref{Storing Variables}.
+
+Also, you can answer the variable-name prompt with an equation or
+assignment: @kbd{s l b=3 @key{RET}} is the same as storing 3 on the stack
+and typing @kbd{s l b @key{RET}}.
+
+The @kbd{a b} (@code{calc-substitute}) command is another way to substitute
+a variable with a value in a formula. It does an actual substitution
+rather than temporarily assigning the variable and evaluating. For
+example, letting @expr{n=2} in @samp{f(n pi)} with @kbd{a b} will
+produce @samp{f(2 pi)}, whereas @kbd{s l} would give @samp{f(6.28)}
+since the evaluation step will also evaluate @code{pi}.
+
address@hidden Evaluates-To Operator, , Let Command, Store and Recall
address@hidden The Evaluates-To Operator
+
address@hidden
address@hidden evalto
address@hidden =>
address@hidden Evaluates-to operator
address@hidden @samp{=>} operator
+The special algebraic symbol @samp{=>} is known as the @dfn{evaluates-to
+operator}. (It will show up as an @code{evalto} function call in
+other language modes like Pascal and address@hidden) This is a binary
+operator, that is, it has a lefthand and a righthand argument,
+although it can be entered with the righthand argument omitted.
+
+A formula like @address@hidden => @var{b}} is evaluated by Calc as
+follows: First, @var{a} is not simplified or modified in any
+way. The previous value of argument @var{b} is thrown away; the
+formula @var{a} is then copied and evaluated as if by the @kbd{=}
+command according to all current modes and stored variable values,
+and the result is installed as the new value of @var{b}.
+
+For example, suppose you enter the algebraic formula @samp{2 + 3 => 17}.
+The number 17 is ignored, and the lefthand argument is left in its
+unevaluated form; the result is the formula @samp{2 + 3 => 5}.
+
address@hidden s =
address@hidden calc-evalto
+You can enter an @samp{=>} formula either directly using algebraic
+entry (in which case the righthand side may be omitted since it is
+going to be replaced right away anyhow), or by using the @kbd{s =}
+(@code{calc-evalto}) command, which takes @var{a} from the stack
+and replaces it with @address@hidden => @var{b}}.
+
+Calc keeps track of all @samp{=>} operators on the stack, and
+recomputes them whenever anything changes that might affect their
+values, i.e., a mode setting or variable value. This occurs only
+if the @samp{=>} operator is at the top level of the formula, or
+if it is part of a top-level vector. In other words, pushing
address@hidden + (a => 17)} will change the 17 to the actual value of
address@hidden when you enter the formula, but the result will not be
+dynamically updated when @samp{a} is changed later because the
address@hidden>} operator is buried inside a sum. However, a vector
+of @samp{=>} operators will be recomputed, since it is convenient
+to push a vector like @samp{[a =>, b =>, c =>]} on the stack to
+make a concise display of all the variables in your problem.
+(Another way to do this would be to use @samp{[a, b, c] =>},
+which provides a slightly different format of display. You
+can use whichever you find easiest to read.)
+
address@hidden m C
address@hidden calc-auto-recompute
+The @kbd{m C} (@code{calc-auto-recompute}) command allows you to
+turn this automatic recomputation on or off. If you turn
+recomputation off, you must explicitly recompute an @samp{=>}
+operator on the stack in one of the usual ways, such as by
+pressing @kbd{=}. Turning recomputation off temporarily can save
+a lot of time if you will be changing several modes or variables
+before you look at the @samp{=>} entries again.
+
+Most commands are not especially useful with @samp{=>} operators
+as arguments. For example, given @samp{x + 2 => 17}, it won't
+work to type @kbd{1 +} to get @samp{x + 3 => 18}. If you want
+to operate on the lefthand side of the @samp{=>} operator on
+the top of the stack, type @kbd{j 1} (that's the digit ``one'')
+to select the lefthand side, execute your commands, then type
address@hidden u} to unselect.
+
+All current modes apply when an @samp{=>} operator is computed,
+including the current simplification mode. Recall that the
+formula @samp{x + y + x} is not handled by Calc's default
+simplifications, but the @kbd{a s} command will reduce it to
+the simpler form @samp{y + 2 x}. You can also type @kbd{m A}
+to enable an Algebraic Simplification mode in which the
+equivalent of @kbd{a s} is used on all of Calc's results.
+If you enter @samp{x + y + x =>} normally, the result will
+be @samp{x + y + x => x + y + x}. If you change to
+Algebraic Simplification mode, the result will be
address@hidden + y + x => y + 2 x}. However, just pressing @kbd{a s}
+once will have no effect on @samp{x + y + x => x + y + x},
+because the righthand side depends only on the lefthand side
+and the current mode settings, and the lefthand side is not
+affected by commands like @kbd{a s}.
+
+The ``let'' command (@kbd{s l}) has an interesting interaction
+with the @samp{=>} operator. The @kbd{s l} command evaluates the
+second-to-top stack entry with the top stack entry supplying
+a temporary value for a given variable. As you might expect,
+if that stack entry is an @samp{=>} operator its righthand
+side will temporarily show this value for the variable. In
+fact, all @samp{=>}s on the stack will be updated if they refer
+to that variable. But this change is temporary in the sense
+that the next command that causes Calc to look at those stack
+entries will make them revert to the old variable value.
+
address@hidden
address@hidden
+2: a => a 2: a => 17 2: a => a
+1: a + 1 => a + 1 1: a + 1 => 18 1: a + 1 => a + 1
+ . . .
+
+ 17 s l a @key{RET} p 8 @key{RET}
address@hidden group
address@hidden smallexample
+
+Here the @kbd{p 8} command changes the current precision,
+thus causing the @samp{=>} forms to be recomputed after the
+influence of the ``let'' is gone. The @kbd{d @key{SPC}} command
+(@code{calc-refresh}) is a handy way to force the @samp{=>}
+operators on the stack to be recomputed without any other
+side effects.
+
address@hidden s :
address@hidden calc-assign
address@hidden assign
address@hidden :=
+Embedded mode also uses @samp{=>} operators. In Embedded mode,
+the lefthand side of an @samp{=>} operator can refer to variables
+assigned elsewhere in the file by @samp{:=} operators. The
+assignment operator @samp{a := 17} does not actually do anything
+by itself. But Embedded mode recognizes it and marks it as a sort
+of file-local definition of the variable. You can enter @samp{:=}
+operators in Algebraic mode, or by using the @kbd{s :}
+(@code{calc-assign}) address@hidden command which takes a variable
+and value from the stack and replaces them with an assignment.
+
address@hidden and LaTeX Language Modes}, for the way @samp{=>} appears in
address@hidden language output. The @dfn{eqn} mode gives similar
+treatment to @samp{=>}.
+
address@hidden Graphics, Kill and Yank, Store and Recall, Top
address@hidden Graphics
+
address@hidden
+The commands for graphing data begin with the @kbd{g} prefix key. Calc
+uses GNUPLOT 2.0 or later to do graphics. These commands will only work
+if GNUPLOT is available on your system. (While GNUPLOT sounds like
+a relative of GNU Emacs, it is actually completely unrelated.
+However, it is free software. It can be obtained from
address@hidden://www.gnuplot.info}.)
+
address@hidden calc-gnuplot-name
+If you have GNUPLOT installed on your system but Calc is unable to
+find it, you may need to set the @code{calc-gnuplot-name} variable
+in your Calc init file or @file{.emacs}. You may also need to set some Lisp
+variables to show Calc how to run GNUPLOT on your system; these
+are described under @kbd{g D} and @kbd{g O} below. If you are
+using the X window system, Calc will configure GNUPLOT for you
+automatically. If you have GNUPLOT 3.0 or later and you are not using X,
+Calc will configure GNUPLOT to display graphs using simple character
+graphics that will work on any terminal.
+
address@hidden
+* Basic Graphics::
+* Three Dimensional Graphics::
+* Managing Curves::
+* Graphics Options::
+* Devices::
address@hidden menu
+
address@hidden Basic Graphics, Three Dimensional Graphics, Graphics, Graphics
address@hidden Basic Graphics
+
address@hidden
address@hidden g f
address@hidden calc-graph-fast
+The easiest graphics command is @kbd{g f} (@code{calc-graph-fast}).
+This command takes two vectors of equal length from the stack.
+The vector at the top of the stack represents the ``y'' values of
+the various data points. The vector in the second-to-top position
+represents the corresponding ``x'' values. This command runs
+GNUPLOT (if it has not already been started by previous graphing
+commands) and displays the set of data points. The points will
+be connected by lines, and there will also be some kind of symbol
+to indicate the points themselves.
+
+The ``x'' entry may instead be an interval form, in which case suitable
+``x'' values are interpolated between the minimum and maximum values of
+the interval (whether the interval is open or closed is ignored).
+
+The ``x'' entry may also be a number, in which case Calc uses the
+sequence of ``x'' values @expr{x}, @expr{x+1}, @expr{x+2}, etc.
+(Generally the number 0 or 1 would be used for @expr{x} in this case.)
+
+The ``y'' entry may be any formula instead of a vector. Calc effectively
+uses @kbd{N} (@code{calc-eval-num}) to evaluate variables in the formula;
+the result of this must be a formula in a single (unassigned) variable.
+The formula is plotted with this variable taking on the various ``x''
+values. Graphs of formulas by default use lines without symbols at the
+computed data points. Note that if neither ``x'' nor ``y'' is a vector,
+Calc guesses at a reasonable number of data points to use. See the
address@hidden N} command below. (The ``x'' values must be either a vector
+or an interval if ``y'' is a formula.)
+
address@hidden
address@hidden
address@hidden ignore
address@hidden xy
+If ``y'' is (or evaluates to) a formula of the form
address@hidden(@var{x}, @var{y})} then the result is a
+parametric plot. The two arguments of the fictitious @code{xy} function
+are used as the ``x'' and ``y'' coordinates of the curve, respectively.
+In this case the ``x'' vector or interval you specified is not directly
+visible in the graph. For example, if ``x'' is the interval @samp{[0..360]}
+and ``y'' is the formula @samp{xy(sin(t), cos(t))}, the resulting graph
+will be a circle.
+
+Also, ``x'' and ``y'' may each be variable names, in which case Calc
+looks for suitable vectors, intervals, or formulas stored in those
+variables.
+
+The ``x'' and ``y'' values for the data points (as pulled from the vectors,
+calculated from the formulas, or interpolated from the intervals) should
+be real numbers (integers, fractions, or floats). One exception to this
+is that the ``y'' entry can consist of a vector of numbers combined with
+error forms, in which case the points will be plotted with the
+appropriate error bars. Other than this, if either the ``x''
+value or the ``y'' value of a given data point is not a real number, that
+data point will be omitted from the graph. The points on either side
+of the invalid point will @emph{not} be connected by a line.
+
+See the documentation for @kbd{g a} below for a description of the way
+numeric prefix arguments affect @kbd{g f}.
+
address@hidden @code{PlotRejects} variable
address@hidden PlotRejects
+If you store an empty vector in the variable @code{PlotRejects}
+(i.e., @kbd{[ ] s t PlotRejects}), Calc will append information to
+this vector for every data point which was rejected because its
+``x'' or ``y'' values were not real numbers. The result will be
+a matrix where each row holds the curve number, data point number,
+``x'' value, and ``y'' value for a rejected data point.
address@hidden Operator}, for a handy way to keep tabs on the
+current value of @code{PlotRejects}. @xref{Operations on Variables},
+for the @kbd{s R} command which is another easy way to examine
address@hidden
+
address@hidden g c
address@hidden calc-graph-clear
+To clear the graphics display, type @kbd{g c} (@code{calc-graph-clear}).
+If the GNUPLOT output device is an X window, the window will go away.
+Effects on other kinds of output devices will vary. You don't need
+to use @kbd{g c} if you don't want to---if you give another @kbd{g f}
+or @kbd{g p} command later on, it will reuse the existing graphics
+window if there is one.
+
address@hidden Three Dimensional Graphics, Managing Curves, Basic Graphics,
Graphics
address@hidden Three-Dimensional Graphics
+
address@hidden g F
address@hidden calc-graph-fast-3d
+The @kbd{g F} (@code{calc-graph-fast-3d}) command makes a three-dimensional
+graph. It works only if you have GNUPLOT 3.0 or later; with GNUPLOT 2.0,
+you will see a GNUPLOT error message if you try this command.
+
+The @kbd{g F} command takes three values from the stack, called ``x'',
+``y'', and ``z'', respectively. As was the case for 2D graphs, there
+are several options for these values.
+
+In the first case, ``x'' and ``y'' are each vectors (not necessarily of
+the same length); either or both may instead be interval forms. The
+``z'' value must be a matrix with the same number of rows as elements
+in ``x'', and the same number of columns as elements in ``y''. The
+result is a surface plot where
address@hidden @math{z_{ij}}
address@hidden @expr{z_ij}
+is the height of the point
+at coordinate @expr{(x_i, y_j)} on the surface. The 3D graph will
+be displayed from a certain default viewpoint; you can change this
+viewpoint by adding a @samp{set view} to the @samp{*Gnuplot Commands*}
+buffer as described later. See the GNUPLOT documentation for a
+description of the @samp{set view} command.
+
+Each point in the matrix will be displayed as a dot in the graph,
+and these points will be connected by a grid of lines (@dfn{isolines}).
+
+In the second case, ``x'', ``y'', and ``z'' are all vectors of equal
+length. The resulting graph displays a 3D line instead of a surface,
+where the coordinates of points along the line are successive triplets
+of values from the input vectors.
+
+In the third case, ``x'' and ``y'' are vectors or interval forms, and
+``z'' is any formula involving two variables (not counting variables
+with assigned values). These variables are sorted into alphabetical
+order; the first takes on values from ``x'' and the second takes on
+values from ``y'' to form a matrix of results that are graphed as a
+3D surface.
+
address@hidden
address@hidden
address@hidden ignore
address@hidden xyz
+If the ``z'' formula evaluates to a call to the fictitious function
address@hidden(@var{x}, @var{y}, @var{z})}, then the result is a
+``parametric surface.'' In this case, the axes of the graph are
+taken from the @var{x} and @var{y} values in these calls, and the
+``x'' and ``y'' values from the input vectors or intervals are used only
+to specify the range of inputs to the formula. For example, plotting
address@hidden, [0..180], xyz(sin(x)*sin(y), cos(x)*sin(y), cos(y))}
+will draw a sphere. (Since the default resolution for 3D plots is
+5 steps in each of ``x'' and ``y'', this will draw a very crude
+sphere. You could use the @kbd{g N} command, described below, to
+increase this resolution, or specify the ``x'' and ``y'' values as
+vectors with more than 5 elements.
+
+It is also possible to have a function in a regular @kbd{g f} plot
+evaluate to an @code{xyz} call. Since @kbd{g f} plots a line, not
+a surface, the result will be a 3D parametric line. For example,
address@hidden, xyz(sin(x), cos(x), x)]} will plot two turns of a
+helix (a three-dimensional spiral).
+
+As for @kbd{g f}, each of ``x'', ``y'', and ``z'' may instead be
+variables containing the relevant data.
+
address@hidden Managing Curves, Graphics Options, Three Dimensional Graphics,
Graphics
address@hidden Managing Curves
+
address@hidden
+The @kbd{g f} command is really shorthand for the following commands:
address@hidden g d g a g p}. Likewise, @address@hidden F}} is shorthand for
address@hidden g d g A g p}. You can gain more control over your graph
+by using these commands directly.
+
address@hidden g a
address@hidden calc-graph-add
+The @kbd{g a} (@code{calc-graph-add}) command adds the ``curve''
+represented by the two values on the top of the stack to the current
+graph. You can have any number of curves in the same graph. When
+you give the @kbd{g p} command, all the curves will be drawn superimposed
+on the same axes.
+
+The @kbd{g a} command (and many others that affect the current graph)
+will cause a special buffer, @samp{*Gnuplot Commands*}, to be displayed
+in another window. This buffer is a template of the commands that will
+be sent to GNUPLOT when it is time to draw the graph. The first
address@hidden a} command adds a @code{plot} command to this buffer. Succeeding
address@hidden a} commands add extra curves onto that @code{plot} command.
+Other graph-related commands put other GNUPLOT commands into this
+buffer. In normal usage you never need to work with this buffer
+directly, but you can if you wish. The only constraint is that there
+must be only one @code{plot} command, and it must be the last command
+in the buffer. If you want to save and later restore a complete graph
+configuration, you can use regular Emacs commands to save and restore
+the contents of the @samp{*Gnuplot Commands*} buffer.
+
address@hidden PlotData1
address@hidden PlotData2
+If the values on the stack are not variable names, @kbd{g a} will invent
+variable names for them (of the form @address@hidden) and store
+the values in those variables. The ``x'' and ``y'' variables are what
+go into the @code{plot} command in the template. If you add a curve
+that uses a certain variable and then later change that variable, you
+can replot the graph without having to delete and re-add the curve.
+That's because the variable name, not the vector, interval or formula
+itself, is what was added by @kbd{g a}.
+
+A numeric prefix argument on @kbd{g a} or @kbd{g f} changes the way
+stack entries are interpreted as curves. With a positive prefix
+argument @expr{n}, the top @expr{n} stack entries are ``y'' values
+for @expr{n} different curves which share a common ``x'' value in
+the @expr{n+1}st stack entry. (Thus @kbd{g a} with no prefix
+argument is equivalent to @kbd{C-u 1 g a}.)
+
+A prefix of zero or plain @kbd{C-u} means to take two stack entries,
+``x'' and ``y'' as usual, but to interpret ``y'' as a vector of
+``y'' values for several curves that share a common ``x''.
+
+A negative prefix argument tells Calc to read @expr{n} vectors from
+the stack; each vector @expr{[x, y]} describes an independent curve.
+This is the only form of @kbd{g a} that creates several curves at once
+that don't have common ``x'' values. (Of course, the range of ``x''
+values covered by all the curves ought to be roughly the same if
+they are to look nice on the same graph.)
+
+For example, to plot
address@hidden @math{\sin n x}
address@hidden @expr{sin(n x)}
+for integers @expr{n}
+from 1 to 5, you could use @kbd{v x} to create a vector of integers
+(@expr{n}), then @kbd{V M '} or @kbd{V M $} to map @samp{sin(n x)}
+across this vector. The resulting vector of formulas is suitable
+for use as the ``y'' argument to a @kbd{C-u g a} or @kbd{C-u g f}
+command.
+
address@hidden g A
address@hidden calc-graph-add-3d
+The @kbd{g A} (@code{calc-graph-add-3d}) command adds a 3D curve
+to the graph. It is not valid to intermix 2D and 3D curves in a
+single graph. This command takes three arguments, ``x'', ``y'',
+and ``z'', from the stack. With a positive prefix @expr{n}, it
+takes @expr{n+2} arguments (common ``x'' and ``y'', plus @expr{n}
+separate ``z''s). With a zero prefix, it takes three stack entries
+but the ``z'' entry is a vector of curve values. With a negative
+prefix @expr{-n}, it takes @expr{n} vectors of the form @expr{[x, y, z]}.
+The @kbd{g A} command works by adding a @code{splot} (surface-plot)
+command to the @samp{*Gnuplot Commands*} buffer.
+
+(Although @kbd{g a} adds a 2D @code{plot} command to the
address@hidden Commands*} buffer, Calc changes this to @code{splot}
+before sending it to GNUPLOT if it notices that the data points are
+evaluating to @code{xyz} calls. It will not work to mix 2D and 3D
address@hidden a} curves in a single graph, although Calc does not currently
+check for this.)
+
address@hidden g d
address@hidden calc-graph-delete
+The @kbd{g d} (@code{calc-graph-delete}) command deletes the most
+recently added curve from the graph. It has no effect if there are
+no curves in the graph. With a numeric prefix argument of any kind,
+it deletes all of the curves from the graph.
+
address@hidden g H
address@hidden calc-graph-hide
+The @kbd{g H} (@code{calc-graph-hide}) command ``hides'' or ``unhides''
+the most recently added curve. A hidden curve will not appear in
+the actual plot, but information about it such as its name and line and
+point styles will be retained.
+
address@hidden g j
address@hidden calc-graph-juggle
+The @kbd{g j} (@code{calc-graph-juggle}) command moves the curve
+at the end of the list (the ``most recently added curve'') to the
+front of the list. The next-most-recent curve is thus exposed for
address@hidden@kbd{g d}} or similar commands to use. With @kbd{g j} you can
work
+with any curve in the graph even though curve-related commands only
+affect the last curve in the list.
+
address@hidden g p
address@hidden calc-graph-plot
+The @kbd{g p} (@code{calc-graph-plot}) command uses GNUPLOT to draw
+the graph described in the @samp{*Gnuplot Commands*} buffer. Any
+GNUPLOT parameters which are not defined by commands in this buffer
+are reset to their default values. The variables named in the @code{plot}
+command are written to a temporary data file and the variable names
+are then replaced by the file name in the template. The resulting
+plotting commands are fed to the GNUPLOT program. See the documentation
+for the GNUPLOT program for more specific information. All temporary
+files are removed when Emacs or GNUPLOT exits.
+
+If you give a formula for ``y'', Calc will remember all the values that
+it calculates for the formula so that later plots can reuse these values.
+Calc throws out these saved values when you change any circumstances
+that may affect the data, such as switching from Degrees to Radians
+mode, or changing the value of a parameter in the formula. You can
+force Calc to recompute the data from scratch by giving a negative
+numeric prefix argument to @kbd{g p}.
+
+Calc uses a fairly rough step size when graphing formulas over intervals.
+This is to ensure quick response. You can ``refine'' a plot by giving
+a positive numeric prefix argument to @kbd{g p}. Calc goes through
+the data points it has computed and saved from previous plots of the
+function, and computes and inserts a new data point midway between
+each of the existing points. You can refine a plot any number of times,
+but beware that the amount of calculation involved doubles each time.
+
+Calc does not remember computed values for 3D graphs. This means the
+numerix prefix argument, if any, to @kbd{g p} is effectively ignored if
+the current graph is three-dimensional.
+
address@hidden g P
address@hidden calc-graph-print
+The @kbd{g P} (@code{calc-graph-print}) command is like @kbd{g p},
+except that it sends the output to a printer instead of to the
+screen. More precisely, @kbd{g p} looks for @samp{set terminal}
+or @samp{set output} commands in the @samp{*Gnuplot Commands*} buffer;
+lacking these it uses the default settings. However, @kbd{g P}
+ignores @samp{set terminal} and @samp{set output} commands and
+uses a different set of default values. All of these values are
+controlled by the @kbd{g D} and @kbd{g O} commands discussed below.
+Provided everything is set up properly, @kbd{g p} will plot to
+the screen unless you have specified otherwise and @kbd{g P} will
+always plot to the printer.
+
address@hidden Graphics Options, Devices, Managing Curves, Graphics
address@hidden Graphics Options
+
address@hidden
address@hidden g g
address@hidden calc-graph-grid
+The @kbd{g g} (@code{calc-graph-grid}) command turns the ``grid''
+on and off. It is off by default; tick marks appear only at the
+edges of the graph. With the grid turned on, dotted lines appear
+across the graph at each tick mark. Note that this command only
+changes the setting in @samp{*Gnuplot Commands*}; to see the effects
+of the change you must give another @kbd{g p} command.
+
address@hidden g b
address@hidden calc-graph-border
+The @kbd{g b} (@code{calc-graph-border}) command turns the border
+(the box that surrounds the graph) on and off. It is on by default.
+This command will only work with GNUPLOT 3.0 and later versions.
+
address@hidden g k
address@hidden calc-graph-key
+The @kbd{g k} (@code{calc-graph-key}) command turns the ``key''
+on and off. The key is a chart in the corner of the graph that
+shows the correspondence between curves and line styles. It is
+off by default, and is only really useful if you have several
+curves on the same graph.
+
address@hidden g N
address@hidden calc-graph-num-points
+The @kbd{g N} (@code{calc-graph-num-points}) command allows you
+to select the number of data points in the graph. This only affects
+curves where neither ``x'' nor ``y'' is specified as a vector.
+Enter a blank line to revert to the default value (initially 15).
+With no prefix argument, this command affects only the current graph.
+With a positive prefix argument this command changes or, if you enter
+a blank line, displays the default number of points used for all
+graphs created by @kbd{g a} that don't specify the resolution explicitly.
+With a negative prefix argument, this command changes or displays
+the default value (initially 5) used for 3D graphs created by @kbd{g A}.
+Note that a 3D setting of 5 means that a total of @expr{5^2 = 25} points
+will be computed for the surface.
+
+Data values in the graph of a function are normally computed to a
+precision of five digits, regardless of the current precision at the
+time. This is usually more than adequate, but there are cases where
+it will not be. For example, plotting @expr{1 + x} with @expr{x} in the
+interval @samp{[0 ..@: 1e-6]} will round all the data points down
+to 1.0! Putting the command @samp{set precision @var{n}} in the
address@hidden Commands*} buffer will cause the data to be computed
+at precision @var{n} instead of 5. Since this is such a rare case,
+there is no keystroke-based command to set the precision.
+
address@hidden g h
address@hidden calc-graph-header
+The @kbd{g h} (@code{calc-graph-header}) command sets the title
+for the graph. This will show up centered above the graph.
+The default title is blank (no title).
+
address@hidden g n
address@hidden calc-graph-name
+The @kbd{g n} (@code{calc-graph-name}) command sets the title of an
+individual curve. Like the other curve-manipulating commands, it
+affects the most recently added curve, i.e., the last curve on the
+list in the @samp{*Gnuplot Commands*} buffer. To set the title of
+the other curves you must first juggle them to the end of the list
+with @kbd{g j}, or edit the @samp{*Gnuplot Commands*} buffer by hand.
+Curve titles appear in the key; if the key is turned off they are
+not used.
+
address@hidden g t
address@hidden g T
address@hidden calc-graph-title-x
address@hidden calc-graph-title-y
+The @kbd{g t} (@code{calc-graph-title-x}) and @kbd{g T}
+(@code{calc-graph-title-y}) commands set the titles on the ``x''
+and ``y'' axes, respectively. These titles appear next to the
+tick marks on the left and bottom edges of the graph, respectively.
+Calc does not have commands to control the tick marks themselves,
+but you can edit them into the @samp{*Gnuplot Commands*} buffer if
+you wish. See the GNUPLOT documentation for details.
+
address@hidden g r
address@hidden g R
address@hidden calc-graph-range-x
address@hidden calc-graph-range-y
+The @kbd{g r} (@code{calc-graph-range-x}) and @kbd{g R}
+(@code{calc-graph-range-y}) commands set the range of values on the
+``x'' and ``y'' axes, respectively. You are prompted to enter a
+suitable range. This should be either a pair of numbers of the
+form, @address@hidden:@var{max}}, or a blank line to revert to the
+default behavior of setting the range based on the range of values
+in the data, or @samp{$} to take the range from the top of the stack.
+Ranges on the stack can be represented as either interval forms or
+vectors: @address@hidden ..@: @var{max}]} or @address@hidden, @var{max}]}.
+
address@hidden g l
address@hidden g L
address@hidden calc-graph-log-x
address@hidden calc-graph-log-y
+The @kbd{g l} (@code{calc-graph-log-x}) and @kbd{g L} (@code{calc-graph-log-y})
+commands allow you to set either or both of the axes of the graph to
+be logarithmic instead of linear.
+
address@hidden g C-l
address@hidden g C-r
address@hidden g C-t
address@hidden calc-graph-log-z
address@hidden calc-graph-range-z
address@hidden calc-graph-title-z
+For 3D plots, @kbd{g C-t}, @kbd{g C-r}, and @kbd{g C-l} (those are
+letters with the Control key held down) are the corresponding commands
+for the ``z'' axis.
+
address@hidden g z
address@hidden g Z
address@hidden calc-graph-zero-x
address@hidden calc-graph-zero-y
+The @kbd{g z} (@code{calc-graph-zero-x}) and @kbd{g Z}
+(@code{calc-graph-zero-y}) commands control whether a dotted line is
+drawn to indicate the ``x'' and/or ``y'' zero axes. (These are the same
+dotted lines that would be drawn there anyway if you used @kbd{g g} to
+turn the ``grid'' feature on.) Zero-axis lines are on by default, and
+may be turned off only in GNUPLOT 3.0 and later versions. They are
+not available for 3D plots.
+
address@hidden g s
address@hidden calc-graph-line-style
+The @kbd{g s} (@code{calc-graph-line-style}) command turns the connecting
+lines on or off for the most recently added curve, and optionally selects
+the style of lines to be used for that curve. Plain @kbd{g s} simply
+toggles the lines on and off. With a numeric prefix argument, @kbd{g s}
+turns lines on and sets a particular line style. Line style numbers
+start at one and their meanings vary depending on the output device.
+GNUPLOT guarantees that there will be at least six different line styles
+available for any device.
+
address@hidden g S
address@hidden calc-graph-point-style
+The @kbd{g S} (@code{calc-graph-point-style}) command similarly turns
+the symbols at the data points on or off, or sets the point style.
+If you turn both lines and points off, the data points will show as
+tiny dots. If the ``y'' values being plotted contain error forms and
+the connecting lines are turned off, then this command will also turn
+the error bars on or off.
+
address@hidden @code{LineStyles} variable
address@hidden @code{PointStyles} variable
address@hidden LineStyles
address@hidden PointStyles
+Another way to specify curve styles is with the @code{LineStyles} and
address@hidden variables. These variables initially have no stored
+values, but if you store a vector of integers in one of these variables,
+the @kbd{g a} and @kbd{g f} commands will use those style numbers
+instead of the defaults for new curves that are added to the graph.
+An entry should be a positive integer for a specific style, or 0 to let
+the style be chosen automatically, or @mathit{-1} to turn off lines or points
+altogether. If there are more curves than elements in the vector, the
+last few curves will continue to have the default styles. Of course,
+you can later use @kbd{g s} and @kbd{g S} to change any of these styles.
+
+For example, @kbd{'[2 -1 3] @key{RET} s t LineStyles} causes the first curve
+to have lines in style number 2, the second curve to have no connecting
+lines, and the third curve to have lines in style 3. Point styles will
+still be assigned automatically, but you could store another vector in
address@hidden to define them, too.
+
address@hidden Devices, , Graphics Options, Graphics
address@hidden Graphical Devices
+
address@hidden
address@hidden g D
address@hidden calc-graph-device
+The @kbd{g D} (@code{calc-graph-device}) command sets the device name
+(or ``terminal name'' in GNUPLOT lingo) to be used by @kbd{g p} commands
+on this graph. It does not affect the permanent default device name.
+If you enter a blank name, the device name reverts to the default.
+Enter @samp{?} to see a list of supported devices.
+
+With a positive numeric prefix argument, @kbd{g D} instead sets
+the default device name, used by all plots in the future which do
+not override it with a plain @kbd{g D} command. If you enter a
+blank line this command shows you the current default. The special
+name @code{default} signifies that Calc should choose @code{x11} if
+the X window system is in use (as indicated by the presence of a
address@hidden environment variable), or otherwise @code{dumb} under
+GNUPLOT 3.0 and later, or @code{postscript} under GNUPLOT 2.0.
+This is the initial default value.
+
+The @code{dumb} device is an interface to ``dumb terminals,'' i.e.,
+terminals with no special graphics facilities. It writes a crude
+picture of the graph composed of characters like @code{-} and @code{|}
+to a buffer called @samp{*Gnuplot Trail*}, which Calc then displays.
+The graph is made the same size as the Emacs screen, which on most
+dumb terminals will be
address@hidden @math{80\times24}
address@hidden 80x24
+characters. The graph is displayed in
+an Emacs ``recursive edit''; type @kbd{q} or @kbd{C-c C-c} to exit
+the recursive edit and return to Calc. Note that the @code{dumb}
+device is present only in GNUPLOT 3.0 and later versions.
+
+The word @code{dumb} may be followed by two numbers separated by
+spaces. These are the desired width and height of the graph in
+characters. Also, the device name @code{big} is like @code{dumb}
+but creates a graph four times the width and height of the Emacs
+screen. You will then have to scroll around to view the entire
+graph. In the @samp{*Gnuplot Trail*} buffer, @key{SPC}, @key{DEL},
address@hidden<}, and @kbd{>} are defined to scroll by one screenful in each
+of the four directions.
+
+With a negative numeric prefix argument, @kbd{g D} sets or displays
+the device name used by @kbd{g P} (@code{calc-graph-print}). This
+is initially @code{postscript}. If you don't have a PostScript
+printer, you may decide once again to use @code{dumb} to create a
+plot on any text-only printer.
+
address@hidden g O
address@hidden calc-graph-output
+The @kbd{g O} (@code{calc-graph-output}) command sets the name of
+the output file used by GNUPLOT. For some devices, notably @code{x11},
+there is no output file and this information is not used. Many other
+``devices'' are really file formats like @code{postscript}; in these
+cases the output in the desired format goes into the file you name
+with @kbd{g O}. Type @kbd{g O stdout @key{RET}} to set GNUPLOT to write
+to its standard output stream, i.e., to @samp{*Gnuplot Trail*}.
+This is the default setting.
+
+Another special output name is @code{tty}, which means that GNUPLOT
+is going to write graphics commands directly to its standard output,
+which you wish Emacs to pass through to your terminal. Tektronix
+graphics terminals, among other devices, operate this way. Calc does
+this by telling GNUPLOT to write to a temporary file, then running a
+sub-shell executing the command @samp{cat tempfile >/dev/tty}. On
+typical Unix systems, this will copy the temporary file directly to
+the terminal, bypassing Emacs entirely. You will have to type @kbd{C-l}
+to Emacs afterwards to refresh the screen.
+
+Once again, @kbd{g O} with a positive or negative prefix argument
+sets the default or printer output file names, respectively. In each
+case you can specify @code{auto}, which causes Calc to invent a temporary
+file name for each @kbd{g p} (or @kbd{g P}) command. This temporary file
+will be deleted once it has been displayed or printed. If the output file
+name is not @code{auto}, the file is not automatically deleted.
+
+The default and printer devices and output files can be saved
+permanently by the @kbd{m m} (@code{calc-save-modes}) command. The
+default number of data points (see @kbd{g N}) and the X geometry
+(see @kbd{g X}) are also saved. Other graph information is @emph{not}
+saved; you can save a graph's configuration simply by saving the contents
+of the @samp{*Gnuplot Commands*} buffer.
+
address@hidden calc-gnuplot-plot-command
address@hidden calc-gnuplot-default-device
address@hidden calc-gnuplot-default-output
address@hidden calc-gnuplot-print-command
address@hidden calc-gnuplot-print-device
address@hidden calc-gnuplot-print-output
+You may wish to configure the default and
+printer devices and output files for the whole system. The relevant
+Lisp variables are @code{calc-gnuplot-default-device} and @code{-output},
+and @code{calc-gnuplot-print-device} and @code{-output}. The output
+file names must be either strings as described above, or Lisp
+expressions which are evaluated on the fly to get the output file names.
+
+Other important Lisp variables are @code{calc-gnuplot-plot-command} and
address@hidden, which give the system commands to
+display or print the output of GNUPLOT, respectively. These may be
address@hidden if no command is necessary, or strings which can include
address@hidden to signify the name of the file to be displayed or printed.
+Or, these variables may contain Lisp expressions which are evaluated
+to display or print the output. These variables are customizable
+(@pxref{Customizing Calc}).
+
address@hidden g x
address@hidden calc-graph-display
+The @kbd{g x} (@code{calc-graph-display}) command lets you specify
+on which X window system display your graphs should be drawn. Enter
+a blank line to see the current display name. This command has no
+effect unless the current device is @code{x11}.
+
address@hidden g X
address@hidden calc-graph-geometry
+The @kbd{g X} (@code{calc-graph-geometry}) command is a similar
+command for specifying the position and size of the X window.
+The normal value is @code{default}, which generally means your
+window manager will let you place the window interactively.
+Entering @samp{800x500+0+0} would create an 800-by-500 pixel
+window in the upper-left corner of the screen.
+
+The buffer called @samp{*Gnuplot Trail*} holds a transcript of the
+session with GNUPLOT. This shows the commands Calc has ``typed'' to
+GNUPLOT and the responses it has received. Calc tries to notice when an
+error message has appeared here and display the buffer for you when
+this happens. You can check this buffer yourself if you suspect
+something has gone wrong.
+
address@hidden g C
address@hidden calc-graph-command
+The @kbd{g C} (@code{calc-graph-command}) command prompts you to
+enter any line of text, then simply sends that line to the current
+GNUPLOT process. The @samp{*Gnuplot Trail*} buffer looks deceptively
+like a Shell buffer but you can't type commands in it yourself.
+Instead, you must use @kbd{g C} for this purpose.
+
address@hidden g v
address@hidden g V
address@hidden calc-graph-view-commands
address@hidden calc-graph-view-trail
+The @kbd{g v} (@code{calc-graph-view-commands}) and @kbd{g V}
+(@code{calc-graph-view-trail}) commands display the @samp{*Gnuplot Commands*}
+and @samp{*Gnuplot Trail*} buffers, respectively, in another window.
+This happens automatically when Calc thinks there is something you
+will want to see in either of these buffers. If you type @kbd{g v}
+or @kbd{g V} when the relevant buffer is already displayed, the
+buffer is hidden again.
+
+One reason to use @kbd{g v} is to add your own commands to the
address@hidden Commands*} buffer. Press @kbd{g v}, then use
address@hidden o} to switch into that window. For example, GNUPLOT has
address@hidden label} and @samp{set arrow} commands that allow you to
+annotate your plots. Since Calc doesn't understand these commands,
+you have to add them to the @samp{*Gnuplot Commands*} buffer
+yourself, then use @address@hidden p}} to replot using these new commands.
Note
+that your commands must appear @emph{before} the @code{plot} command.
+To get help on any GNUPLOT feature, type, e.g., @kbd{g C help set label}.
+You may have to type @kbd{g C @key{RET}} a few times to clear the
+``press return for more'' or ``subtopic of @dots{}'' requests.
+Note that Calc always sends commands (like @samp{set nolabel}) to
+reset all plotting parameters to the defaults before each plot, so
+to delete a label all you need to do is delete the @samp{set label}
+line you added (or comment it out with @samp{#}) and then replot
+with @kbd{g p}.
+
address@hidden g q
address@hidden calc-graph-quit
+You can use @kbd{g q} (@code{calc-graph-quit}) to kill the GNUPLOT
+process that is running. The next graphing command you give will
+start a fresh GNUPLOT process. The word @samp{Graph} appears in
+the Calc window's mode line whenever a GNUPLOT process is currently
+running. The GNUPLOT process is automatically killed when you
+exit Emacs if you haven't killed it manually by then.
+
address@hidden g K
address@hidden calc-graph-kill
+The @kbd{g K} (@code{calc-graph-kill}) command is like @kbd{g q}
+except that it also views the @samp{*Gnuplot Trail*} buffer so that
+you can see the process being killed. This is better if you are
+killing GNUPLOT because you think it has gotten stuck.
+
address@hidden Kill and Yank, Keypad Mode, Graphics, Top
address@hidden Kill and Yank Functions
+
address@hidden
+The commands in this chapter move information between the Calculator and
+other Emacs editing buffers.
+
+In many cases Embedded mode is an easier and more natural way to
+work with Calc from a regular editing buffer. @xref{Embedded Mode}.
+
address@hidden
+* Killing From Stack::
+* Yanking Into Stack::
+* Grabbing From Buffers::
+* Yanking Into Buffers::
+* X Cut and Paste::
address@hidden menu
+
address@hidden Killing From Stack, Yanking Into Stack, Kill and Yank, Kill and
Yank
address@hidden Killing from the Stack
+
address@hidden
address@hidden C-k
address@hidden calc-kill
address@hidden M-k
address@hidden calc-copy-as-kill
address@hidden C-w
address@hidden calc-kill-region
address@hidden M-w
address@hidden calc-copy-region-as-kill
address@hidden Kill ring
address@hidden commands are Emacs commands that insert text into the
+``kill ring,'' from which it can later be ``yanked'' by a @kbd{C-y}
+command. Three common kill commands in normal Emacs are @kbd{C-k}, which
+kills one line, @kbd{C-w}, which kills the region between mark and point,
+and @kbd{M-w}, which puts the region into the kill ring without actually
+deleting it. All of these commands work in the Calculator, too. Also,
address@hidden has been provided to complete the set; it puts the current line
+into the kill ring without deleting anything.
+
+The kill commands are unusual in that they pay attention to the location
+of the cursor in the Calculator buffer. If the cursor is on or below the
+bottom line, the kill commands operate on the top of the stack. Otherwise,
+they operate on whatever stack element the cursor is on. Calc's kill
+commands always operate on whole stack entries. (They act the same as their
+standard Emacs cousins except they ``round up'' the specified region to
+encompass full lines.) The text is copied into the kill ring exactly as
+it appears on the screen, including line numbers if they are enabled.
+
+A numeric prefix argument to @kbd{C-k} or @kbd{M-k} affects the number
+of lines killed. A positive argument kills the current line and @expr{n-1}
+lines below it. A negative argument kills the @expr{-n} lines above the
+current line. Again this mirrors the behavior of the standard Emacs
address@hidden command. Although a whole line is always deleted, @kbd{C-k}
+with no argument copies only the number itself into the kill ring, whereas
address@hidden with a prefix argument of 1 copies the number with its trailing
+newline.
+
address@hidden Yanking Into Stack, Grabbing From Buffers, Killing From Stack,
Kill and Yank
address@hidden Yanking into the Stack
+
address@hidden
address@hidden C-y
address@hidden calc-yank
+The @kbd{C-y} command yanks the most recently killed text back into the
+Calculator. It pushes this value onto the top of the stack regardless of
+the cursor position. In general it re-parses the killed text as a number
+or formula (or a list of these separated by commas or newlines). However if
+the thing being yanked is something that was just killed from the Calculator
+itself, its full internal structure is yanked. For example, if you have
+set the floating-point display mode to show only four significant digits,
+then killing and re-yanking 3.14159 (which displays as 3.142) will yank the
+full 3.14159, even though yanking it into any other buffer would yank the
+number in its displayed form, 3.142. (Since the default display modes
+show all objects to their full precision, this feature normally makes no
+difference.)
+
address@hidden Grabbing From Buffers, Yanking Into Buffers, Yanking Into Stack,
Kill and Yank
address@hidden Grabbing from Other Buffers
+
address@hidden
address@hidden C-x * g
address@hidden calc-grab-region
+The @kbd{C-x * g} (@code{calc-grab-region}) command takes the text between
+point and mark in the current buffer and attempts to parse it as a
+vector of values. Basically, it wraps the text in vector brackets
address@hidden ]} unless the text already is enclosed in vector brackets,
+then reads the text as if it were an algebraic entry. The contents
+of the vector may be numbers, formulas, or any other Calc objects.
+If the @kbd{C-x * g} command works successfully, it does an automatic
address@hidden * c} to enter the Calculator buffer.
+
+A numeric prefix argument grabs the specified number of lines around
+point, ignoring the mark. A positive prefix grabs from point to the
address@hidden following newline (so that @kbd{M-1 C-x * g} grabs from point
+to the end of the current line); a negative prefix grabs from point
+back to the @expr{n+1}st preceding newline. In these cases the text
+that is grabbed is exactly the same as the text that @kbd{C-k} would
+delete given that prefix argument.
+
+A prefix of zero grabs the current line; point may be anywhere on the
+line.
+
+A plain @kbd{C-u} prefix interprets the region between point and mark
+as a single number or formula rather than a vector. For example,
address@hidden * g} on the text @samp{2 a b} produces the vector of three
+values @samp{[2, a, b]}, but @kbd{C-u C-x * g} on the same region
+reads a formula which is a product of three things: @samp{2 a b}.
+(The text @samp{a + b}, on the other hand, will be grabbed as a
+vector of one element by plain @kbd{C-x * g} because the interpretation
address@hidden, +, b]} would be a syntax error.)
+
+If a different language has been specified (@pxref{Language Modes}),
+the grabbed text will be interpreted according to that language.
+
address@hidden C-x * r
address@hidden calc-grab-rectangle
+The @kbd{C-x * r} (@code{calc-grab-rectangle}) command takes the text between
+point and mark and attempts to parse it as a matrix. If point and mark
+are both in the leftmost column, the lines in between are parsed in their
+entirety. Otherwise, point and mark define the corners of a rectangle
+whose contents are parsed.
+
+Each line of the grabbed area becomes a row of the matrix. The result
+will actually be a vector of vectors, which Calc will treat as a matrix
+only if every row contains the same number of values.
+
+If a line contains a portion surrounded by square brackets (or curly
+braces), that portion is interpreted as a vector which becomes a row
+of the matrix. Any text surrounding the bracketed portion on the line
+is ignored.
+
+Otherwise, the entire line is interpreted as a row vector as if it
+were surrounded by square brackets. Leading line numbers (in the
+format used in the Calc stack buffer) are ignored. If you wish to
+force this interpretation (even if the line contains bracketed
+portions), give a negative numeric prefix argument to the
address@hidden * r} command.
+
+If you give a numeric prefix argument of zero or plain @kbd{C-u}, each
+line is instead interpreted as a single formula which is converted into
+a one-element vector. Thus the result of @kbd{C-u C-x * r} will be a
+one-column matrix. For example, suppose one line of the data is the
+expression @samp{2 a}. A plain @address@hidden * r}} will interpret this as
address@hidden a]}, which in turn is read as a two-element vector that forms
+one row of the matrix. But a @kbd{C-u C-x * r} will interpret this row
+as @samp{[2*a]}.
+
+If you give a positive numeric prefix argument @var{n}, then each line
+will be split up into columns of width @var{n}; each column is parsed
+separately as a matrix element. If a line contained
address@hidden@samp{2 +/- 3 4 +/- 5}}, then grabbing with a prefix argument of 8
+would correctly split the line into two error forms.
+
address@hidden Functions}, to see how to pull the matrix apart into its
+constituent rows and columns. (If it is a
address@hidden @math{1\times1}
address@hidden 1x1
+matrix, just hit @kbd{v u} (@code{calc-unpack}) twice.)
+
address@hidden C-x * :
address@hidden C-x * _
address@hidden calc-grab-sum-across
address@hidden calc-grab-sum-down
address@hidden Summing rows and columns of data
+The @kbd{C-x * :} (@code{calc-grab-sum-down}) command is a handy way to
+grab a rectangle of data and sum its columns. It is equivalent to
+typing @kbd{C-x * r}, followed by @kbd{V R : +} (the vector reduction
+command that sums the columns of a matrix; @pxref{Reducing}). The
+result of the command will be a vector of numbers, one for each column
+in the input data. The @kbd{C-x * _} (@code{calc-grab-sum-across}) command
+similarly grabs a rectangle and sums its rows by executing @address@hidden R _
+}}.
+
+As well as being more convenient, @kbd{C-x * :} and @kbd{C-x * _} are also
+much faster because they don't actually place the grabbed vector on
+the stack. In a @kbd{C-x * r V R : +} sequence, formatting the vector
+for display on the stack takes a large fraction of the total time
+(unless you have planned ahead and used @kbd{v .} and @kbd{t .} modes).
+
+For example, suppose we have a column of numbers in a file which we
+wish to sum. Go to one corner of the column and press @kbd{C-@@} to
+set the mark; go to the other corner and type @kbd{C-x * :}. Since there
+is only one column, the result will be a vector of one number, the sum.
+(You can type @kbd{v u} to unpack this vector into a plain number if
+you want to do further arithmetic with it.)
+
+To compute the product of the column of numbers, we would have to do
+it ``by hand'' since there's no special grab-and-multiply command.
+Use @kbd{C-x * r} to grab the column of numbers into the calculator in
+the form of a column matrix. The statistics command @kbd{u *} is a
+handy way to find the product of a vector or matrix of numbers.
address@hidden Operations}. Another approach would be to use
+an explicit column reduction command, @kbd{V R : *}.
+
address@hidden Yanking Into Buffers, X Cut and Paste, Grabbing From Buffers,
Kill and Yank
address@hidden Yanking into Other Buffers
+
address@hidden
address@hidden y
address@hidden calc-copy-to-buffer
+The plain @kbd{y} (@code{calc-copy-to-buffer}) command inserts the number
+at the top of the stack into the most recently used normal editing buffer.
+(More specifically, this is the most recently used buffer which is displayed
+in a window and whose name does not begin with @samp{*}. If there is no
+such buffer, this is the most recently used buffer except for Calculator
+and Calc Trail buffers.) The number is inserted exactly as it appears and
+without a newline. (If line-numbering is enabled, the line number is
+normally not included.) The number is @emph{not} removed from the stack.
+
+With a prefix argument, @kbd{y} inserts several numbers, one per line.
+A positive argument inserts the specified number of values from the top
+of the stack. A negative argument inserts the @expr{n}th value from the
+top of the stack. An argument of zero inserts the entire stack. Note
+that @kbd{y} with an argument of 1 is slightly different from @kbd{y}
+with no argument; the former always copies full lines, whereas the
+latter strips off the trailing newline.
+
+With a lone @kbd{C-u} as a prefix argument, @kbd{y} @emph{replaces} the
+region in the other buffer with the yanked text, then quits the
+Calculator, leaving you in that buffer. A typical use would be to use
address@hidden * g} to read a region of data into the Calculator, operate on the
+data to produce a new matrix, then type @kbd{C-u y} to replace the
+original data with the new data. One might wish to alter the matrix
+display style (@pxref{Vector and Matrix Formats}) or change the current
+display language (@pxref{Language Modes}) before doing this. Also, note
+that this command replaces a linear region of text (as grabbed by
address@hidden * g}), not a rectangle (as grabbed by @kbd{C-x * r}).
+
+If the editing buffer is in overwrite (as opposed to insert) mode,
+and the @kbd{C-u} prefix was not used, then the yanked number will
+overwrite the characters following point rather than being inserted
+before those characters. The usual conventions of overwrite mode
+are observed; for example, characters will be inserted at the end of
+a line rather than overflowing onto the next line. Yanking a multi-line
+object such as a matrix in overwrite mode overwrites the next @var{n}
+lines in the buffer, lengthening or shortening each line as necessary.
+Finally, if the thing being yanked is a simple integer or floating-point
+number (like @samp{-1.2345e-3}) and the characters following point also
+make up such a number, then Calc will replace that number with the new
+number, lengthening or shortening as necessary. The concept of
+``overwrite mode'' has thus been generalized from overwriting characters
+to overwriting one complete number with another.
+
address@hidden C-x * y
+The @kbd{C-x * y} key sequence is equivalent to @kbd{y} except that
+it can be typed anywhere, not just in Calc. This provides an easy
+way to guarantee that Calc knows which editing buffer you want to use!
+
address@hidden X Cut and Paste, , Yanking Into Buffers, Kill and Yank
address@hidden X Cut and Paste
+
address@hidden
+If you are using Emacs with the X window system, there is an easier
+way to move small amounts of data into and out of the calculator:
+Use the mouse-oriented cut and paste facilities of X.
+
+The default bindings for a three-button mouse cause the left button
+to move the Emacs cursor to the given place, the right button to
+select the text between the cursor and the clicked location, and
+the middle button to yank the selection into the buffer at the
+clicked location. So, if you have a Calc window and an editing
+window on your Emacs screen, you can use left-click/right-click
+to select a number, vector, or formula from one window, then
+middle-click to paste that value into the other window. When you
+paste text into the Calc window, Calc interprets it as an algebraic
+entry. It doesn't matter where you click in the Calc window; the
+new value is always pushed onto the top of the stack.
+
+The @code{xterm} program that is typically used for general-purpose
+shell windows in X interprets the mouse buttons in the same way.
+So you can use the mouse to move data between Calc and any other
+Unix program. One nice feature of @code{xterm} is that a double
+left-click selects one word, and a triple left-click selects a
+whole line. So you can usually transfer a single number into Calc
+just by double-clicking on it in the shell, then middle-clicking
+in the Calc window.
+
address@hidden Keypad Mode, Embedded Mode, Kill and Yank, Top
address@hidden Keypad Mode
+
address@hidden
address@hidden C-x * k
address@hidden calc-keypad
+The @kbd{C-x * k} (@code{calc-keypad}) command starts the Calculator
+and displays a picture of a calculator-style keypad. If you are using
+the X window system, you can click on any of the ``keys'' in the
+keypad using the left mouse button to operate the calculator.
+The original window remains the selected window; in Keypad mode
+you can type in your file while simultaneously performing
+calculations with the mouse.
+
address@hidden full-calc-keypad
+If you have used @kbd{C-x * b} first, @kbd{C-x * k} instead invokes
+the @code{full-calc-keypad} command, which takes over the whole
+Emacs screen and displays the keypad, the Calc stack, and the Calc
+trail all at once. This mode would normally be used when running
+Calc standalone (@pxref{Standalone Operation}).
+
+If you aren't using the X window system, you must switch into
+the @samp{*Calc Keypad*} window, place the cursor on the desired
+``key,'' and type @key{SPC} or @key{RET}. If you think this
+is easier than using Calc normally, go right ahead.
+
+Calc commands are more or less the same in Keypad mode. Certain
+keypad keys differ slightly from the corresponding normal Calc
+keystrokes; all such deviations are described below.
+
+Keypad mode includes many more commands than will fit on the keypad
+at once. Click the right mouse button address@hidden
+to switch to the next menu. The bottom five rows of the keypad
+stay the same; the top three rows change to a new set of commands.
+To return to earlier menus, click the middle mouse button
address@hidden or simply advance through the menus
+until you wrap around. Typing @key{TAB} inside the keypad window
+is equivalent to clicking the right mouse button there.
+
+You can always click the @key{EXEC} button and type any normal
+Calc key sequence. This is equivalent to switching into the
+Calc buffer, typing the keys, then switching back to your
+original buffer.
+
address@hidden
+* Keypad Main Menu::
+* Keypad Functions Menu::
+* Keypad Binary Menu::
+* Keypad Vectors Menu::
+* Keypad Modes Menu::
address@hidden menu
+
address@hidden Keypad Main Menu, Keypad Functions Menu, Keypad Mode, Keypad Mode
address@hidden Main Menu
+
address@hidden
address@hidden
+|----+-----Calc 2.1------+----1
+|FLR |CEIL|RND |TRNC|CLN2|FLT |
+|----+----+----+----+----+----|
+| LN |EXP | |ABS |IDIV|MOD |
+|----+----+----+----+----+----|
+|SIN |COS |TAN |SQRT|y^x |1/x |
+|----+----+----+----+----+----|
+| ENTER |+/- |EEX |UNDO| <- |
+|-----+---+-+--+--+-+---++----|
+| INV | 7 | 8 | 9 | / |
+|-----+-----+-----+-----+-----|
+| HYP | 4 | 5 | 6 | * |
+|-----+-----+-----+-----+-----|
+|EXEC | 1 | 2 | 3 | - |
+|-----+-----+-----+-----+-----|
+| OFF | 0 | . | PI | + |
+|-----+-----+-----+-----+-----+
address@hidden group
address@hidden smallexample
+
address@hidden
+This is the menu that appears the first time you start Keypad mode.
+It will show up in a vertical window on the right side of your screen.
+Above this menu is the traditional Calc stack display. On a 24-line
+screen you will be able to see the top three stack entries.
+
+The ten digit keys, decimal point, and @key{EEX} key are used for
+entering numbers in the obvious way. @key{EEX} begins entry of an
+exponent in scientific notation. Just as with regular Calc, the
+number is pushed onto the stack as soon as you press @key{ENTER}
+or any other function key.
+
+The @key{+/-} key corresponds to normal Calc's @kbd{n} key. During
+numeric entry it changes the sign of the number or of the exponent.
+At other times it changes the sign of the number on the top of the
+stack.
+
+The @key{INV} and @key{HYP} keys modify other keys. As well as
+having the effects described elsewhere in this manual, Keypad mode
+defines several other ``inverse'' operations. These are described
+below and in the following sections.
+
+The @key{ENTER} key finishes the current numeric entry, or otherwise
+duplicates the top entry on the stack.
+
+The @key{UNDO} key undoes the most recent Calc operation.
address@hidden UNDO} is the ``redo'' command, and @kbd{HYP UNDO} is
+``last arguments'' (@address@hidden).
+
+The @key{<-} key acts as a ``backspace'' during numeric entry.
+At other times it removes the top stack entry. @kbd{INV <-}
+clears the entire stack. @kbd{HYP <-} takes an integer from
+the stack, then removes that many additional stack elements.
+
+The @key{EXEC} key prompts you to enter any keystroke sequence
+that would normally work in Calc mode. This can include a
+numeric prefix if you wish. It is also possible simply to
+switch into the Calc window and type commands in it; there is
+nothing ``magic'' about this window when Keypad mode is active.
+
+The other keys in this display perform their obvious calculator
+functions. @key{CLN2} rounds the top-of-stack by temporarily
+reducing the precision by 2 digits. @key{FLT} converts an
+integer or fraction on the top of the stack to floating-point.
+
+The @key{INV} and @key{HYP} keys combined with several of these keys
+give you access to some common functions even if the appropriate menu
+is not displayed. Obviously you don't need to learn these keys
+unless you find yourself wasting time switching among the menus.
+
address@hidden @kbd
address@hidden INV +/-
+is the same as @key{1/x}.
address@hidden INV +
+is the same as @key{SQRT}.
address@hidden INV -
+is the same as @key{CONJ}.
address@hidden INV *
+is the same as @key{y^x}.
address@hidden INV /
+is the same as @key{INV y^x} (the @expr{x}th root of @expr{y}).
address@hidden HYP/INV 1
+are the same as @key{SIN} / @kbd{INV SIN}.
address@hidden HYP/INV 2
+are the same as @key{COS} / @kbd{INV COS}.
address@hidden HYP/INV 3
+are the same as @key{TAN} / @kbd{INV TAN}.
address@hidden INV/HYP 4
+are the same as @key{LN} / @kbd{HYP LN}.
address@hidden INV/HYP 5
+are the same as @key{EXP} / @kbd{HYP EXP}.
address@hidden INV 6
+is the same as @key{ABS}.
address@hidden INV 7
+is the same as @key{RND} (@code{calc-round}).
address@hidden INV 8
+is the same as @key{CLN2}.
address@hidden INV 9
+is the same as @key{FLT} (@code{calc-float}).
address@hidden INV 0
+is the same as @key{IMAG}.
address@hidden INV .
+is the same as @key{PREC}.
address@hidden INV ENTER
+is the same as @key{SWAP}.
address@hidden HYP ENTER
+is the same as @key{RLL3}.
address@hidden INV HYP ENTER
+is the same as @key{OVER}.
address@hidden HYP +/-
+packs the top two stack entries as an error form.
address@hidden HYP EEX
+packs the top two stack entries as a modulo form.
address@hidden INV EEX
+creates an interval form; this removes an integer which is one
+of 0 @samp{[]}, 1 @samp{[)}, 2 @samp{(]} or 3 @samp{()}, followed
+by the two limits of the interval.
address@hidden table
+
+The @kbd{OFF} key turns Calc off; typing @kbd{C-x * k} or @kbd{C-x * *}
+again has the same effect. This is analogous to typing @kbd{q} or
+hitting @kbd{C-x * c} again in the normal calculator. If Calc is
+running standalone (the @code{full-calc-keypad} command appeared in the
+command line that started Emacs), then @kbd{OFF} is replaced with
address@hidden; clicking on this actually exits Emacs itself.
+
address@hidden Keypad Functions Menu, Keypad Binary Menu, Keypad Main Menu,
Keypad Mode
address@hidden Functions Menu
+
address@hidden
address@hidden
+|----+----+----+----+----+----2
+|IGAM|BETA|IBET|ERF |BESJ|BESY|
+|----+----+----+----+----+----|
+|IMAG|CONJ| RE |ATN2|RAND|RAGN|
+|----+----+----+----+----+----|
+|GCD |FACT|DFCT|BNOM|PERM|NXTP|
+|----+----+----+----+----+----|
address@hidden group
address@hidden smallexample
+
address@hidden
+This menu provides various operations from the @kbd{f} and @kbd{k}
+prefix keys.
+
address@hidden multiplies the number on the stack by the imaginary
+number @expr{i = (0, 1)}.
+
address@hidden extracts the real part a complex number. @kbd{INV RE}
+extracts the imaginary part.
+
address@hidden takes a number from the top of the stack and computes
+a random number greater than or equal to zero but less than that
+number. (@xref{Random Numbers}.) @key{RAGN} is the ``random
+again'' command; it computes another random number using the
+same limit as last time.
+
address@hidden GCD} computes the LCM (least common multiple) function.
+
address@hidden FACT} is the gamma function.
address@hidden @math{\Gamma(x) = (x-1)!}.
address@hidden @expr{gamma(x) = (x-1)!}.
+
address@hidden is the number-of-permutations function, which is on the
address@hidden k c} key in normal Calc.
+
address@hidden finds the next prime after a number. @kbd{INV NXTP}
+finds the previous prime.
+
address@hidden Keypad Binary Menu, Keypad Vectors Menu, Keypad Functions Menu,
Keypad Mode
address@hidden Binary Menu
+
address@hidden
address@hidden
+|----+----+----+----+----+----3
+|AND | OR |XOR |NOT |LSH |RSH |
+|----+----+----+----+----+----|
+|DEC |HEX |OCT |BIN |WSIZ|ARSH|
+|----+----+----+----+----+----|
+| A | B | C | D | E | F |
+|----+----+----+----+----+----|
address@hidden group
address@hidden smallexample
+
address@hidden
+The keys in this menu perform operations on binary integers.
+Note that both logical and arithmetic right-shifts are provided.
address@hidden LSH} rotates one bit to the left.
+
+The ``difference'' function (normally on @kbd{b d}) is on @key{INV AND}.
+The ``clip'' function (normally on @address@hidden c}}) is on @key{INV NOT}.
+
+The @key{DEC}, @key{HEX}, @key{OCT}, and @key{BIN} keys select the
+current radix for display and entry of numbers: Decimal, hexadecimal,
+octal, or binary. The six letter keys @key{A} through @key{F} are used
+for entering hexadecimal numbers.
+
+The @key{WSIZ} key displays the current word size for binary operations
+and allows you to enter a new word size. You can respond to the prompt
+using either the keyboard or the digits and @key{ENTER} from the keypad.
+The initial word size is 32 bits.
+
address@hidden Keypad Vectors Menu, Keypad Modes Menu, Keypad Binary Menu,
Keypad Mode
address@hidden Vectors Menu
+
address@hidden
address@hidden
+|----+----+----+----+----+----4
+|SUM |PROD|MAX |MAP*|MAP^|MAP$|
+|----+----+----+----+----+----|
+|MINV|MDET|MTRN|IDNT|CRSS|"x" |
+|----+----+----+----+----+----|
+|PACK|UNPK|INDX|BLD |LEN |... |
+|----+----+----+----+----+----|
address@hidden group
address@hidden smallexample
+
address@hidden
+The keys in this menu operate on vectors and matrices.
+
address@hidden removes an integer @var{n} from the top of the stack;
+the next @var{n} stack elements are removed and packed into a vector,
+which is replaced onto the stack. Thus the sequence
address@hidden ENTER 3 ENTER 5 ENTER 3 PACK} enters the vector
address@hidden, 3, 5]} onto the stack. To enter a matrix, build each row
+on the stack as a vector, then use a final @key{PACK} to collect the
+rows into a matrix.
+
address@hidden unpacks the vector on the stack, pushing each of its
+components separately.
+
address@hidden removes an integer @var{n}, then builds a vector of
+integers from 1 to @var{n}. @kbd{INV INDX} takes three numbers
+from the stack: The vector size @var{n}, the starting number,
+and the increment. @kbd{BLD} takes an integer @var{n} and any
+value @var{x} and builds a vector of @var{n} copies of @var{x}.
+
address@hidden removes an integer @var{n}, then builds an @address@hidden
+identity matrix.
+
address@hidden replaces a vector by its length, an integer.
+
address@hidden turns on or off ``abbreviated'' display mode for large vectors.
+
address@hidden, @key{MDET}, @key{MTRN}, and @key{CROSS} are the matrix
+inverse, determinant, and transpose, and vector cross product.
+
address@hidden replaces a vector by the sum of its elements. It is
+equivalent to @kbd{u +} in normal Calc (@pxref{Statistical Operations}).
address@hidden computes the product of the elements of a vector, and
address@hidden computes the maximum of all the elements of a vector.
+
address@hidden SUM} computes the alternating sum of the first element
+minus the second, plus the third, minus the fourth, and so on.
address@hidden MAX} computes the minimum of the vector elements.
+
address@hidden SUM} computes the mean of the vector elements.
address@hidden PROD} computes the sample standard deviation.
address@hidden MAX} computes the median.
+
address@hidden multiplies two vectors elementwise. It is equivalent
+to the @kbd{V M *} command. @key{MAP^} computes powers elementwise.
+The arguments must be vectors of equal length, or one must be a vector
+and the other must be a plain number. For example, @kbd{2 MAP^} squares
+all the elements of a vector.
+
address@hidden maps the formula on the top of the stack across the
+vector in the second-to-top position. If the formula contains
+several variables, Calc takes that many vectors starting at the
+second-to-top position and matches them to the variables in
+alphabetical order. The result is a vector of the same size as
+the input vectors, whose elements are the formula evaluated with
+the variables set to the various sets of numbers in those vectors.
+For example, you could simulate @key{MAP^} using @key{MAP$} with
+the formula @samp{x^y}.
+
+The @kbd{"x"} key pushes the variable name @expr{x} onto the
+stack. To build the formula @expr{x^2 + 6}, you would use the
+key sequence @kbd{"x" 2 y^x 6 +}. This formula would then be
+suitable for use with the @key{MAP$} key described above.
+With @key{INV}, @key{HYP}, or @key{INV} and @key{HYP}, the
address@hidden"x"} key pushes the variable names @expr{y}, @expr{z}, and
address@hidden, respectively.
+
address@hidden Keypad Modes Menu, , Keypad Vectors Menu, Keypad Mode
address@hidden Modes Menu
+
address@hidden
address@hidden
+|----+----+----+----+----+----5
+|FLT |FIX |SCI |ENG |GRP | |
+|----+----+----+----+----+----|
+|RAD |DEG |FRAC|POLR|SYMB|PREC|
+|----+----+----+----+----+----|
+|SWAP|RLL3|RLL4|OVER|STO |RCL |
+|----+----+----+----+----+----|
address@hidden group
address@hidden smallexample
+
address@hidden
+The keys in this menu manipulate modes, variables, and the stack.
+
+The @key{FLT}, @key{FIX}, @key{SCI}, and @key{ENG} keys select
+floating-point, fixed-point, scientific, or engineering notation.
address@hidden displays two digits after the decimal by default; the
+others display full precision. With the @key{INV} prefix, these
+keys pop a number-of-digits argument from the stack.
+
+The @key{GRP} key turns grouping of digits with commas on or off.
address@hidden GRP} enables grouping to the right of the decimal point as
+well as to the left.
+
+The @key{RAD} and @key{DEG} keys switch between radians and degrees
+for trigonometric functions.
+
+The @key{FRAC} key turns Fraction mode on or off. This affects
+whether commands like @kbd{/} with integer arguments produce
+fractional or floating-point results.
+
+The @key{POLR} key turns Polar mode on or off, determining whether
+polar or rectangular complex numbers are used by default.
+
+The @key{SYMB} key turns Symbolic mode on or off, in which
+operations that would produce inexact floating-point results
+are left unevaluated as algebraic formulas.
+
+The @key{PREC} key selects the current precision. Answer with
+the keyboard or with the keypad digit and @key{ENTER} keys.
+
+The @key{SWAP} key exchanges the top two stack elements.
+The @key{RLL3} key rotates the top three stack elements upwards.
+The @key{RLL4} key rotates the top four stack elements upwards.
+The @key{OVER} key duplicates the second-to-top stack element.
+
+The @key{STO} and @key{RCL} keys are analogous to @kbd{s t} and
address@hidden r} in regular Calc. @xref{Store and Recall}. Click the
address@hidden or @key{RCL} key, then one of the ten digits. (Named
+variables are not available in Keypad mode.) You can also use,
+for example, @kbd{STO + 3} to add to register 3.
+
address@hidden Embedded Mode, Programming, Keypad Mode, Top
address@hidden Embedded Mode
+
address@hidden
+Embedded mode in Calc provides an alternative to copying numbers
+and formulas back and forth between editing buffers and the Calc
+stack. In Embedded mode, your editing buffer becomes temporarily
+linked to the stack and this copying is taken care of automatically.
+
address@hidden
+* Basic Embedded Mode::
+* More About Embedded Mode::
+* Assignments in Embedded Mode::
+* Mode Settings in Embedded Mode::
+* Customizing Embedded Mode::
address@hidden menu
+
address@hidden Basic Embedded Mode, More About Embedded Mode, Embedded Mode,
Embedded Mode
address@hidden Basic Embedded Mode
+
address@hidden
address@hidden C-x * e
address@hidden calc-embedded
+To enter Embedded mode, position the Emacs point (cursor) on a
+formula in any buffer and press @kbd{C-x * e} (@code{calc-embedded}).
+Note that @kbd{C-x * e} is not to be used in the Calc stack buffer
+like most Calc commands, but rather in regular editing buffers that
+are visiting your own files.
+
+Calc will try to guess an appropriate language based on the major mode
+of the editing buffer. (@xref{Language Modes}.) If the current buffer is
+in @code{latex-mode}, for example, Calc will set its language to address@hidden
+Similarly, Calc will use @TeX{} language for @code{tex-mode},
address@hidden and @code{context-mode}, C language for
address@hidden and @code{c++-mode}, FORTRAN language for
address@hidden and @code{f90-mode}, Pascal for @code{pascal-mode},
+and eqn for @code{nroff-mode} (@pxref{Customizing Calc}).
+These can be overridden with Calc's mode
+changing commands (@pxref{Mode Settings in Embedded Mode}). If no
+suitable language is available, Calc will continue with its current language.
+
+Calc normally scans backward and forward in the buffer for the
+nearest opening and closing @dfn{formula delimiters}. The simplest
+delimiters are blank lines. Other delimiters that Embedded mode
+understands are:
+
address@hidden
address@hidden
+The @TeX{} and address@hidden math delimiters @samp{$ $}, @samp{$$ $$},
address@hidden \]}, and @samp{\( \)};
address@hidden
+Lines beginning with @samp{\begin} and @samp{\end} (except matrix delimiters);
address@hidden
+Lines beginning with @samp{@@} (Texinfo delimiters).
address@hidden
+Lines beginning with @samp{.EQ} and @samp{.EN} (@dfn{eqn} delimiters);
address@hidden
+Lines containing a single @samp{%} or @samp{.\"} symbol and nothing else.
address@hidden enumerate
+
address@hidden Embedded Mode}, to see how to make Calc recognize
+your own favorite delimiters. Delimiters like @samp{$ $} can appear
+on their own separate lines or in-line with the formula.
+
+If you give a positive or negative numeric prefix argument, Calc
+instead uses the current point as one end of the formula, and includes
+that many lines forward or backward (respectively, including the current
+line). Explicit delimiters are not necessary in this case.
+
+With a prefix argument of zero, Calc uses the current region (delimited
+by point and mark) instead of formula delimiters. With a prefix
+argument of @kbd{C-u} only, Calc uses the current line as the formula.
+
address@hidden C-x * w
address@hidden calc-embedded-word
+The @kbd{C-x * w} (@code{calc-embedded-word}) command will start Embedded
+mode on the current ``word''; in this case Calc will scan for the first
+non-numeric character (i.e., the first character that is not a digit,
+sign, decimal point, or upper- or lower-case @samp{e}) forward and
+backward to delimit the formula.
+
+When you enable Embedded mode for a formula, Calc reads the text
+between the delimiters and tries to interpret it as a Calc formula.
+Calc can generally identify @TeX{} formulas and
+Big-style formulas even if the language mode is wrong. If Calc
+can't make sense of the formula, it beeps and refuses to enter
+Embedded mode. But if the current language is wrong, Calc can
+sometimes parse the formula successfully (but incorrectly);
+for example, the C expression @samp{atan(a[1])} can be parsed
+in Normal language mode, but the @code{atan} won't correspond to
+the built-in @code{arctan} function, and the @samp{a[1]} will be
+interpreted as @samp{a} times the vector @samp{[1]}!
+
+If you press @kbd{C-x * e} or @kbd{C-x * w} to activate an embedded
+formula which is blank, say with the cursor on the space between
+the two delimiters @samp{$ $}, Calc will immediately prompt for
+an algebraic entry.
+
+Only one formula in one buffer can be enabled at a time. If you
+move to another area of the current buffer and give Calc commands,
+Calc turns Embedded mode off for the old formula and then tries
+to restart Embedded mode at the new position. Other buffers are
+not affected by Embedded mode.
+
+When Embedded mode begins, Calc pushes the current formula onto
+the stack. No Calc stack window is created; however, Calc copies
+the top-of-stack position into the original buffer at all times.
+You can create a Calc window by hand with @kbd{C-x * o} if you
+find you need to see the entire stack.
+
+For example, typing @kbd{C-x * e} while somewhere in the formula
address@hidden>2} in the following line enables Embedded mode on that
+inequality:
+
address@hidden
+We define $F_n = F_(n-1)+F_(n-2)$ for all $n>2$.
address@hidden example
+
address@hidden
+The formula @expr{n>2} will be pushed onto the Calc stack, and
+the top of stack will be copied back into the editing buffer.
+This means that spaces will appear around the @samp{>} symbol
+to match Calc's usual display style:
+
address@hidden
+We define $F_n = F_(n-1)+F_(n-2)$ for all $n > 2$.
address@hidden example
+
address@hidden
+No spaces have appeared around the @samp{+} sign because it's
+in a different formula, one which we have not yet touched with
+Embedded mode.
+
+Now that Embedded mode is enabled, keys you type in this buffer
+are interpreted as Calc commands. At this point we might use
+the ``commute'' command @kbd{j C} to reverse the inequality.
+This is a selection-based command for which we first need to
+move the cursor onto the operator (@samp{>} in this case) that
+needs to be commuted.
+
address@hidden
+We define $F_n = F_(n-1)+F_(n-2)$ for all $2 < n$.
address@hidden example
+
+The @kbd{C-x * o} command is a useful way to open a Calc window
+without actually selecting that window. Giving this command
+verifies that @samp{2 < n} is also on the Calc stack. Typing
address@hidden @key{RET}} would produce:
+
address@hidden
+We define $F_n = F_(n-1)+F_(n-2)$ for all $17$.
address@hidden example
+
address@hidden
+with @samp{2 < n} and @samp{17} on the stack; typing @key{TAB}
+at this point will exchange the two stack values and restore
address@hidden < n} to the embedded formula. Even though you can't
+normally see the stack in Embedded mode, it is still there and
+it still operates in the same way. But, as with old-fashioned
+RPN calculators, you can only see the value at the top of the
+stack at any given time (unless you use @kbd{C-x * o}).
+
+Typing @kbd{C-x * e} again turns Embedded mode off. The Calc
+window reveals that the formula @address@hidden < n}} is automatically
+removed from the stack, but the @samp{17} is not. Entering
+Embedded mode always pushes one thing onto the stack, and
+leaving Embedded mode always removes one thing. Anything else
+that happens on the stack is entirely your business as far as
+Embedded mode is concerned.
+
+If you press @kbd{C-x * e} in the wrong place by accident, it is
+possible that Calc will be able to parse the nearby text as a
+formula and will mangle that text in an attempt to redisplay it
+``properly'' in the current language mode. If this happens,
+press @kbd{C-x * e} again to exit Embedded mode, then give the
+regular Emacs ``undo'' command (@kbd{C-_} or @kbd{C-x u}) to put
+the text back the way it was before Calc edited it. Note that Calc's
+own Undo command (typed before you turn Embedded mode back off)
+will not do you any good, because as far as Calc is concerned
+you haven't done anything with this formula yet.
+
address@hidden More About Embedded Mode, Assignments in Embedded Mode, Basic
Embedded Mode, Embedded Mode
address@hidden More About Embedded Mode
+
address@hidden
+When Embedded mode ``activates'' a formula, i.e., when it examines
+the formula for the first time since the buffer was created or
+loaded, Calc tries to sense the language in which the formula was
+written. If the formula contains any address@hidden @samp{\} sequences,
+it is parsed (i.e., read) in address@hidden mode. If the formula appears to
+be written in multi-line Big mode, it is parsed in Big mode. Otherwise,
+it is parsed according to the current language mode.
+
+Note that Calc does not change the current language mode according
+the formula it reads in. Even though it can read a address@hidden formula when
+not in address@hidden mode, it will immediately rewrite this formula using
+whatever language mode is in effect.
+
address@hidden
+\bigskip
address@hidden tex
+
address@hidden d p
address@hidden calc-show-plain
+Calc's parser is unable to read certain kinds of formulas. For
+example, with @kbd{v ]} (@code{calc-matrix-brackets}) you can
+specify matrix display styles which the parser is unable to
+recognize as matrices. The @kbd{d p} (@code{calc-show-plain})
+command turns on a mode in which a ``plain'' version of a
+formula is placed in front of the fully-formatted version.
+When Calc reads a formula that has such a plain version in
+front, it reads the plain version and ignores the formatted
+version.
+
+Plain formulas are preceded and followed by @samp{%%%} signs
+by default. This notation has the advantage that the @samp{%}
+character begins a comment in @TeX{} and address@hidden, so if your formula is
+embedded in a @TeX{} or address@hidden document its plain version will be
+invisible in the final printed copy. Certain major modes have different
+delimiters to ensure that the ``plain'' version will be
+in a comment for those modes, also.
+See @ref{Customizing Embedded Mode} to see how to change the ``plain''
+formula delimiters.
+
+There are several notations which Calc's parser for ``big''
+formatted formulas can't yet recognize. In particular, it can't
+read the large symbols for @code{sum}, @code{prod}, and @code{integ},
+and it can't handle @samp{=>} with the righthand argument omitted.
+Also, Calc won't recognize special formats you have defined with
+the @kbd{Z C} command (@pxref{User-Defined Compositions}). In
+these cases it is important to use ``plain'' mode to make sure
+Calc will be able to read your formula later.
+
+Another example where ``plain'' mode is important is if you have
+specified a float mode with few digits of precision. Normally
+any digits that are computed but not displayed will simply be
+lost when you save and re-load your embedded buffer, but ``plain''
+mode allows you to make sure that the complete number is present
+in the file as well as the rounded-down number.
+
address@hidden
+\bigskip
address@hidden tex
+
+Embedded buffers remember active formulas for as long as they
+exist in Emacs memory. Suppose you have an embedded formula
+which is @cpi{} to the normal 12 decimal places, and then
+type @address@hidden 5 d n}} to display only five decimal places.
+If you then type @kbd{d n}, all 12 places reappear because the
+full number is still there on the Calc stack. More surprisingly,
+even if you exit Embedded mode and later re-enter it for that
+formula, typing @kbd{d n} will restore all 12 places because
+each buffer remembers all its active formulas. However, if you
+save the buffer in a file and reload it in a new Emacs session,
+all non-displayed digits will have been lost unless you used
+``plain'' mode.
+
address@hidden
+\bigskip
address@hidden tex
+
+In some applications of Embedded mode, you will want to have a
+sequence of copies of a formula that show its evolution as you
+work on it. For example, you might want to have a sequence
+like this in your file (elaborating here on the example from
+the ``Getting Started'' chapter):
+
address@hidden
+The derivative of
+
+ ln(ln(x))
+
+is
+
+ @r{(the derivative of }ln(ln(x))@r{)}
+
+whose value at x = 2 is
+
+ @r{(the value)}
+
+and at x = 3 is
+
+ @r{(the value)}
address@hidden smallexample
+
address@hidden C-x * d
address@hidden calc-embedded-duplicate
+The @kbd{C-x * d} (@code{calc-embedded-duplicate}) command is a
+handy way to make sequences like this. If you type @kbd{C-x * d},
+the formula under the cursor (which may or may not have Embedded
+mode enabled for it at the time) is copied immediately below and
+Embedded mode is then enabled for that copy.
+
+For this example, you would start with just
+
address@hidden
+The derivative of
+
+ ln(ln(x))
address@hidden smallexample
+
address@hidden
+and press @kbd{C-x * d} with the cursor on this formula. The result
+is
+
address@hidden
+The derivative of
+
+ ln(ln(x))
+
+
+ ln(ln(x))
address@hidden smallexample
+
address@hidden
+with the second copy of the formula enabled in Embedded mode.
+You can now press @kbd{a d x @key{RET}} to take the derivative, and
address@hidden * d C-x * d} to make two more copies of the derivative.
+To complete the computations, type @kbd{3 s l x @key{RET}} to evaluate
+the last formula, then move up to the second-to-last formula
+and type @kbd{2 s l x @key{RET}}.
+
+Finally, you would want to press @kbd{C-x * e} to exit Embedded
+mode, then go up and insert the necessary text in between the
+various formulas and numbers.
+
address@hidden
+\bigskip
address@hidden tex
+
address@hidden C-x * f
address@hidden C-x * '
address@hidden calc-embedded-new-formula
+The @kbd{C-x * f} (@code{calc-embedded-new-formula}) command
+creates a new embedded formula at the current point. It inserts
+some default delimiters, which are usually just blank lines,
+and then does an algebraic entry to get the formula (which is
+then enabled for Embedded mode). This is just shorthand for
+typing the delimiters yourself, positioning the cursor between
+the new delimiters, and pressing @kbd{C-x * e}. The key sequence
address@hidden * '} is equivalent to @kbd{C-x * f}.
+
address@hidden C-x * n
address@hidden C-x * p
address@hidden calc-embedded-next
address@hidden calc-embedded-previous
+The @kbd{C-x * n} (@code{calc-embedded-next}) and @kbd{C-x * p}
+(@code{calc-embedded-previous}) commands move the cursor to the
+next or previous active embedded formula in the buffer. They
+can take positive or negative prefix arguments to move by several
+formulas. Note that these commands do not actually examine the
+text of the buffer looking for formulas; they only see formulas
+which have previously been activated in Embedded mode. In fact,
address@hidden * n} and @kbd{C-x * p} are a useful way to tell which
+embedded formulas are currently active. Also, note that these
+commands do not enable Embedded mode on the next or previous
+formula, they just move the cursor.
+
address@hidden C-x * `
address@hidden calc-embedded-edit
+The @kbd{C-x * `} (@code{calc-embedded-edit}) command edits the
+embedded formula at the current point as if by @kbd{`} (@code{calc-edit}).
+Embedded mode does not have to be enabled for this to work. Press
address@hidden C-c} to finish the edit, or @kbd{C-x k} to cancel.
+
address@hidden Assignments in Embedded Mode, Mode Settings in Embedded Mode,
More About Embedded Mode, Embedded Mode
address@hidden Assignments in Embedded Mode
+
address@hidden
+The @samp{:=} (assignment) and @samp{=>} (``evaluates-to'') operators
+are especially useful in Embedded mode. They allow you to make
+a definition in one formula, then refer to that definition in
+other formulas embedded in the same buffer.
+
+An embedded formula which is an assignment to a variable, as in
+
address@hidden
+foo := 5
address@hidden example
+
address@hidden
+records @expr{5} as the stored value of @code{foo} for the
+purposes of Embedded mode operations in the current buffer. It
+does @emph{not} actually store @expr{5} as the ``global'' value
+of @code{foo}, however. Regular Calc operations, and Embedded
+formulas in other buffers, will not see this assignment.
+
+One way to use this assigned value is simply to create an
+Embedded formula elsewhere that refers to @code{foo}, and to press
address@hidden in that formula. However, this permanently replaces the
address@hidden in the formula with its current value. More interesting
+is to use @samp{=>} elsewhere:
+
address@hidden
+foo + 7 => 12
address@hidden example
+
address@hidden Operator}, for a general discussion of @samp{=>}.
+
+If you move back and change the assignment to @code{foo}, any
address@hidden>} formulas which refer to it are automatically updated.
+
address@hidden
+foo := 17
+
+foo + 7 => 24
address@hidden example
+
+The obvious question then is, @emph{how} can one easily change the
+assignment to @code{foo}? If you simply select the formula in
+Embedded mode and type 17, the assignment itself will be replaced
+by the 17. The effect on the other formula will be that the
+variable @code{foo} becomes unassigned:
+
address@hidden
+17
+
+foo + 7 => foo + 7
address@hidden example
+
+The right thing to do is first to use a selection command (@kbd{j 2}
+will do the trick) to select the righthand side of the assignment.
+Then, @kbd{17 @key{TAB} @key{DEL}} will swap the 17 into place
(@pxref{Selecting
+Subformulas}, to see how this works).
+
address@hidden C-x * j
address@hidden calc-embedded-select
+The @kbd{C-x * j} (@code{calc-embedded-select}) command provides an
+easy way to operate on assignments. It is just like @kbd{C-x * e},
+except that if the enabled formula is an assignment, it uses
address@hidden 2} to select the righthand side. If the enabled formula
+is an evaluates-to, it uses @kbd{j 1} to select the lefthand side.
+A formula can also be a combination of both:
+
address@hidden
+bar := foo + 3 => 20
address@hidden example
+
address@hidden
+in which case @kbd{C-x * j} will select the middle part (@samp{foo + 3}).
+
+The formula is automatically deselected when you leave Embedded
+mode.
+
address@hidden C-x * u
address@hidden calc-embedded-update-formula
+Another way to change the assignment to @code{foo} would simply be
+to edit the number using regular Emacs editing rather than Embedded
+mode. Then, we have to find a way to get Embedded mode to notice
+the change. The @kbd{C-x * u} (@code{calc-embedded-update-formula})
+command is a convenient way to do this.
+
address@hidden
+foo := 6
+
+foo + 7 => 13
address@hidden example
+
+Pressing @kbd{C-x * u} is much like pressing @kbd{C-x * e = C-x * e}, that
+is, temporarily enabling Embedded mode for the formula under the
+cursor and then evaluating it with @kbd{=}. But @kbd{C-x * u} does
+not actually use @kbd{C-x * e}, and in fact another formula somewhere
+else can be enabled in Embedded mode while you use @kbd{C-x * u} and
+that formula will not be disturbed.
+
+With a numeric prefix argument, @kbd{C-x * u} updates all active
address@hidden>} formulas in the buffer. Formulas which have not yet
+been activated in Embedded mode, and formulas which do not have
address@hidden>} as their top-level operator, are not affected by this.
+(This is useful only if you have used @kbd{m C}; see below.)
+
+With a plain @kbd{C-u} prefix, @kbd{C-u C-x * u} updates only in the
+region between mark and point rather than in the whole buffer.
+
address@hidden * u} is also a handy way to activate a formula, such as an
address@hidden>} formula that has freshly been typed in or loaded from a
+file.
+
address@hidden C-x * a
address@hidden calc-embedded-activate
+The @kbd{C-x * a} (@code{calc-embedded-activate}) command scans
+through the current buffer and activates all embedded formulas
+that contain @samp{:=} or @samp{=>} symbols. This does not mean
+that Embedded mode is actually turned on, but only that the
+formulas' positions are registered with Embedded mode so that
+the @samp{=>} values can be properly updated as assignments are
+changed.
+
+It is a good idea to type @kbd{C-x * a} right after loading a file
+that uses embedded @samp{=>} operators. Emacs includes a nifty
+``buffer-local variables'' feature that you can use to do this
+automatically. The idea is to place near the end of your file
+a few lines that look like this:
+
address@hidden
+--- Local Variables: ---
+--- eval:(calc-embedded-activate) ---
+--- End: ---
address@hidden example
+
address@hidden
+where the leading and trailing @samp{---} can be replaced by
+any suitable strings (which must be the same on all three lines)
+or omitted altogether; in a @TeX{} or address@hidden file, @samp{%} would be a
good
+leading string and no trailing string would be necessary. In a
+C program, @samp{/*} and @samp{*/} would be good leading and
+trailing strings.
+
+When Emacs loads a file into memory, it checks for a Local Variables
+section like this one at the end of the file. If it finds this
+section, it does the specified things (in this case, running
address@hidden * a} automatically) before editing of the file begins.
+The Local Variables section must be within 3000 characters of the
+end of the file for Emacs to find it, and it must be in the last
+page of the file if the file has any page separators.
address@hidden Variables, , Local Variables in Files, emacs, the
+Emacs manual}.
+
+Note that @kbd{C-x * a} does not update the formulas it finds.
+To do this, type, say, @kbd{M-1 C-x * u} after @address@hidden * a}}.
+Generally this should not be a problem, though, because the
+formulas will have been up-to-date already when the file was
+saved.
+
+Normally, @kbd{C-x * a} activates all the formulas it finds, but
+any previous active formulas remain active as well. With a
+positive numeric prefix argument, @kbd{C-x * a} first deactivates
+all current active formulas, then actives the ones it finds in
+its scan of the buffer. With a negative prefix argument,
address@hidden * a} simply deactivates all formulas.
+
+Embedded mode has two symbols, @samp{Active} and @samp{~Active},
+which it puts next to the major mode name in a buffer's mode line.
+It puts @samp{Active} if it has reason to believe that all
+formulas in the buffer are active, because you have typed @kbd{C-x * a}
+and Calc has not since had to deactivate any formulas (which can
+happen if Calc goes to update an @samp{=>} formula somewhere because
+a variable changed, and finds that the formula is no longer there
+due to some kind of editing outside of Embedded mode). Calc puts
address@hidden in the mode line if some, but probably not all,
+formulas in the buffer are active. This happens if you activate
+a few formulas one at a time but never use @kbd{C-x * a}, or if you
+used @kbd{C-x * a} but then Calc had to deactivate a formula
+because it lost track of it. If neither of these symbols appears
+in the mode line, no embedded formulas are active in the buffer
+(e.g., before Embedded mode has been used, or after a @kbd{M-- C-x * a}).
+
+Embedded formulas can refer to assignments both before and after them
+in the buffer. If there are several assignments to a variable, the
+nearest preceding assignment is used if there is one, otherwise the
+following assignment is used.
+
address@hidden
+x => 1
+
+x := 1
+
+x => 1
+
+x := 2
+
+x => 2
address@hidden example
+
+As well as simple variables, you can also assign to subscript
+expressions of the form @address@hidden@var{number}} (as in
address@hidden), or @address@hidden@var{var}} (as in @code{x_max}).
+Assignments to other kinds of objects can be represented by Calc,
+but the automatic linkage between assignments and references works
+only for plain variables and these two kinds of subscript expressions.
+
+If there are no assignments to a given variable, the global
+stored value for the variable is used (@pxref{Storing Variables}),
+or, if no value is stored, the variable is left in symbolic form.
+Note that global stored values will be lost when the file is saved
+and loaded in a later Emacs session, unless you have used the
address@hidden p} (@code{calc-permanent-variable}) command to save them;
address@hidden on Variables}.
+
+The @kbd{m C} (@code{calc-auto-recompute}) command turns automatic
+recomputation of @samp{=>} forms on and off. If you turn automatic
+recomputation off, you will have to use @kbd{C-x * u} to update these
+formulas manually after an assignment has been changed. If you
+plan to change several assignments at once, it may be more efficient
+to type @kbd{m C}, change all the assignments, then use @kbd{M-1 C-x * u}
+to update the entire buffer afterwards. The @kbd{m C} command also
+controls @samp{=>} formulas on the stack; @pxref{Evaluates-To
+Operator}. When you turn automatic recomputation back on, the
+stack will be updated but the Embedded buffer will not; you must
+use @kbd{C-x * u} to update the buffer by hand.
+
address@hidden Mode Settings in Embedded Mode, Customizing Embedded Mode,
Assignments in Embedded Mode, Embedded Mode
address@hidden Mode Settings in Embedded Mode
+
address@hidden m e
address@hidden calc-embedded-preserve-modes
address@hidden
+The mode settings can be changed while Calc is in embedded mode, but
+by default they will revert to their original values when embedded mode
+is ended. However, the modes saved when the mode-recording mode is
address@hidden (see below) and the modes in effect when the @kbd{m e}
+(@code{calc-embedded-preserve-modes}) command is given
+will be preserved when embedded mode is ended.
+
+Embedded mode has a rather complicated mechanism for handling mode
+settings in Embedded formulas. It is possible to put annotations
+in the file that specify mode settings either global to the entire
+file or local to a particular formula or formulas. In the latter
+case, different modes can be specified for use when a formula
+is the enabled Embedded mode formula.
+
+When you give any mode-setting command, like @kbd{m f} (for Fraction
+mode) or @kbd{d s} (for scientific notation), Embedded mode adds
+a line like the following one to the file just before the opening
+delimiter of the formula.
+
address@hidden
+% [calc-mode: fractions: t]
+% [calc-mode: float-format: (sci 0)]
address@hidden example
+
+When Calc interprets an embedded formula, it scans the text before
+the formula for mode-setting annotations like these and sets the
+Calc buffer to match these modes. Modes not explicitly described
+in the file are not changed. Calc scans all the way to the top of
+the file, or up to a line of the form
+
address@hidden
+% [calc-defaults]
address@hidden example
+
address@hidden
+which you can insert at strategic places in the file if this backward
+scan is getting too slow, or just to provide a barrier between one
+``zone'' of mode settings and another.
+
+If the file contains several annotations for the same mode, the
+closest one before the formula is used. Annotations after the
+formula are never used (except for global annotations, described
+below).
+
+The scan does not look for the leading @samp{% }, only for the
+square brackets and the text they enclose. In fact, the leading
+characters are different for different major modes. You can edit the
+mode annotations to a style that works better in context if you wish.
address@hidden Embedded Mode}, to see how to change the style
+that Calc uses when it generates the annotations. You can write
+mode annotations into the file yourself if you know the syntax;
+the easiest way to find the syntax for a given mode is to let
+Calc write the annotation for it once and see what it does.
+
+If you give a mode-changing command for a mode that already has
+a suitable annotation just above the current formula, Calc will
+modify that annotation rather than generating a new, conflicting
+one.
+
+Mode annotations have three parts, separated by colons. (Spaces
+after the colons are optional.) The first identifies the kind
+of mode setting, the second is a name for the mode itself, and
+the third is the value in the form of a Lisp symbol, number,
+or list. Annotations with unrecognizable text in the first or
+second parts are ignored. The third part is not checked to make
+sure the value is of a valid type or range; if you write an
+annotation by hand, be sure to give a proper value or results
+will be unpredictable. Mode-setting annotations are case-sensitive.
+
+While Embedded mode is enabled, the word @code{Local} appears in
+the mode line. This is to show that mode setting commands generate
+annotations that are ``local'' to the current formula or set of
+formulas. The @kbd{m R} (@code{calc-mode-record-mode}) command
+causes Calc to generate different kinds of annotations. Pressing
address@hidden R} repeatedly cycles through the possible modes.
+
address@hidden and @code{LocPerm} modes generate annotations
+that look like this, respectively:
+
address@hidden
+% [calc-edit-mode: float-format: (sci 0)]
+% [calc-perm-mode: float-format: (sci 5)]
address@hidden example
+
+The first kind of annotation will be used only while a formula
+is enabled in Embedded mode. The second kind will be used only
+when the formula is @emph{not} enabled. (Whether the formula
+is ``active'' or not, i.e., whether Calc has seen this formula
+yet, is not relevant here.)
+
address@hidden mode generates an annotation like this at the end
+of the file:
+
address@hidden
+% [calc-global-mode: fractions t]
address@hidden example
+
+Global mode annotations affect all formulas throughout the file,
+and may appear anywhere in the file. This allows you to tuck your
+mode annotations somewhere out of the way, say, on a new page of
+the file, as long as those mode settings are suitable for all
+formulas in the file.
+
+Enabling a formula with @kbd{C-x * e} causes a fresh scan for local
+mode annotations; you will have to use this after adding annotations
+above a formula by hand to get the formula to notice them. Updating
+a formula with @kbd{C-x * u} will also re-scan the local modes, but
+global modes are only re-scanned by @kbd{C-x * a}.
+
+Another way that modes can get out of date is if you add a local
+mode annotation to a formula that has another formula after it.
+In this example, we have used the @kbd{d s} command while the
+first of the two embedded formulas is active. But the second
+formula has not changed its style to match, even though by the
+rules of reading annotations the @samp{(sci 0)} applies to it, too.
+
address@hidden
+% [calc-mode: float-format: (sci 0)]
+1.23e2
+
+456.
address@hidden example
+
+We would have to go down to the other formula and press @kbd{C-x * u}
+on it in order to get it to notice the new annotation.
+
+Two more mode-recording modes selectable by @kbd{m R} are available
+which are also available outside of Embedded mode.
+(@pxref{General Mode Commands}.) They are @code{Save}, in which mode
+settings are recorded permanently in your Calc init file (the file given
+by the variable @code{calc-settings-file}, typically @file{~/.calc.el})
+rather than by annotating the current document, and no-recording
+mode (where there is no symbol like @code{Save} or @code{Local} in
+the mode line), in which mode-changing commands do not leave any
+annotations at all.
+
+When Embedded mode is not enabled, mode-recording modes except
+for @code{Save} have no effect.
+
address@hidden Customizing Embedded Mode, , Mode Settings in Embedded Mode,
Embedded Mode
address@hidden Customizing Embedded Mode
+
address@hidden
+You can modify Embedded mode's behavior by setting various Lisp
+variables described here. These variables are customizable
+(@pxref{Customizing Calc}), or you can use @kbd{M-x set-variable}
+or @kbd{M-x edit-options} to adjust a variable on the fly.
+(Another possibility would be to use a file-local variable annotation at
+the end of the file;
address@hidden Variables, , Local Variables in Files, emacs, the Emacs manual}.)
+Many of the variables given mentioned here can be set to depend on the
+major mode of the editing buffer (@pxref{Customizing Calc}).
+
address@hidden calc-embedded-open-formula
+The @code{calc-embedded-open-formula} variable holds a regular
+expression for the opening delimiter of a formula. @xref{Regexp Search,
+, Regular Expression Search, emacs, the Emacs manual}, to see
+how regular expressions work. Basically, a regular expression is a
+pattern that Calc can search for. A regular expression that considers
+blank lines, @samp{$}, and @samp{$$} to be opening delimiters is
address@hidden"\\`\\|^\n\\|\\$\\$?"}. Just in case the meaning of this
+regular expression is not completely plain, let's go through it
+in detail.
+
+The surrounding @samp{" "} marks quote the text between them as a
+Lisp string. If you left them off, @code{set-variable} or
address@hidden would try to read the regular expression as a
+Lisp program.
+
+The most obvious property of this regular expression is that it
+contains indecently many backslashes. There are actually two levels
+of backslash usage going on here. First, when Lisp reads a quoted
+string, all pairs of characters beginning with a backslash are
+interpreted as special characters. Here, @code{\n} changes to a
+new-line character, and @code{\\} changes to a single backslash.
+So the actual regular expression seen by Calc is
address@hidden|^ @r{(newline)} \|\$\$?}.
+
+Regular expressions also consider pairs beginning with backslash
+to have special meanings. Sometimes the backslash is used to quote
+a character that otherwise would have a special meaning in a regular
+expression, like @samp{$}, which normally means ``end-of-line,''
+or @samp{?}, which means that the preceding item is optional. So
address@hidden matches either one or two dollar signs.
+
+The other codes in this regular expression are @samp{^}, which matches
+``beginning-of-line,'' @samp{\|}, which means ``or,'' and @samp{\`},
+which matches ``beginning-of-buffer.'' So the whole pattern means
+that a formula begins at the beginning of the buffer, or on a newline
+that occurs at the beginning of a line (i.e., a blank line), or at
+one or two dollar signs.
+
+The default value of @code{calc-embedded-open-formula} looks just
+like this example, with several more alternatives added on to
+recognize various other common kinds of delimiters.
+
+By the way, the reason to use @samp{^\n} rather than @samp{^$}
+or @samp{\n\n}, which also would appear to match blank lines,
+is that the former expression actually ``consumes'' only one
+newline character as @emph{part of} the delimiter, whereas the
+latter expressions consume zero or two newlines, respectively.
+The former choice gives the most natural behavior when Calc
+must operate on a whole formula including its delimiters.
+
+See the Emacs manual for complete details on regular expressions.
+But just for your convenience, here is a list of all characters
+which must be quoted with backslash (like @samp{\$}) to avoid
+some special interpretation: @samp{. * + ? [ ] ^ $ \}. (Note
+the backslash in this list; for example, to match @samp{\[} you
+must use @code{"\\\\\\["}. An exercise for the reader is to
+account for each of these six backslashes!)
+
address@hidden calc-embedded-close-formula
+The @code{calc-embedded-close-formula} variable holds a regular
+expression for the closing delimiter of a formula. A closing
+regular expression to match the above example would be
address@hidden"\\'\\|\n$\\|\\$\\$?"}. This is almost the same as the
+other one, except it now uses @samp{\'} (``end-of-buffer'') and
address@hidden (newline occurring at end of line, yet another way
+of describing a blank line that is more appropriate for this
+case).
+
address@hidden calc-embedded-open-word
address@hidden calc-embedded-close-word
+The @code{calc-embedded-open-word} and @code{calc-embedded-close-word}
+variables are similar expressions used when you type @kbd{C-x * w}
+instead of @kbd{C-x * e} to enable Embedded mode.
+
address@hidden calc-embedded-open-plain
+The @code{calc-embedded-open-plain} variable is a string which
+begins a ``plain'' formula written in front of the formatted
+formula when @kbd{d p} mode is turned on. Note that this is an
+actual string, not a regular expression, because Calc must be able
+to write this string into a buffer as well as to recognize it.
+The default string is @code{"%%% "} (note the trailing space), but may
+be different for certain major modes.
+
address@hidden calc-embedded-close-plain
+The @code{calc-embedded-close-plain} variable is a string which
+ends a ``plain'' formula. The default is @code{" %%%\n"}, but may be
+different for different major modes. Without
+the trailing newline here, the first line of a Big mode formula
+that followed might be shifted over with respect to the other lines.
+
address@hidden calc-embedded-open-new-formula
+The @code{calc-embedded-open-new-formula} variable is a string
+which is inserted at the front of a new formula when you type
address@hidden * f}. Its default value is @code{"\n\n"}. If this
+string begins with a newline character and the @kbd{C-x * f} is
+typed at the beginning of a line, @kbd{C-x * f} will skip this
+first newline to avoid introducing unnecessary blank lines in
+the file.
+
address@hidden calc-embedded-close-new-formula
+The @code{calc-embedded-close-new-formula} variable is the corresponding
+string which is inserted at the end of a new formula. Its default
+value is also @code{"\n\n"}. The final newline is omitted by
address@hidden@kbd{C-x * f}} if typed at the end of a line. (It follows that if
address@hidden * f} is typed on a blank line, both a leading opening
+newline and a trailing closing newline are omitted.)
+
address@hidden calc-embedded-announce-formula
+The @code{calc-embedded-announce-formula} variable is a regular
+expression which is sure to be followed by an embedded formula.
+The @kbd{C-x * a} command searches for this pattern as well as for
address@hidden>} and @samp{:=} operators. Note that @kbd{C-x * a} will
+not activate just anything surrounded by formula delimiters; after
+all, blank lines are considered formula delimiters by default!
+But if your language includes a delimiter which can only occur
+actually in front of a formula, you can take advantage of it here.
+The default pattern is @code{"%Embed\n\\(% .*\n\\)*"}, but may be
+different for different major modes.
+This pattern will check for @samp{%Embed} followed by any number of
+lines beginning with @samp{%} and a space. This last is important to
+make Calc consider mode annotations part of the pattern, so that the
+formula's opening delimiter really is sure to follow the pattern.
+
address@hidden calc-embedded-open-mode
+The @code{calc-embedded-open-mode} variable is a string (not a
+regular expression) which should precede a mode annotation.
+Calc never scans for this string; Calc always looks for the
+annotation itself. But this is the string that is inserted before
+the opening bracket when Calc adds an annotation on its own.
+The default is @code{"% "}, but may be different for different major
+modes.
+
address@hidden calc-embedded-close-mode
+The @code{calc-embedded-close-mode} variable is a string which
+follows a mode annotation written by Calc. Its default value
+is simply a newline, @code{"\n"}, but may be different for different
+major modes. If you change this, it is a good idea still to end with a
+newline so that mode annotations will appear on lines by themselves.
+
address@hidden Programming, Copying, Embedded Mode, Top
address@hidden Programming
+
address@hidden
+There are several ways to ``program'' the Emacs Calculator, depending
+on the nature of the problem you need to solve.
+
address@hidden
address@hidden
address@hidden macros} allow you to record a sequence of keystrokes
+and play them back at a later time. This is just the standard Emacs
+keyboard macro mechanism, dressed up with a few more features such
+as loops and conditionals.
+
address@hidden
address@hidden definitions} allow you to use any formula to define a
+new function. This function can then be used in algebraic formulas or
+as an interactive command.
+
address@hidden
address@hidden rules} are discussed in the section on algebra commands.
address@hidden Rules}. If you put your rewrite rules in the variable
address@hidden, they will be applied automatically to all Calc
+results in just the same way as an internal ``rule'' is applied to
+evaluate @samp{sqrt(9)} to 3 and so on. @xref{Automatic Rewrites}.
+
address@hidden
address@hidden is the programming language that Calc (and most of Emacs)
+is written in. If the above techniques aren't powerful enough, you
+can write Lisp functions to do anything that built-in Calc commands
+can do. Lisp code is also somewhat faster than keyboard macros or
+rewrite rules.
address@hidden enumerate
+
address@hidden z
+Programming features are available through the @kbd{z} and @kbd{Z}
+prefix keys. New commands that you define are two-key sequences
+beginning with @kbd{z}. Commands for managing these definitions
+use the address@hidden prefix. (The @kbd{Z T} (@code{calc-timing})
+command is described elsewhere; @pxref{Troubleshooting Commands}.
+The @kbd{Z C} (@code{calc-user-define-composition}) command is also
+described elsewhere; @pxref{User-Defined Compositions}.)
+
address@hidden
+* Creating User Keys::
+* Keyboard Macros::
+* Invocation Macros::
+* Algebraic Definitions::
+* Lisp Definitions::
address@hidden menu
+
address@hidden Creating User Keys, Keyboard Macros, Programming, Programming
address@hidden Creating User Keys
+
address@hidden
address@hidden Z D
address@hidden calc-user-define
+Any Calculator command may be bound to a key using the @kbd{Z D}
+(@code{calc-user-define}) command. Actually, it is bound to a two-key
+sequence beginning with the lower-case @kbd{z} prefix.
+
+The @kbd{Z D} command first prompts for the key to define. For example,
+press @kbd{Z D a} to define the new key sequence @kbd{z a}. You are then
+prompted for the name of the Calculator command that this key should
+run. For example, the @code{calc-sincos} command is not normally
+available on a key. Typing @kbd{Z D s sincos @key{RET}} programs the
address@hidden s} key sequence to run @code{calc-sincos}. This definition will
remain
+in effect for the rest of this Emacs session, or until you redefine
address@hidden s} to be something else.
+
+You can actually bind any Emacs command to a @kbd{z} key sequence by
+backspacing over the @samp{calc-} when you are prompted for the command name.
+
+As with any other prefix key, you can type @kbd{z ?} to see a list of
+all the two-key sequences you have defined that start with @kbd{z}.
+Initially, no @kbd{z} sequences (except @kbd{z ?} itself) are defined.
+
+User keys are typically letters, but may in fact be any key.
+(@key{META}-keys are not permitted, nor are a terminal's special
+function keys which generate multi-character sequences when pressed.)
+You can define different commands on the shifted and unshifted versions
+of a letter if you wish.
+
address@hidden Z U
address@hidden calc-user-undefine
+The @kbd{Z U} (@code{calc-user-undefine}) command unbinds a user key.
+For example, the key sequence @kbd{Z U s} will undefine the @code{sincos}
+key we defined above.
+
address@hidden Z P
address@hidden calc-user-define-permanent
address@hidden Storing user definitions
address@hidden Permanent user definitions
address@hidden Calc init file, user-defined commands
+The @kbd{Z P} (@code{calc-user-define-permanent}) command makes a key
+binding permanent so that it will remain in effect even in future Emacs
+sessions. (It does this by adding a suitable bit of Lisp code into
+your Calc init file; that is, the file given by the variable
address@hidden, typically @file{~/.calc.el}.) For example,
address@hidden P s} would register our @code{sincos} command permanently. If
+you later wish to unregister this command you must edit your Calc init
+file by hand. (@xref{General Mode Commands}, for a way to tell Calc to
+use a different file for the Calc init file.)
+
+The @kbd{Z P} command also saves the user definition, if any, for the
+command bound to the key. After @kbd{Z F} and @kbd{Z C}, a given user
+key could invoke a command, which in turn calls an algebraic function,
+which might have one or more special display formats. A single @kbd{Z P}
+command will save all of these definitions.
+To save an algebraic function, type @kbd{'} (the apostrophe)
+when prompted for a key, and type the function name. To save a command
+without its key binding, type @kbd{M-x} and enter a function name. (The
address@hidden prefix will automatically be inserted for you.)
+(If the command you give implies a function, the function will be saved,
+and if the function has any display formats, those will be saved, but
+not the other way around: Saving a function will not save any commands
+or key bindings associated with the function.)
+
address@hidden Z E
address@hidden calc-user-define-edit
address@hidden Editing user definitions
+The @kbd{Z E} (@code{calc-user-define-edit}) command edits the definition
+of a user key. This works for keys that have been defined by either
+keyboard macros or formulas; further details are contained in the relevant
+following sections.
+
address@hidden Keyboard Macros, Invocation Macros, Creating User Keys,
Programming
address@hidden Programming with Keyboard Macros
+
address@hidden
address@hidden X
address@hidden Programming with keyboard macros
address@hidden Keyboard macros
+The easiest way to ``program'' the Emacs Calculator is to use standard
+keyboard macros. Press @address@hidden (}} to begin recording a macro. From
+this point on, keystrokes you type will be saved away as well as
+performing their usual functions. Press @kbd{C-x )} to end recording.
+Press address@hidden (or the standard Emacs key sequence @kbd{C-x e}) to
+execute your keyboard macro by replaying the recorded keystrokes.
address@hidden Macros, , , emacs, the Emacs Manual}, for further
+information.
+
+When you use @kbd{X} to invoke a keyboard macro, the entire macro is
+treated as a single command by the undo and trail features. The stack
+display buffer is not updated during macro execution, but is instead
+fixed up once the macro completes. Thus, commands defined with keyboard
+macros are convenient and efficient. The @kbd{C-x e} command, on the
+other hand, invokes the keyboard macro with no special treatment: Each
+command in the macro will record its own undo information and trail entry,
+and update the stack buffer accordingly. If your macro uses features
+outside of Calc's control to operate on the contents of the Calc stack
+buffer, or if it includes Undo, Redo, or last-arguments commands, you
+must use @kbd{C-x e} to make sure the buffer and undo list are up-to-date
+at all times. You could also consider using @kbd{K} (@code{calc-keep-args})
+instead of @address@hidden (@code{calc-last-args}).
+
+Calc extends the standard Emacs keyboard macros in several ways.
+Keyboard macros can be used to create user-defined commands. Keyboard
+macros can include conditional and iteration structures, somewhat
+analogous to those provided by a traditional programmable calculator.
+
address@hidden
+* Naming Keyboard Macros::
+* Conditionals in Macros::
+* Loops in Macros::
+* Local Values in Macros::
+* Queries in Macros::
address@hidden menu
+
address@hidden Naming Keyboard Macros, Conditionals in Macros, Keyboard Macros,
Keyboard Macros
address@hidden Naming Keyboard Macros
+
address@hidden
address@hidden Z K
address@hidden calc-user-define-kbd-macro
+Once you have defined a keyboard macro, you can bind it to a @kbd{z}
+key sequence with the @kbd{Z K} (@code{calc-user-define-kbd-macro}) command.
+This command prompts first for a key, then for a command name. For
+example, if you type @kbd{C-x ( n @key{TAB} n @key{TAB} C-x )} you will
+define a keyboard macro which negates the top two numbers on the stack
+(@key{TAB} swaps the top two stack elements). Now you can type
address@hidden K n @key{RET}} to define this keyboard macro onto the @kbd{z n}
key
+sequence. The default command name (if you answer the second prompt with
+just the @key{RET} key as in this example) will be something like
address@hidden The keyboard macro will now be available as both
address@hidden n} and @kbd{M-x calc-User-n}. You can backspace and enter a more
+descriptive command name if you wish.
+
+Macros defined by @kbd{Z K} act like single commands; they are executed
+in the same way as by the @kbd{X} key. If you wish to define the macro
+as a standard no-frills Emacs macro (to be executed as if by @kbd{C-x e}),
+give a negative prefix argument to @kbd{Z K}.
+
+Once you have bound your keyboard macro to a key, you can use
address@hidden P} to register it permanently with Emacs. @xref{Creating User
Keys}.
+
address@hidden Keyboard macros, editing
+The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has
+been defined by a keyboard macro tries to use the @code{edmacro} package
+edit the macro. Type @kbd{C-c C-c} to finish editing and update
+the definition stored on the key, or, to cancel the edit, kill the
+buffer with @kbd{C-x k}.
+The special characters @code{RET}, @code{LFD}, @code{TAB}, @code{SPC},
address@hidden, and @code{NUL} must be entered as these three character
+sequences, written in all uppercase, as must the prefixes @code{C-} and
address@hidden Spaces and line breaks are ignored. Other characters are
+copied verbatim into the keyboard macro. Basically, the notation is the
+same as is used in all of this manual's examples, except that the manual
+takes some liberties with spaces: When we say @kbd{' [1 2 3] @key{RET}},
+we take it for granted that it is clear we really mean
address@hidden' [1 @key{SPC} 2 @key{SPC} 3] @key{RET}}.
+
address@hidden C-x * m
address@hidden read-kbd-macro
+The @kbd{C-x * m} (@code{read-kbd-macro}) command reads an Emacs ``region''
+of spelled-out keystrokes and defines it as the current keyboard macro.
+It is a convenient way to define a keyboard macro that has been stored
+in a file, or to define a macro without executing it at the same time.
+
address@hidden Conditionals in Macros, Loops in Macros, Naming Keyboard Macros,
Keyboard Macros
address@hidden Conditionals in Keyboard Macros
+
address@hidden
address@hidden Z [
address@hidden Z ]
address@hidden calc-kbd-if
address@hidden calc-kbd-else
address@hidden calc-kbd-else-if
address@hidden calc-kbd-end-if
address@hidden Conditional structures
+The @kbd{Z [} (@code{calc-kbd-if}) and @kbd{Z ]} (@code{calc-kbd-end-if})
+commands allow you to put simple tests in a keyboard macro. When Calc
+sees the @kbd{Z [}, it pops an object from the stack and, if the object is
+a non-zero value, continues executing keystrokes. But if the object is
+zero, or if it is not provably nonzero, Calc skips ahead to the matching
address@hidden ]} keystroke. @xref{Logical Operations}, for a set of commands
for
+performing tests which conveniently produce 1 for true and 0 for false.
+
+For example, @address@hidden 0 a < Z [ n Z ]} implements an absolute-value
+function in the form of a keyboard macro. This macro duplicates the
+number on the top of the stack, pushes zero and compares using @kbd{a <}
+(@code{calc-less-than}), then, if the number was less than zero,
+executes @kbd{n} (@code{calc-change-sign}). Otherwise, the change-sign
+command is skipped.
+
+To program this macro, type @kbd{C-x (}, type the above sequence of
+keystrokes, then type @kbd{C-x )}. Note that the keystrokes will be
+executed while you are making the definition as well as when you later
+re-execute the macro by typing @kbd{X}. Thus you should make sure a
+suitable number is on the stack before defining the macro so that you
+don't get a stack-underflow error during the definition process.
+
+Conditionals can be nested arbitrarily. However, there should be exactly
+one @kbd{Z ]} for each @kbd{Z [} in a keyboard macro.
+
address@hidden Z :
+The @kbd{Z :} (@code{calc-kbd-else}) command allows you to choose between
+two keystroke sequences. The general format is @address@hidden Z [
address@hidden Z : @var{else-part} Z ]}. If @var{cond} is true
+(i.e., if the top of stack contains a non-zero number after @var{cond}
+has been executed), the @var{then-part} will be executed and the
address@hidden will be skipped. Otherwise, the @var{then-part} will
+be skipped and the @var{else-part} will be executed.
+
address@hidden Z |
+The @kbd{Z |} (@code{calc-kbd-else-if}) command allows you to choose
+between any number of alternatives. For example,
address@hidden@var{cond1} Z [ @var{part1} Z : @var{cond2} Z | @var{part2} Z :
address@hidden Z ]} will execute @var{part1} if @var{cond1} is true,
+otherwise it will execute @var{part2} if @var{cond2} is true, otherwise
+it will execute @var{part3}.
+
+More precisely, @kbd{Z [} pops a number and conditionally skips to the
+next matching @kbd{Z :} or @kbd{Z ]} key. @address@hidden ]}} has no effect
when
+actually executed. @kbd{Z :} skips to the next matching @kbd{Z ]}.
address@hidden |} pops a number and conditionally skips to the next matching
address@hidden :} or @kbd{Z ]}; thus, @kbd{Z [} and @kbd{Z |} are functionally
+equivalent except that @kbd{Z [} participates in nesting but @kbd{Z |}
+does not.
+
+Calc's conditional and looping constructs work by scanning the
+keyboard macro for occurrences of character sequences like @samp{Z:}
+and @samp{Z]}. One side-effect of this is that if you use these
+constructs you must be careful that these character pairs do not
+occur by accident in other parts of the macros. Since Calc rarely
+uses address@hidden for any purpose except as a prefix character, this
+is not likely to be a problem. Another side-effect is that it will
+not work to define your own custom key bindings for these commands.
+Only the standard address@hidden bindings will work correctly.
+
address@hidden Z C-g
+If Calc gets stuck while skipping characters during the definition of a
+macro, type @kbd{Z C-g} to cancel the definition. (Typing plain @kbd{C-g}
+actually adds a @kbd{C-g} keystroke to the macro.)
+
address@hidden Loops in Macros, Local Values in Macros, Conditionals in Macros,
Keyboard Macros
address@hidden Loops in Keyboard Macros
+
address@hidden
address@hidden Z <
address@hidden Z >
address@hidden calc-kbd-repeat
address@hidden calc-kbd-end-repeat
address@hidden Looping structures
address@hidden Iterative structures
+The @kbd{Z <} (@code{calc-kbd-repeat}) and @kbd{Z >}
+(@code{calc-kbd-end-repeat}) commands pop a number from the stack,
+which must be an integer, then repeat the keystrokes between the brackets
+the specified number of times. If the integer is zero or negative, the
+body is skipped altogether. For example, @kbd{1 @key{TAB} Z < 2 * Z >}
+computes two to a nonnegative integer power. First, we push 1 on the
+stack and then swap the integer argument back to the top. The @kbd{Z <}
+pops that argument leaving the 1 back on top of the stack. Then, we
+repeat a multiply-by-two step however many times.
+
+Once again, the keyboard macro is executed as it is being entered.
+In this case it is especially important to set up reasonable initial
+conditions before making the definition: Suppose the integer 1000 just
+happened to be sitting on the stack before we typed the above definition!
+Another approach is to enter a harmless dummy definition for the macro,
+then go back and edit in the real one with a @kbd{Z E} command. Yet
+another approach is to type the macro as written-out keystroke names
+in a buffer, then use @kbd{C-x * m} (@code{read-kbd-macro}) to read the
+macro.
+
address@hidden Z /
address@hidden calc-break
+The @kbd{Z /} (@code{calc-kbd-break}) command allows you to break out
+of a keyboard macro loop prematurely. It pops an object from the stack;
+if that object is true (a non-zero number), control jumps out of the
+innermost enclosing @kbd{Z <} @dots{} @kbd{Z >} loop and continues
+after the @kbd{Z >}. If the object is false, the @kbd{Z /} has no
+effect. Thus @address@hidden Z /} is similar to @samp{if (@var{cond}) break;}
+in the C language.
+
address@hidden Z (
address@hidden Z )
address@hidden calc-kbd-for
address@hidden calc-kbd-end-for
+The @kbd{Z (} (@code{calc-kbd-for}) and @kbd{Z )} (@code{calc-kbd-end-for})
+commands are similar to @kbd{Z <} and @kbd{Z >}, except that they make the
+value of the counter available inside the loop. The general layout is
address@hidden@var{init} @var{final} Z ( @var{body} @var{step} Z )}. The
@kbd{Z (}
+command pops initial and final values from the stack. It then creates
+a temporary internal counter and initializes it with the value @var{init}.
+The @kbd{Z (} command then repeatedly pushes the counter value onto the
+stack and executes @var{body} and @var{step}, adding @var{step} to the
+counter each time until the loop finishes.
+
address@hidden Summations (by keyboard macros)
+By default, the loop finishes when the counter becomes greater than (or
+less than) @var{final}, assuming @var{initial} is less than (greater
+than) @var{final}. If @var{initial} is equal to @var{final}, the body
+executes exactly once. The body of the loop always executes at least
+once. For example, @kbd{0 1 10 Z ( 2 ^ + 1 Z )} computes the sum of the
+squares of the integers from 1 to 10, in steps of 1.
+
+If you give a numeric prefix argument of 1 to @kbd{Z (}, the loop is
+forced to use upward-counting conventions. In this case, if @var{initial}
+is greater than @var{final} the body will not be executed at all.
+Note that @var{step} may still be negative in this loop; the prefix
+argument merely constrains the loop-finished test. Likewise, a prefix
+argument of @mathit{-1} forces downward-counting conventions.
+
address@hidden Z @{
address@hidden Z @}
address@hidden calc-kbd-loop
address@hidden calc-kbd-end-loop
+The @kbd{Z @{} (@code{calc-kbd-loop}) and @kbd{Z @}}
+(@code{calc-kbd-end-loop}) commands are similar to @kbd{Z <} and
address@hidden >}, except that they do not pop a count from the stack---they
+effectively create an infinite loop. Every @kbd{Z @{} @dots{} @kbd{Z @}}
+loop ought to include at least one @kbd{Z /} to make sure the loop
+doesn't run forever. (If any error message occurs which causes Emacs
+to beep, the keyboard macro will also be halted; this is a standard
+feature of Emacs. You can also generally press @kbd{C-g} to halt a
+running keyboard macro, although not all versions of Unix support
+this feature.)
+
+The conditional and looping constructs are not actually tied to
+keyboard macros, but they are most often used in that context.
+For example, the keystrokes @kbd{10 Z < 23 @key{RET} Z >} push
+ten copies of 23 onto the stack. This can be typed ``live'' just
+as easily as in a macro definition.
+
address@hidden in Macros}, for some additional notes about
+conditional and looping commands.
+
address@hidden Local Values in Macros, Queries in Macros, Loops in Macros,
Keyboard Macros
address@hidden Local Values in Macros
+
address@hidden
address@hidden Local variables
address@hidden Restoring saved modes
+Keyboard macros sometimes want to operate under known conditions
+without affecting surrounding conditions. For example, a keyboard
+macro may wish to turn on Fraction mode, or set a particular
+precision, independent of the user's normal setting for those
+modes.
+
address@hidden Z `
address@hidden Z '
address@hidden calc-kbd-push
address@hidden calc-kbd-pop
+Macros also sometimes need to use local variables. Assignments to
+local variables inside the macro should not affect any variables
+outside the macro. The @kbd{Z `} (@code{calc-kbd-push}) and @kbd{Z '}
+(@code{calc-kbd-pop}) commands give you both of these capabilities.
+
+When you type @kbd{Z `} (with a backquote or accent grave character),
+the values of various mode settings are saved away. The ten ``quick''
+variables @code{q0} through @code{q9} are also saved. When
+you type @address@hidden '}} (with an apostrophe), these values are restored.
+Pairs of @kbd{Z `} and @kbd{Z '} commands may be nested.
+
+If a keyboard macro halts due to an error in between a @kbd{Z `} and
+a @kbd{Z '}, the saved values will be restored correctly even though
+the macro never reaches the @kbd{Z '} command. Thus you can use
address@hidden `} and @kbd{Z '} without having to worry about what happens
+in exceptional conditions.
+
+If you type @kbd{Z `} ``live'' (not in a keyboard macro), Calc puts
+you into a ``recursive edit.'' You can tell you are in a recursive
+edit because there will be extra square brackets in the mode line,
+as in @samp{[(Calculator)]}. These brackets will go away when you
+type the matching @kbd{Z '} command. The modes and quick variables
+will be saved and restored in just the same way as if actual keyboard
+macros were involved.
+
+The modes saved by @kbd{Z `} and @kbd{Z '} are the current precision
+and binary word size, the angular mode (Deg, Rad, or HMS), the
+simplification mode, Algebraic mode, Symbolic mode, Infinite mode,
+Matrix or Scalar mode, Fraction mode, and the current complex mode
+(Polar or Rectangular). The ten ``quick'' variables' values (or lack
+thereof) are also saved.
+
+Most mode-setting commands act as toggles, but with a numeric prefix
+they force the mode either on (positive prefix) or off (negative
+or zero prefix). Since you don't know what the environment might
+be when you invoke your macro, it's best to use prefix arguments
+for all mode-setting commands inside the macro.
+
+In fact, @kbd{C-u Z `} is like @kbd{Z `} except that it sets the modes
+listed above to their default values. As usual, the matching @kbd{Z '}
+will restore the modes to their settings from before the @kbd{C-u Z `}.
+Also, @address@hidden `}} with a negative prefix argument resets the algebraic
mode
+to its default (off) but leaves the other modes the same as they were
+outside the construct.
+
+The contents of the stack and trail, values of non-quick variables, and
+other settings such as the language mode and the various display modes,
+are @emph{not} affected by @kbd{Z `} and @kbd{Z '}.
+
address@hidden Queries in Macros, , Local Values in Macros, Keyboard Macros
address@hidden Queries in Keyboard Macros
+
address@hidden @noindent
address@hidden @kindex Z =
address@hidden @pindex calc-kbd-report
address@hidden The @kbd{Z =} (@code{calc-kbd-report}) command displays an
informative
address@hidden message including the value on the top of the stack. You are
prompted
address@hidden to enter a string. That string, along with the top-of-stack
value,
address@hidden is displayed unless @kbd{m w} (@code{calc-working}) has been used
address@hidden to turn such messages off.
+
address@hidden
address@hidden Z #
address@hidden calc-kbd-query
+The @kbd{Z #} (@code{calc-kbd-query}) command prompts for an algebraic
+entry which takes its input from the keyboard, even during macro
+execution. All the normal conventions of algebraic input, including the
+use of @kbd{$} characters, are supported. The prompt message itself is
+taken from the top of the stack, and so must be entered (as a string)
+before the @kbd{Z #} command. (Recall, as a string it can be entered by
+pressing the @kbd{"} key and will appear as a vector when it is put on
+the stack. The prompt message is only put on the stack to provide a
+prompt for the @kbd{Z #} command; it will not play any role in any
+subsequent calculations.) This command allows your keyboard macros to
+accept numbers or formulas as interactive input.
+
+As an example,
address@hidden @key{RET} "Power: " @key{RET} Z # 3 @key{RET} ^} will prompt for
+input with ``Power: '' in the minibuffer, then return 2 to the provided
+power. (The response to the prompt that's given, 3 in this example,
+will not be part of the macro.)
+
address@hidden Macro Query, , , emacs, the Emacs Manual}, for a description of
address@hidden q} (@code{kbd-macro-query}), the standard Emacs way to accept
+keyboard input during a keyboard macro. In particular, you can use
address@hidden q} to enter a recursive edit, which allows the user to perform
+any Calculator operations interactively before pressing @kbd{C-M-c} to
+return control to the keyboard macro.
+
address@hidden Invocation Macros, Algebraic Definitions, Keyboard Macros,
Programming
address@hidden Invocation Macros
+
address@hidden C-x * z
address@hidden Z I
address@hidden calc-user-invocation
address@hidden calc-user-define-invocation
+Calc provides one special keyboard macro, called up by @kbd{C-x * z}
+(@code{calc-user-invocation}), that is intended to allow you to define
+your own special way of starting Calc. To define this ``invocation
+macro,'' create the macro in the usual way with @kbd{C-x (} and
address@hidden )}, then type @kbd{Z I} (@code{calc-user-define-invocation}).
+There is only one invocation macro, so you don't need to type any
+additional letters after @kbd{Z I}. From now on, you can type
address@hidden * z} at any time to execute your invocation macro.
+
+For example, suppose you find yourself often grabbing rectangles of
+numbers into Calc and multiplying their columns. You can do this
+by typing @kbd{C-x * r} to grab, and @kbd{V R : *} to multiply columns.
+To make this into an invocation macro, just type @kbd{C-x ( C-x * r
+V R : * C-x )}, then @kbd{Z I}. Then, to multiply a rectangle of data,
+just mark the data in its buffer in the usual way and type @kbd{C-x * z}.
+
+Invocation macros are treated like regular Emacs keyboard macros;
+all the special features described above for @kbd{Z K}-style macros
+do not apply. @kbd{C-x * z} is just like @kbd{C-x e}, except that it
+uses the macro that was last stored by @kbd{Z I}. (In fact, the
+macro does not even have to have anything to do with Calc!)
+
+The @kbd{m m} command saves the last invocation macro defined by
address@hidden I} along with all the other Calc mode settings.
address@hidden Mode Commands}.
+
address@hidden Algebraic Definitions, Lisp Definitions, Invocation Macros,
Programming
address@hidden Programming with Formulas
+
address@hidden
address@hidden Z F
address@hidden calc-user-define-formula
address@hidden Programming with algebraic formulas
+Another way to create a new Calculator command uses algebraic formulas.
+The @kbd{Z F} (@code{calc-user-define-formula}) command stores the
+formula at the top of the stack as the definition for a key. This
+command prompts for five things: The key, the command name, the function
+name, the argument list, and the behavior of the command when given
+non-numeric arguments.
+
+For example, suppose we type @kbd{' a+2b @key{RET}} to push the formula
address@hidden + 2*b} onto the stack. We now type @kbd{Z F m} to define this
+formula on the @kbd{z m} key sequence. The next prompt is for a command
+name, beginning with @samp{calc-}, which should be the long (@kbd{M-x}) form
+for the new command. If you simply press @key{RET}, a default name like
address@hidden will be constructed. In our example, suppose we enter
address@hidden @key{RET}} to define the new command as @code{calc-spam}.
+
+If you want to give the formula a long-style name only, you can press
address@hidden or @key{RET} when asked which single key to use. For example
address@hidden F @key{RET} spam @key{RET}} defines the new command as
address@hidden calc-spam}, with no keyboard equivalent.
+
+The third prompt is for an algebraic function name. The default is to
+use the same name as the command name but without the @samp{calc-}
+prefix. (If this is of the form @samp{User-m}, the hyphen is removed so
+it won't be taken for a minus sign in algebraic formulas.)
+This is the name you will use if you want to enter your
+new function in an algebraic formula. Suppose we enter @kbd{yow @key{RET}}.
+Then the new function can be invoked by pushing two numbers on the
+stack and typing @kbd{z m} or @kbd{x spam}, or by entering the algebraic
+formula @samp{yow(x,y)}.
+
+The fourth prompt is for the function's argument list. This is used to
+associate values on the stack with the variables that appear in the formula.
+The default is a list of all variables which appear in the formula, sorted
+into alphabetical order. In our case, the default would be @samp{(a b)}.
+This means that, when the user types @kbd{z m}, the Calculator will remove
+two numbers from the stack, substitute these numbers for @samp{a} and
address@hidden (respectively) in the formula, then simplify the formula and
+push the result on the stack. In other words, @kbd{10 @key{RET} 100 z m}
+would replace the 10 and 100 on the stack with the number 210, which is
address@hidden + 2 b} with @expr{a=10} and @expr{b=100}. Likewise, the formula
address@hidden(10, 100)} will be evaluated by substituting @expr{a=10} and
address@hidden in the definition.
+
+You can rearrange the order of the names before pressing @key{RET} to
+control which stack positions go to which variables in the formula. If
+you remove a variable from the argument list, that variable will be left
+in symbolic form by the command. Thus using an argument list of @samp{(b)}
+for our function would cause @kbd{10 z m} to replace the 10 on the stack
+with the formula @samp{a + 20}. If we had used an argument list of
address@hidden(b a)}, the result with inputs 10 and 100 would have been 120.
+
+You can also put a nameless function on the stack instead of just a
+formula, as in @samp{<a, b : a + 2 b>}. @xref{Specifying Operators}.
+In this example, the command will be defined by the formula @samp{a + 2 b}
+using the argument list @samp{(a b)}.
+
+The final prompt is a y-or-n question concerning what to do if symbolic
+arguments are given to your function. If you answer @kbd{y}, then
+executing @kbd{z m} (using the original argument list @samp{(a b)}) with
+arguments @expr{10} and @expr{x} will leave the function in symbolic
+form, i.e., @samp{yow(10,x)}. On the other hand, if you answer @kbd{n},
+then the formula will always be expanded, even for non-constant
+arguments: @samp{10 + 2 x}. If you never plan to feed algebraic
+formulas to your new function, it doesn't matter how you answer this
+question.
+
+If you answered @kbd{y} to this question you can still cause a function
+call to be expanded by typing @kbd{a "} (@code{calc-expand-formula}).
+Also, Calc will expand the function if necessary when you take a
+derivative or integral or solve an equation involving the function.
+
address@hidden Z G
address@hidden calc-get-user-defn
+Once you have defined a formula on a key, you can retrieve this formula
+with the @kbd{Z G} (@code{calc-user-define-get-defn}) command. Press a
+key, and this command pushes the formula that was used to define that
+key onto the stack. Actually, it pushes a nameless function that
+specifies both the argument list and the defining formula. You will get
+an error message if the key is undefined, or if the key was not defined
+by a @kbd{Z F} command.
+
+The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has
+been defined by a formula uses a variant of the @code{calc-edit} command
+to edit the defining formula. Press @kbd{C-c C-c} to finish editing and
+store the new formula back in the definition, or kill the buffer with
address@hidden k} to
+cancel the edit. (The argument list and other properties of the
+definition are unchanged; to adjust the argument list, you can use
address@hidden G} to grab the function onto the stack, edit with @kbd{`}, and
+then re-execute the @kbd{Z F} command.)
+
+As usual, the @kbd{Z P} command records your definition permanently.
+In this case it will permanently record all three of the relevant
+definitions: the key, the command, and the function.
+
+You may find it useful to turn off the default simplifications with
address@hidden O} (@code{calc-no-simplify-mode}) when entering a formula to be
+used as a function definition. For example, the formula @samp{deriv(a^2,v)}
+which might be used to define a new function @samp{dsqr(a,v)} will be
+``simplified'' to 0 immediately upon entry since @code{deriv} considers
address@hidden to be constant with respect to @expr{v}. Turning off
+default simplifications cures this problem: The definition will be stored
+in symbolic form without ever activating the @code{deriv} function. Press
address@hidden D} to turn the default simplifications back on afterwards.
+
address@hidden Lisp Definitions, , Algebraic Definitions, Programming
address@hidden Programming with Lisp
+
address@hidden
+The Calculator can be programmed quite extensively in Lisp. All you
+do is write a normal Lisp function definition, but with @code{defmath}
+in place of @code{defun}. This has the same form as @code{defun}, but it
+automagically replaces calls to standard Lisp functions like @code{+} and
address@hidden with calls to the corresponding functions in Calc's own library.
+Thus you can write natural-looking Lisp code which operates on all of the
+standard Calculator data types. You can then use @kbd{Z D} if you wish to
+bind your new command to a @kbd{z}-prefix key sequence. The @kbd{Z E} command
+will not edit a Lisp-based definition.
+
+Emacs Lisp is described in the GNU Emacs Lisp Reference Manual. This section
+assumes a familiarity with Lisp programming concepts; if you do not know
+Lisp, you may find keyboard macros or rewrite rules to be an easier way
+to program the Calculator.
+
+This section first discusses ways to write commands, functions, or
+small programs to be executed inside of Calc. Then it discusses how
+your own separate programs are able to call Calc from the outside.
+Finally, there is a list of internal Calc functions and data structures
+for the true Lisp enthusiast.
+
address@hidden
+* Defining Functions::
+* Defining Simple Commands::
+* Defining Stack Commands::
+* Argument Qualifiers::
+* Example Definitions::
+
+* Calling Calc from Your Programs::
+* Internals::
address@hidden menu
+
address@hidden Defining Functions, Defining Simple Commands, Lisp Definitions,
Lisp Definitions
address@hidden Defining New Functions
+
address@hidden
address@hidden defmath
+The @code{defmath} function (actually a Lisp macro) is like @code{defun}
+except that code in the body of the definition can make use of the full
+range of Calculator data types. The prefix @samp{calcFunc-} is added
+to the specified name to get the actual Lisp function name. As a simple
+example,
+
address@hidden
+(defmath myfact (n)
+ (if (> n 0)
+ (* n (myfact (1- n)))
+ 1))
address@hidden example
+
address@hidden
+This actually expands to the code,
+
address@hidden
+(defun calcFunc-myfact (n)
+ (if (math-posp n)
+ (math-mul n (calcFunc-myfact (math-add n -1)))
+ 1))
address@hidden example
+
address@hidden
+This function can be used in algebraic expressions, e.g., @samp{myfact(5)}.
+
+The @samp{myfact} function as it is defined above has the bug that an
+expression @samp{myfact(a+b)} will be simplified to 1 because the
+formula @samp{a+b} is not considered to be @code{posp}. A robust
+factorial function would be written along the following lines:
+
address@hidden
+(defmath myfact (n)
+ (if (> n 0)
+ (* n (myfact (1- n)))
+ (if (= n 0)
+ 1
+ nil))) ; this could be simplified as: (and (= n 0) 1)
address@hidden smallexample
+
+If a function returns @code{nil}, it is left unsimplified by the Calculator
+(except that its arguments will be simplified). Thus, @samp{myfact(a+1+2)}
+will be simplified to @samp{myfact(a+3)} but no further. Beware that every
+time the Calculator reexamines this formula it will attempt to resimplify
+it, so your function ought to detect the address@hidden case as
+efficiently as possible.
+
+The following standard Lisp functions are treated by @code{defmath}:
address@hidden, @code{-}, @code{*}, @code{/}, @code{%}, @code{^} or
address@hidden, @code{=}, @code{<}, @code{>}, @code{<=}, @code{>=},
address@hidden/=}, @code{1+}, @code{1-}, @code{logand}, @code{logior},
@code{logxor},
address@hidden, @code{lognot}. Also, @code{~=} is an abbreviation for
address@hidden, which is useful in implementing Taylor series.
+
+For other functions @var{func}, if a function by the name
address@hidden@var{func}} exists it is used, otherwise if a function by the
+name @address@hidden exists it is used, otherwise if @var{func} itself
+is defined as a function it is used, otherwise @address@hidden is
+used on the assumption that this is a to-be-defined math function. Also, if
+the function name is quoted as in @samp{('integerp a)} the function name is
+always used exactly as written (but not quoted).
+
+Variable names have @samp{var-} prepended to them unless they appear in
+the function's argument list or in an enclosing @code{let}, @code{let*},
address@hidden, or @code{foreach} form,
+or their names already contain a @samp{-} character. Thus a reference to
address@hidden is the same as a reference to @samp{var-foo}.
+
+A few other Lisp extensions are available in @code{defmath} definitions:
+
address@hidden @bullet
address@hidden
+The @code{elt} function accepts any number of index variables.
+Note that Calc vectors are stored as Lisp lists whose first
+element is the symbol @code{vec}; thus, @samp{(elt v 2)} yields
+the second element of vector @code{v}, and @samp{(elt m i j)}
+yields one element of a Calc matrix.
+
address@hidden
+The @code{setq} function has been extended to act like the Common
+Lisp @code{setf} function. (The name @code{setf} is recognized as
+a synonym of @code{setq}.) Specifically, the first argument of
address@hidden can be an @code{nth}, @code{elt}, @code{car}, or @code{cdr} form,
+in which case the effect is to store into the specified
+element of a list. Thus, @samp{(setq (elt m i j) x)} stores @expr{x}
+into one element of a matrix.
+
address@hidden
+A @code{for} looping construct is available. For example,
address@hidden(for ((i 0 10)) body)} executes @code{body} once for each
+binding of @expr{i} from zero to 10. This is like a @code{let}
+form in that @expr{i} is temporarily bound to the loop count
+without disturbing its value outside the @code{for} construct.
+Nested loops, as in @samp{(for ((i 0 10) (j 0 (1- i) 2)) body)},
+are also available. For each value of @expr{i} from zero to 10,
address@hidden counts from 0 to @expr{i-1} in steps of two. Note that
address@hidden has the same general outline as @code{let*}, except
+that each element of the header is a list of three or four
+things, not just two.
+
address@hidden
+The @code{foreach} construct loops over elements of a list.
+For example, @samp{(foreach ((x (cdr v))) body)} executes
address@hidden with @expr{x} bound to each element of Calc vector
address@hidden in turn. The purpose of @code{cdr} here is to skip over
+the initial @code{vec} symbol in the vector.
+
address@hidden
+The @code{break} function breaks out of the innermost enclosing
address@hidden, @code{for}, or @code{foreach} loop. If given a
+value, as in @samp{(break x)}, this value is returned by the
+loop. (Lisp loops otherwise always return @code{nil}.)
+
address@hidden
+The @code{return} function prematurely returns from the enclosing
+function. For example, @samp{(return (+ x y))} returns @expr{x+y}
+as the value of a function. You can use @code{return} anywhere
+inside the body of the function.
address@hidden itemize
+
+Non-integer numbers (and extremely large integers) cannot be included
+directly into a @code{defmath} definition. This is because the Lisp
+reader will fail to parse them long before @code{defmath} ever gets control.
+Instead, use the notation, @samp{:"3.1415"}. In fact, any algebraic
+formula can go between the quotes. For example,
+
address@hidden
+(defmath sqexp (x) ; sqexp(x) == sqrt(exp(x)) == exp(x*0.5)
+ (and (numberp x)
+ (exp :"x * 0.5")))
address@hidden smallexample
+
+expands to
+
address@hidden
+(defun calcFunc-sqexp (x)
+ (and (math-numberp x)
+ (calcFunc-exp (math-mul x '(float 5 -1)))))
address@hidden smallexample
+
+Note the use of @code{numberp} as a guard to ensure that the argument is
+a number first, returning @code{nil} if not. The exponential function
+could itself have been included in the expression, if we had preferred:
address@hidden:"exp(x * 0.5)"}. As another example, the
multiplication-and-recursion
+step of @code{myfact} could have been written
+
address@hidden
+:"n * myfact(n-1)"
address@hidden example
+
+A good place to put your @code{defmath} commands is your Calc init file
+(the file given by @code{calc-settings-file}, typically
address@hidden/.calc.el}), which will not be loaded until Calc starts.
+If a file named @file{.emacs} exists in your home directory, Emacs reads
+and executes the Lisp forms in this file as it starts up. While it may
+seem reasonable to put your favorite @code{defmath} commands there,
+this has the unfortunate side-effect that parts of the Calculator must be
+loaded in to process the @code{defmath} commands whether or not you will
+actually use the Calculator! If you want to put the @code{defmath}
+commands there (for example, if you redefine @code{calc-settings-file}
+to be @file{.emacs}), a better effect can be had by writing
+
address@hidden
+(put 'calc-define 'thing '(progn
+ (defmath ... )
+ (defmath ... )
+))
address@hidden example
+
address@hidden
address@hidden calc-define
+The @code{put} function adds a @dfn{property} to a symbol. Each Lisp
+symbol has a list of properties associated with it. Here we add a
+property with a name of @code{thing} and a @samp{(progn ...)} form as
+its value. When Calc starts up, and at the start of every Calc command,
+the property list for the symbol @code{calc-define} is checked and the
+values of any properties found are evaluated as Lisp forms. The
+properties are removed as they are evaluated. The property names
+(like @code{thing}) are not used; you should choose something like the
+name of your project so as not to conflict with other properties.
+
+The net effect is that you can put the above code in your @file{.emacs}
+file and it will not be executed until Calc is loaded. Or, you can put
+that same code in another file which you load by hand either before or
+after Calc itself is loaded.
+
+The properties of @code{calc-define} are evaluated in the same order
+that they were added. They can assume that the Calc modules @file{calc.el},
address@hidden, and @file{calc-macs.el} have been fully loaded, and
+that the @samp{*Calculator*} buffer will be the current buffer.
+
+If your @code{calc-define} property only defines algebraic functions,
+you can be sure that it will have been evaluated before Calc tries to
+call your function, even if the file defining the property is loaded
+after Calc is loaded. But if the property defines commands or key
+sequences, it may not be evaluated soon enough. (Suppose it defines the
+new command @code{tweak-calc}; the user can load your file, then type
address@hidden tweak-calc} before Calc has had chance to do anything.) To
+protect against this situation, you can put
+
address@hidden
+(run-hooks 'calc-check-defines)
address@hidden example
+
address@hidden calc-check-defines
address@hidden
+at the end of your file. The @code{calc-check-defines} function is what
+looks for and evaluates properties on @code{calc-define}; @code{run-hooks}
+has the advantage that it is quietly ignored if @code{calc-check-defines}
+is not yet defined because Calc has not yet been loaded.
+
+Examples of things that ought to be enclosed in a @code{calc-define}
+property are @code{defmath} calls, @code{define-key} calls that modify
+the Calc key map, and any calls that redefine things defined inside Calc.
+Ordinary @code{defun}s need not be enclosed with @code{calc-define}.
+
address@hidden Defining Simple Commands, Defining Stack Commands, Defining
Functions, Lisp Definitions
address@hidden Defining New Simple Commands
+
address@hidden
address@hidden interactive
+If a @code{defmath} form contains an @code{interactive} clause, it defines
+a Calculator command. Actually such a @code{defmath} results in @emph{two}
+function definitions: One, a @samp{calcFunc-} function as was just described,
+with the @code{interactive} clause removed. Two, a @samp{calc-} function
+with a suitable @code{interactive} clause and some sort of wrapper to make
+the command work in the Calc environment.
+
+In the simple case, the @code{interactive} clause has the same form as
+for normal Emacs Lisp commands:
+
address@hidden
+(defmath increase-precision (delta)
+ "Increase precision by DELTA." ; This is the "documentation string"
+ (interactive "p") ; Register this as a M-x-able command
+ (setq calc-internal-prec (+ calc-internal-prec delta)))
address@hidden smallexample
+
+This expands to the pair of definitions,
+
address@hidden
+(defun calc-increase-precision (delta)
+ "Increase precision by DELTA."
+ (interactive "p")
+ (calc-wrapper
+ (setq calc-internal-prec (math-add calc-internal-prec delta))))
+
+(defun calcFunc-increase-precision (delta)
+ "Increase precision by DELTA."
+ (setq calc-internal-prec (math-add calc-internal-prec delta)))
address@hidden smallexample
+
address@hidden
+where in this case the latter function would never really be used! Note
+that since the Calculator stores small integers as plain Lisp integers,
+the @code{math-add} function will work just as well as the native
address@hidden even when the intent is to operate on native Lisp integers.
+
address@hidden calc-wrapper
+The @samp{calc-wrapper} call invokes a macro which surrounds the body of
+the function with code that looks roughly like this:
+
address@hidden
+(let ((calc-command-flags nil))
+ (unwind-protect
+ (save-excursion
+ (calc-select-buffer)
+ @emph{body of function}
+ @emph{renumber stack}
+ @emph{clear} Working @emph{message})
+ @emph{realign cursor and window}
+ @emph{clear Inverse, Hyperbolic, and Keep Args flags}
+ @emph{update Emacs mode line}))
address@hidden smallexample
+
address@hidden calc-select-buffer
+The @code{calc-select-buffer} function selects the @samp{*Calculator*}
+buffer if necessary, say, because the command was invoked from inside
+the @samp{*Calc Trail*} window.
+
address@hidden calc-set-command-flag
+You can call, for example, @code{(calc-set-command-flag 'no-align)} to
+set the above-mentioned command flags. Calc routines recognize the
+following command flags:
+
address@hidden @code
address@hidden renum-stack
+Stack line numbers @samp{1:}, @samp{2:}, and so on must be renumbered
+after this command completes. This is set by routines like
address@hidden
+
address@hidden clear-message
+Calc should call @samp{(message "")} if this command completes normally
+(to clear a address@hidden'' message out of the echo area).
+
address@hidden no-align
+Do not move the cursor back to the @samp{.} top-of-stack marker.
+
address@hidden position-point
+Use the variables @code{calc-position-point-line} and
address@hidden to position the cursor after
+this command finishes.
+
address@hidden keep-flags
+Do not clear @code{calc-inverse-flag}, @code{calc-hyperbolic-flag},
+and @code{calc-keep-args-flag} at the end of this command.
+
address@hidden do-edit
+Switch to buffer @samp{*Calc Edit*} after this command.
+
address@hidden hold-trail
+Do not move trail pointer to end of trail when something is recorded
+there.
address@hidden table
+
address@hidden Y
address@hidden Y ?
address@hidden calc-Y-help-msgs
+Calc reserves a special prefix key, address@hidden, for user-written
+extensions to Calc. There are no built-in commands that work with
+this prefix key; you must call @code{define-key} from Lisp (probably
+from inside a @code{calc-define} property) to add to it. Initially only
address@hidden ?} is defined; it takes help messages from a list of strings
+(initially @code{nil}) in the variable @code{calc-Y-help-msgs}. All
+other undefined keys except for @kbd{Y} are reserved for use by
+future versions of Calc.
+
+If you are writing a Calc enhancement which you expect to give to
+others, it is best to minimize the number of @kbd{Y}-key sequences
+you use. In fact, if you have more than one key sequence you should
+consider defining three-key sequences with a @kbd{Y}, then a key that
+stands for your package, then a third key for the particular command
+within your package.
+
+Users may wish to install several Calc enhancements, and it is possible
+that several enhancements will choose to use the same key. In the
+example below, a variable @code{inc-prec-base-key} has been defined
+to contain the key that identifies the @code{inc-prec} package. Its
+value is initially @code{"P"}, but a user can change this variable
+if necessary without having to modify the file.
+
+Here is a complete file, @file{inc-prec.el}, which makes a @kbd{Y P I}
+command that increases the precision, and a @kbd{Y P D} command that
+decreases the precision.
+
address@hidden
+;;; Increase and decrease Calc precision. Dave Gillespie, 5/31/91.
+;; (Include copyright or copyleft stuff here.)
+
+(defvar inc-prec-base-key "P"
+ "Base key for inc-prec.el commands.")
+
+(put 'calc-define 'inc-prec '(progn
+
+(define-key calc-mode-map (format "Y%sI" inc-prec-base-key)
+ 'increase-precision)
+(define-key calc-mode-map (format "Y%sD" inc-prec-base-key)
+ 'decrease-precision)
+
+(setq calc-Y-help-msgs
+ (cons (format "%s + Inc-prec, Dec-prec" inc-prec-base-key)
+ calc-Y-help-msgs))
+
+(defmath increase-precision (delta)
+ "Increase precision by DELTA."
+ (interactive "p")
+ (setq calc-internal-prec (+ calc-internal-prec delta)))
+
+(defmath decrease-precision (delta)
+ "Decrease precision by DELTA."
+ (interactive "p")
+ (setq calc-internal-prec (- calc-internal-prec delta)))
+
+)) ; end of calc-define property
+
+(run-hooks 'calc-check-defines)
address@hidden smallexample
+
address@hidden Defining Stack Commands, Argument Qualifiers, Defining Simple
Commands, Lisp Definitions
address@hidden Defining New Stack-Based Commands
+
address@hidden
+To define a new computational command which takes and/or leaves arguments
+on the stack, a special form of @code{interactive} clause is used.
+
address@hidden
+(interactive @var{num} @var{tag})
address@hidden example
+
address@hidden
+where @var{num} is an integer, and @var{tag} is a string. The effect is
+to pop @var{num} values off the stack, resimplify them by calling
address@hidden, and hand them to your function according to the
+function's argument list. Your function may include @code{&optional} and
address@hidden&rest} parameters, so long as calling the function with @var{num}
+parameters is valid.
+
+Your function must return either a number or a formula in a form
+acceptable to Calc, or a list of such numbers or formulas. These value(s)
+are pushed onto the stack when the function completes. They are also
+recorded in the Calc Trail buffer on a line beginning with @var{tag},
+a string of (normally) four characters or less. If you omit @var{tag}
+or use @code{nil} as a tag, the result is not recorded in the trail.
+
+As an example, the definition
+
address@hidden
+(defmath myfact (n)
+ "Compute the factorial of the integer at the top of the stack."
+ (interactive 1 "fact")
+ (if (> n 0)
+ (* n (myfact (1- n)))
+ (and (= n 0) 1)))
address@hidden smallexample
+
address@hidden
+is a version of the factorial function shown previously which can be used
+as a command as well as an algebraic function. It expands to
+
address@hidden
+(defun calc-myfact ()
+ "Compute the factorial of the integer at the top of the stack."
+ (interactive)
+ (calc-slow-wrapper
+ (calc-enter-result 1 "fact"
+ (cons 'calcFunc-myfact (calc-top-list-n 1)))))
+
+(defun calcFunc-myfact (n)
+ "Compute the factorial of the integer at the top of the stack."
+ (if (math-posp n)
+ (math-mul n (calcFunc-myfact (math-add n -1)))
+ (and (math-zerop n) 1)))
address@hidden smallexample
+
address@hidden calc-slow-wrapper
+The @code{calc-slow-wrapper} function is a version of @code{calc-wrapper}
+that automatically puts up a @samp{Working...} message before the
+computation begins. (This message can be turned off by the user
+with an @kbd{m w} (@code{calc-working}) command.)
+
address@hidden calc-top-list-n
+The @code{calc-top-list-n} function returns a list of the specified number
+of values from the top of the stack. It resimplifies each value by
+calling @code{calc-normalize}. If its argument is zero it returns an
+empty list. It does not actually remove these values from the stack.
+
address@hidden calc-enter-result
+The @code{calc-enter-result} function takes an integer @var{num} and string
address@hidden as described above, plus a third argument which is either a
+Calculator data object or a list of such objects. These objects are
+resimplified and pushed onto the stack after popping the specified number
+of values from the stack. If @var{tag} is address@hidden, the values
+being pushed are also recorded in the trail.
+
+Note that if @code{calcFunc-myfact} returns @code{nil} this represents
+``leave the function in symbolic form.'' To return an actual empty list,
+in the sense that @code{calc-enter-result} will push zero elements back
+onto the stack, you should return the special value @samp{'(nil)}, a list
+containing the single symbol @code{nil}.
+
+The @code{interactive} declaration can actually contain a limited
+Emacs-style code string as well which comes just before @var{num} and
address@hidden Currently the only Emacs code supported is @samp{"p"}, as in
+
address@hidden
+(defmath foo (a b &optional c)
+ (interactive "p" 2 "foo")
+ @var{body})
address@hidden example
+
+In this example, the command @code{calc-foo} will evaluate the expression
address@hidden(a,b)} if executed with no argument, or @samp{foo(a,b,n)} if
+executed with a numeric prefix argument of @expr{n}.
+
+The other code string allowed is @samp{"m"} (unrelated to the usual @samp{"m"}
+code as used with @code{defun}). It uses the numeric prefix argument as the
+number of objects to remove from the stack and pass to the function.
+In this case, the integer @var{num} serves as a default number of
+arguments to be used when no prefix is supplied.
+
address@hidden Argument Qualifiers, Example Definitions, Defining Stack
Commands, Lisp Definitions
address@hidden Argument Qualifiers
+
address@hidden
+Anywhere a parameter name can appear in the parameter list you can also use
+an @dfn{argument qualifier}. Thus the general form of a definition is:
+
address@hidden
+(defmath @var{name} (@var{param} @var{param...}
+ &optional @var{param} @var{param...}
+ &rest @var{param})
+ @var{body})
address@hidden example
+
address@hidden
+where each @var{param} is either a symbol or a list of the form
+
address@hidden
+(@var{qual} @var{param})
address@hidden example
+
+The following qualifiers are recognized:
+
address@hidden @samp
address@hidden complete
address@hidden complete
+The argument must not be an incomplete vector, interval, or complex number.
+(This is rarely needed since the Calculator itself will never call your
+function with an incomplete argument. But there is nothing stopping your
+own Lisp code from calling your function with an incomplete argument.)
+
address@hidden integer
address@hidden integer
+The argument must be an integer. If it is an integer-valued float
+it will be accepted but converted to integer form. Non-integers and
+formulas are rejected.
+
address@hidden natnum
address@hidden natnum
+Like @samp{integer}, but the argument must be non-negative.
+
address@hidden fixnum
address@hidden fixnum
+Like @samp{integer}, but the argument must fit into a native Lisp integer,
+which on most systems means less than 2^23 in absolute value. The
+argument is converted into Lisp-integer form if necessary.
+
address@hidden float
address@hidden float
+The argument is converted to floating-point format if it is a number or
+vector. If it is a formula it is left alone. (The argument is never
+actually rejected by this qualifier.)
+
address@hidden @var{pred}
+The argument must satisfy predicate @var{pred}, which is one of the
+standard Calculator predicates. @xref{Predicates}.
+
address@hidden address@hidden
+The argument must @emph{not} satisfy predicate @var{pred}.
address@hidden table
+
+For example,
+
address@hidden
+(defmath foo (a (constp (not-matrixp b)) &optional (float c)
+ &rest (integer d))
+ @var{body})
address@hidden example
+
address@hidden
+expands to
+
address@hidden
+(defun calcFunc-foo (a b &optional c &rest d)
+ (and (math-matrixp b)
+ (math-reject-arg b 'not-matrixp))
+ (or (math-constp b)
+ (math-reject-arg b 'constp))
+ (and c (setq c (math-check-float c)))
+ (setq d (mapcar 'math-check-integer d))
+ @var{body})
address@hidden example
+
address@hidden
+which performs the necessary checks and conversions before executing the
+body of the function.
+
address@hidden Example Definitions, Calling Calc from Your Programs, Argument
Qualifiers, Lisp Definitions
address@hidden Example Definitions
+
address@hidden
+This section includes some Lisp programming examples on a larger scale.
+These programs make use of some of the Calculator's internal functions;
address@hidden
+
address@hidden
+* Bit Counting Example::
+* Sine Example::
address@hidden menu
+
address@hidden Bit Counting Example, Sine Example, Example Definitions, Example
Definitions
address@hidden Bit-Counting
+
address@hidden
address@hidden
address@hidden
address@hidden ignore
address@hidden bcount
+Calc does not include a built-in function for counting the number of
+``one'' bits in a binary integer. It's easy to invent one using @kbd{b u}
+to convert the integer to a set, and @kbd{V #} to count the elements of
+that set; let's write a function that counts the bits without having to
+create an intermediate set.
+
address@hidden
+(defmath bcount ((natnum n))
+ (interactive 1 "bcnt")
+ (let ((count 0))
+ (while (> n 0)
+ (if (oddp n)
+ (setq count (1+ count)))
+ (setq n (lsh n -1)))
+ count))
address@hidden smallexample
+
address@hidden
+When this is expanded by @code{defmath}, it will become the following
+Emacs Lisp function:
+
address@hidden
+(defun calcFunc-bcount (n)
+ (setq n (math-check-natnum n))
+ (let ((count 0))
+ (while (math-posp n)
+ (if (math-oddp n)
+ (setq count (math-add count 1)))
+ (setq n (calcFunc-lsh n -1)))
+ count))
address@hidden smallexample
+
+If the input numbers are large, this function involves a fair amount
+of arithmetic. A binary right shift is essentially a division by two;
+recall that Calc stores integers in decimal form so bit shifts must
+involve actual division.
+
+To gain a bit more efficiency, we could divide the integer into
address@hidden chunks, each of which can be handled quickly because
+they fit into Lisp integers. It turns out that Calc's arithmetic
+routines are especially fast when dividing by an integer less than
+1000, so we can set @var{n = 9} bits and use repeated division by 512:
+
address@hidden
+(defmath bcount ((natnum n))
+ (interactive 1 "bcnt")
+ (let ((count 0))
+ (while (not (fixnump n))
+ (let ((qr (idivmod n 512)))
+ (setq count (+ count (bcount-fixnum (cdr qr)))
+ n (car qr))))
+ (+ count (bcount-fixnum n))))
+
+(defun bcount-fixnum (n)
+ (let ((count 0))
+ (while (> n 0)
+ (setq count (+ count (logand n 1))
+ n (lsh n -1)))
+ count))
address@hidden smallexample
+
address@hidden
+Note that the second function uses @code{defun}, not @code{defmath}.
+Because this function deals only with native Lisp integers (``fixnums''),
+it can use the actual Emacs @code{+} and related functions rather
+than the slower but more general Calc equivalents which @code{defmath}
+uses.
+
+The @code{idivmod} function does an integer division, returning both
+the quotient and the remainder at once. Again, note that while it
+might seem that @samp{(logand n 511)} and @samp{(lsh n -9)} are
+more efficient ways to split off the bottom nine bits of @code{n},
+actually they are less efficient because each operation is really
+a division by 512 in disguise; @code{idivmod} allows us to do the
+same thing with a single division by 512.
+
address@hidden Sine Example, , Bit Counting Example, Example Definitions
address@hidden The Sine Function
+
address@hidden
address@hidden
address@hidden
address@hidden ignore
address@hidden mysin
+A somewhat limited sine function could be defined as follows, using the
+well-known Taylor series expansion for
address@hidden @math{\sin x}:
address@hidden @samp{sin(x)}:
+
address@hidden
+(defmath mysin ((float (anglep x)))
+ (interactive 1 "mysn")
+ (setq x (to-radians x)) ; Convert from current angular mode.
+ (let ((sum x) ; Initial term of Taylor expansion of sin.
+ newsum
+ (nfact 1) ; "nfact" equals "n" factorial at all times.
+ (xnegsqr :"-(x^2)")) ; "xnegsqr" equals -x^2.
+ (for ((n 3 100 2)) ; Upper limit of 100 is a good precaution.
+ (working "mysin" sum) ; Display "Working" message, if enabled.
+ (setq nfact (* nfact (1- n) n)
+ x (* x xnegsqr)
+ newsum (+ sum (/ x nfact)))
+ (if (~= newsum sum) ; If newsum is "nearly equal to" sum,
+ (break)) ; then we are done.
+ (setq sum newsum))
+ sum))
address@hidden smallexample
+
+The actual @code{sin} function in Calc works by first reducing the problem
+to a sine or cosine of a nonnegative number less than @cpiover{4}. This
+ensures that the Taylor series will converge quickly. Also, the calculation
+is carried out with two extra digits of precision to guard against cumulative
+round-off in @samp{sum}. Finally, complex arguments are allowed and handled
+by a separate algorithm.
+
address@hidden
+(defmath mysin ((float (scalarp x)))
+ (interactive 1 "mysn")
+ (setq x (to-radians x)) ; Convert from current angular mode.
+ (with-extra-prec 2 ; Evaluate with extra precision.
+ (cond ((complexp x)
+ (mysin-complex x))
+ ((< x 0)
+ (- (mysin-raw (- x))) ; Always call mysin-raw with x >= 0.
+ (t (mysin-raw x))))))
+
+(defmath mysin-raw (x)
+ (cond ((>= x 7)
+ (mysin-raw (% x (two-pi)))) ; Now x < 7.
+ ((> x (pi-over-2))
+ (- (mysin-raw (- x (pi))))) ; Now -pi/2 <= x <= pi/2.
+ ((> x (pi-over-4))
+ (mycos-raw (- x (pi-over-2)))) ; Now -pi/2 <= x <= pi/4.
+ ((< x (- (pi-over-4)))
+ (- (mycos-raw (+ x (pi-over-2))))) ; Now -pi/4 <= x <= pi/4,
+ (t (mysin-series x)))) ; so the series will be efficient.
address@hidden smallexample
+
address@hidden
+where @code{mysin-complex} is an appropriate function to handle complex
+numbers, @code{mysin-series} is the routine to compute the sine Taylor
+series as before, and @code{mycos-raw} is a function analogous to
address@hidden for cosines.
+
+The strategy is to ensure that @expr{x} is nonnegative before calling
address@hidden This function then recursively reduces its argument
+to a suitable range, namely, plus-or-minus @cpiover{4}. Note that each
+test, and particularly the first comparison against 7, is designed so
+that small roundoff errors cannot produce an infinite loop. (Suppose
+we compared with @samp{(two-pi)} instead; if due to roundoff problems
+the modulo operator ever returned @samp{(two-pi)} exactly, an infinite
+recursion could result!) We use modulo only for arguments that will
+clearly get reduced, knowing that the next rule will catch any reductions
+that this rule misses.
+
+If a program is being written for general use, it is important to code
+it carefully as shown in this second example. For quick-and-dirty programs,
+when you know that your own use of the sine function will never encounter
+a large argument, a simpler program like the first one shown is fine.
+
address@hidden Calling Calc from Your Programs, Internals, Example Definitions,
Lisp Definitions
address@hidden Calling Calc from Your Lisp Programs
+
address@hidden
+A later section (@pxref{Internals}) gives a full description of
+Calc's internal Lisp functions. It's not hard to call Calc from
+inside your programs, but the number of these functions can be daunting.
+So Calc provides one special ``programmer-friendly'' function called
address@hidden that can be made to do just about everything you
+need. It's not as fast as the low-level Calc functions, but it's
+much simpler to use!
+
+It may seem that @code{calc-eval} itself has a daunting number of
+options, but they all stem from one simple operation.
+
+In its simplest manifestation, @samp{(calc-eval "1+2")} parses the
+string @code{"1+2"} as if it were a Calc algebraic entry and returns
+the result formatted as a string: @code{"3"}.
+
+Since @code{calc-eval} is on the list of recommended @code{autoload}
+functions, you don't need to make any special preparations to load
+Calc before calling @code{calc-eval} the first time. Calc will be
+loaded and initialized for you.
+
+All the Calc modes that are currently in effect will be used when
+evaluating the expression and formatting the result.
+
address@hidden
address@hidden
+
address@hidden example
address@hidden ifinfo
address@hidden Additional Arguments to @code{calc-eval}
+
address@hidden
+If the input string parses to a list of expressions, Calc returns
+the results separated by @code{", "}. You can specify a different
+separator by giving a second string argument to @code{calc-eval}:
address@hidden(calc-eval "1+2,3+4" ";")} returns @code{"3;7"}.
+
+The ``separator'' can also be any of several Lisp symbols which
+request other behaviors from @code{calc-eval}. These are discussed
+one by one below.
+
+You can give additional arguments to be substituted for
address@hidden, @samp{$$}, and so on in the main expression. For
+example, @samp{(calc-eval "$/$$" nil "7" "1+1")} evaluates the
+expression @code{"7/(1+1)"} to yield the result @code{"3.5"}
+(assuming Fraction mode is not in effect). Note the @code{nil}
+used as a placeholder for the item-separator argument.
+
address@hidden
address@hidden
+
address@hidden example
address@hidden ifinfo
address@hidden Error Handling
+
address@hidden
+If @code{calc-eval} encounters an error, it returns a list containing
+the character position of the error, plus a suitable message as a
+string. Note that @samp{1 / 0} is @emph{not} an error by Calc's
+standards; it simply returns the string @code{"1 / 0"} which is the
+division left in symbolic form. But @samp{(calc-eval "1/")} will
+return the list @samp{(2 "Expected a number")}.
+
+If you bind the variable @code{calc-eval-error} to @code{t}
+using a @code{let} form surrounding the call to @code{calc-eval},
+errors instead call the Emacs @code{error} function which aborts
+to the Emacs command loop with a beep and an error message.
+
+If you bind this variable to the symbol @code{string}, error messages
+are returned as strings instead of lists. The character position is
+ignored.
+
+As a courtesy to other Lisp code which may be using Calc, be sure
+to bind @code{calc-eval-error} using @code{let} rather than changing
+it permanently with @code{setq}.
+
address@hidden
address@hidden
+
address@hidden example
address@hidden ifinfo
address@hidden Numbers Only
+
address@hidden
+Sometimes it is preferable to treat @samp{1 / 0} as an error
+rather than returning a symbolic result. If you pass the symbol
address@hidden as the second argument to @code{calc-eval}, results
+that are not constants are treated as errors. The error message
+reported is the first @code{calc-why} message if there is one,
+or otherwise ``Number expected.''
+
+A result is ``constant'' if it is a number, vector, or other
+object that does not include variables or function calls. If it
+is a vector, the components must themselves be constants.
+
address@hidden
address@hidden
+
address@hidden example
address@hidden ifinfo
address@hidden Default Modes
+
address@hidden
+If the first argument to @code{calc-eval} is a list whose first
+element is a formula string, then @code{calc-eval} sets all the
+various Calc modes to their default values while the formula is
+evaluated and formatted. For example, the precision is set to 12
+digits, digit grouping is turned off, and the Normal language
+mode is used.
+
+This same principle applies to the other options discussed below.
+If the first argument would normally be @var{x}, then it can also
+be the list @samp{(@var{x})} to use the default mode settings.
+
+If there are other elements in the list, they are taken as
+variable-name/value pairs which override the default mode
+settings. Look at the documentation at the front of the
address@hidden file to find the names of the Lisp variables for
+the various modes. The mode settings are restored to their
+original values when @code{calc-eval} is done.
+
+For example, @samp{(calc-eval '("$+$$" calc-internal-prec 8) 'num a b)}
+computes the sum of two numbers, requiring a numeric result, and
+using default mode settings except that the precision is 8 instead
+of the default of 12.
+
+It's usually best to use this form of @code{calc-eval} unless your
+program actually considers the interaction with Calc's mode settings
+to be a feature. This will avoid all sorts of potential ``gotchas'';
+consider what happens with @samp{(calc-eval "sqrt(2)" 'num)}
+when the user has left Calc in Symbolic mode or No-Simplify mode.
+
+As another example, @samp{(equal (calc-eval '("$<$$") nil a b) "1")}
+checks if the number in string @expr{a} is less than the one in
+string @expr{b}. Without using a list, the integer 1 might
+come out in a variety of formats which would be hard to test for
+conveniently: @code{"1"}, @code{"8#1"}, @code{"00001"}. (But
+see ``Predicates'' mode, below.)
+
address@hidden
address@hidden
+
address@hidden example
address@hidden ifinfo
address@hidden Raw Numbers
+
address@hidden
+Normally all input and output for @code{calc-eval} is done with strings.
+You can do arithmetic with, say, @samp{(calc-eval "$+$$" nil a b)}
+in place of @samp{(+ a b)}, but this is very inefficient since the
+numbers must be converted to and from string format as they are passed
+from one @code{calc-eval} to the next.
+
+If the separator is the symbol @code{raw}, the result will be returned
+as a raw Calc data structure rather than a string. You can read about
+how these objects look in the following sections, but usually you can
+treat them as ``black box'' objects with no important internal
+structure.
+
+There is also a @code{rawnum} symbol, which is a combination of
address@hidden (returning a raw Calc object) and @code{num} (signaling
+an error if that object is not a constant).
+
+You can pass a raw Calc object to @code{calc-eval} in place of a
+string, either as the formula itself or as one of the @samp{$}
+arguments. Thus @samp{(calc-eval "$+$$" 'raw a b)} is an
+addition function that operates on raw Calc objects. Of course
+in this case it would be easier to call the low-level @code{math-add}
+function in Calc, if you can remember its name.
+
+In particular, note that a plain Lisp integer is acceptable to Calc
+as a raw object. (All Lisp integers are accepted on input, but
+integers of more than six decimal digits are converted to ``big-integer''
+form for output. @xref{Data Type Formats}.)
+
+When it comes time to display the object, just use @samp{(calc-eval a)}
+to format it as a string.
+
+It is an error if the input expression evaluates to a list of
+values. The separator symbol @code{list} is like @code{raw}
+except that it returns a list of one or more raw Calc objects.
+
+Note that a Lisp string is not a valid Calc object, nor is a list
+containing a string. Thus you can still safely distinguish all the
+various kinds of error returns discussed above.
+
address@hidden
address@hidden
+
address@hidden example
address@hidden ifinfo
address@hidden Predicates
+
address@hidden
+If the separator symbol is @code{pred}, the result of the formula is
+treated as a true/false value; @code{calc-eval} returns @code{t} or
address@hidden, respectively. A value is considered ``true'' if it is a
+non-zero number, or false if it is zero or if it is not a number.
+
+For example, @samp{(calc-eval "$<$$" 'pred a b)} tests whether
+one value is less than another.
+
+As usual, it is also possible for @code{calc-eval} to return one of
+the error indicators described above. Lisp will interpret such an
+indicator as ``true'' if you don't check for it explicitly. If you
+wish to have an error register as ``false'', use something like
address@hidden(eq (calc-eval ...) t)}.
+
address@hidden
address@hidden
+
address@hidden example
address@hidden ifinfo
address@hidden Variable Values
+
address@hidden
+Variables in the formula passed to @code{calc-eval} are not normally
+replaced by their values. If you wish this, you can use the
address@hidden function (@pxref{Algebraic Manipulation}). For example,
+if 4 is stored in Calc variable @code{a} (i.e., in Lisp variable
address@hidden), then @samp{(calc-eval "a+pi")} will return the
+formula @code{"a + pi"}, but @samp{(calc-eval "evalv(a+pi)")}
+will return @code{"7.14159265359"}.
+
+To store in a Calc variable, just use @code{setq} to store in the
+corresponding Lisp variable. (This is obtained by prepending
address@hidden to the Calc variable name.) Calc routines will
+understand either string or raw form values stored in variables,
+although raw data objects are much more efficient. For example,
+to increment the Calc variable @code{a}:
+
address@hidden
+(setq var-a (calc-eval "evalv(a+1)" 'raw))
address@hidden example
+
address@hidden
address@hidden
+
address@hidden example
address@hidden ifinfo
address@hidden Stack Access
+
address@hidden
+If the separator symbol is @code{push}, the formula argument is
+evaluated (with possible @samp{$} expansions, as usual). The
+result is pushed onto the Calc stack. The return value is @code{nil}
+(unless there is an error from evaluating the formula, in which
+case the return value depends on @code{calc-eval-error} in the
+usual way).
+
+If the separator symbol is @code{pop}, the first argument to
address@hidden must be an integer instead of a string. That
+many values are popped from the stack and thrown away. A negative
+argument deletes the entry at that stack level. The return value
+is the number of elements remaining in the stack after popping;
address@hidden(calc-eval 0 'pop)} is a good way to measure the size of
+the stack.
+
+If the separator symbol is @code{top}, the first argument to
address@hidden must again be an integer. The value at that
+stack level is formatted as a string and returned. Thus
address@hidden(calc-eval 1 'top)} returns the top-of-stack value. If the
+integer is out of range, @code{nil} is returned.
+
+The separator symbol @code{rawtop} is just like @code{top} except
+that the stack entry is returned as a raw Calc object instead of
+as a string.
+
+In all of these cases the first argument can be made a list in
+order to force the default mode settings, as described above.
+Thus @samp{(calc-eval '(2 calc-number-radix 16) 'top)} returns the
+second-to-top stack entry, formatted as a string using the default
+instead of current display modes, except that the radix is
+hexadecimal instead of decimal.
+
+It is, of course, polite to put the Calc stack back the way you
+found it when you are done, unless the user of your program is
+actually expecting it to affect the stack.
+
+Note that you do not actually have to switch into the @samp{*Calculator*}
+buffer in order to use @code{calc-eval}; it temporarily switches into
+the stack buffer if necessary.
+
address@hidden
address@hidden
+
address@hidden example
address@hidden ifinfo
address@hidden Keyboard Macros
+
address@hidden
+If the separator symbol is @code{macro}, the first argument must be a
+string of characters which Calc can execute as a sequence of keystrokes.
+This switches into the Calc buffer for the duration of the macro.
+For example, @samp{(calc-eval "vx5\rVR+" 'macro)} pushes the
+vector @samp{[1,2,3,4,5]} on the stack and then replaces it
+with the sum of those numbers. Note that @samp{\r} is the Lisp
+notation for the carriage-return, @key{RET}, character.
+
+If your keyboard macro wishes to pop the stack, @samp{\C-d} is
+safer than @samp{\177} (the @key{DEL} character) because some
+installations may have switched the meanings of @key{DEL} and
address@hidden Calc always interprets @kbd{C-d} as a synonym for
+``pop-stack'' regardless of key mapping.
+
+If you provide a third argument to @code{calc-eval}, evaluation
+of the keyboard macro will leave a record in the Trail using
+that argument as a tag string. Normally the Trail is unaffected.
+
+The return value in this case is always @code{nil}.
+
address@hidden
address@hidden
+
address@hidden example
address@hidden ifinfo
address@hidden Lisp Evaluation
+
address@hidden
+Finally, if the separator symbol is @code{eval}, then the Lisp
address@hidden function is called on the first argument, which must
+be a Lisp expression rather than a Calc formula. Remember to
+quote the expression so that it is not evaluated until inside
address@hidden
+
+The difference from plain @code{eval} is that @code{calc-eval}
+switches to the Calc buffer before evaluating the expression.
+For example, @samp{(calc-eval '(setq calc-internal-prec 17) 'eval)}
+will correctly affect the buffer-local Calc precision variable.
+
+An alternative would be @samp{(calc-eval '(calc-precision 17) 'eval)}.
+This is evaluating a call to the function that is normally invoked
+by the @kbd{p} key, giving it 17 as its ``numeric prefix argument.''
+Note that this function will leave a message in the echo area as
+a side effect. Also, all Calc functions switch to the Calc buffer
+automatically if not invoked from there, so the above call is
+also equivalent to @samp{(calc-precision 17)} by itself.
+In all cases, Calc uses @code{save-excursion} to switch back to
+your original buffer when it is done.
+
+As usual the first argument can be a list that begins with a Lisp
+expression to use default instead of current mode settings.
+
+The result of @code{calc-eval} in this usage is just the result
+returned by the evaluated Lisp expression.
+
address@hidden
address@hidden
+
address@hidden example
address@hidden ifinfo
address@hidden Example
+
address@hidden
address@hidden convert-temp
+Here is a sample Emacs command that uses @code{calc-eval}. Suppose
+you have a document with lots of references to temperatures on the
+Fahrenheit scale, say ``98.6 F'', and you wish to convert these
+references to Centigrade. The following command does this conversion.
+Place the Emacs cursor right after the letter ``F'' and invoke the
+command to change ``98.6 F'' to ``37 C''. Or, if the temperature is
+already in Centigrade form, the command changes it back to Fahrenheit.
+
address@hidden
+(defun convert-temp ()
+ (interactive)
+ (save-excursion
+ (re-search-backward "[^-.0-9]\\([-.0-9]+\\) *\\([FC]\\)")
+ (let* ((top1 (match-beginning 1))
+ (bot1 (match-end 1))
+ (number (buffer-substring top1 bot1))
+ (top2 (match-beginning 2))
+ (bot2 (match-end 2))
+ (type (buffer-substring top2 bot2)))
+ (if (equal type "F")
+ (setq type "C"
+ number (calc-eval "($ - 32)*5/9" nil number))
+ (setq type "F"
+ number (calc-eval "$*9/5 + 32" nil number)))
+ (goto-char top2)
+ (delete-region top2 bot2)
+ (insert-before-markers type)
+ (goto-char top1)
+ (delete-region top1 bot1)
+ (if (string-match "\\.$" number) ; change "37." to "37"
+ (setq number (substring number 0 -1)))
+ (insert number))))
address@hidden example
+
+Note the use of @code{insert-before-markers} when changing between
+``F'' and ``C'', so that the character winds up before the cursor
+instead of after it.
+
address@hidden Internals, , Calling Calc from Your Programs, Lisp Definitions
address@hidden Calculator Internals
+
address@hidden
+This section describes the Lisp functions defined by the Calculator that
+may be of use to user-written Calculator programs (as described in the
+rest of this chapter). These functions are shown by their names as they
+conventionally appear in @code{defmath}. Their full Lisp names are
+generally gotten by prepending @samp{calcFunc-} or @samp{math-} to their
+apparent names. (Names that begin with @samp{calc-} are already in
+their full Lisp form.) You can use the actual full names instead if you
+prefer them, or if you are calling these functions from regular Lisp.
+
+The functions described here are scattered throughout the various
+Calc component files. Note that @file{calc.el} includes @code{autoload}s
+for only a few component files; when Calc wants to call an advanced
+function it calls @samp{(calc-extensions)} first; this function
+autoloads @file{calc-ext.el}, which in turn autoloads all the functions
+in the remaining component files.
+
+Because @code{defmath} itself uses the extensions, user-written code
+generally always executes with the extensions already loaded, so
+normally you can use any Calc function and be confident that it will
+be autoloaded for you when necessary. If you are doing something
+special, check carefully to make sure each function you are using is
+from @file{calc.el} or its components, and call @samp{(calc-extensions)}
+before using any function based in @file{calc-ext.el} if you can't
+prove this file will already be loaded.
+
address@hidden
+* Data Type Formats::
+* Interactive Lisp Functions::
+* Stack Lisp Functions::
+* Predicates::
+* Computational Lisp Functions::
+* Vector Lisp Functions::
+* Symbolic Lisp Functions::
+* Formatting Lisp Functions::
+* Hooks::
address@hidden menu
+
address@hidden Data Type Formats, Interactive Lisp Functions, Internals,
Internals
address@hidden Data Type Formats
+
address@hidden
+Integers are stored in either of two ways, depending on their magnitude.
+Integers less than one million in absolute value are stored as standard
+Lisp integers. This is the only storage format for Calc data objects
+which is not a Lisp list.
+
+Large integers are stored as lists of the form @samp{(bigpos @var{d0}
address@hidden @var{d2} @dots{})} for positive integers 1000000 or more, or
address@hidden(bigneg @var{d0} @var{d1} @var{d2} @dots{})} for negative integers
address@hidden or less. Each @var{d} is a base-1000 ``digit,'' a Lisp integer
+from 0 to 999. The least significant digit is @var{d0}; the last digit,
address@hidden, which is always nonzero, is the most significant digit. For
+example, the integer @mathit{-12345678} is stored as @samp{(bigneg 678 345
12)}.
+
+The distinction between small and large integers is entirely hidden from
+the user. In @code{defmath} definitions, the Lisp predicate @code{integerp}
+returns true for either kind of integer, and in general both big and small
+integers are accepted anywhere the word ``integer'' is used in this manual.
+If the distinction must be made, native Lisp integers are called @dfn{fixnums}
+and large integers are called @dfn{bignums}.
+
+Fractions are stored as a list of the form, @samp{(frac @var{n} @var{d})}
+where @var{n} is an integer (big or small) numerator, @var{d} is an
+integer denominator greater than one, and @var{n} and @var{d} are relatively
+prime. Note that fractions where @var{d} is one are automatically converted
+to plain integers by all math routines; fractions where @var{d} is negative
+are normalized by negating the numerator and denominator.
+
+Floating-point numbers are stored in the form, @samp{(float @var{mant}
address@hidden)}, where @var{mant} (the ``mantissa'') is an integer less than
address@hidden@var{p}} in absolute value (@var{p} represents the current
+precision), and @var{exp} (the ``exponent'') is a fixnum. The value of
+the float is @address@hidden * address@hidden For example, the number
address@hidden is stored as @samp{(float -314 -2) = -314*10^-2}. Other
constraints
+are that the number 0.0 is always stored as @samp{(float 0 0)}, and,
+except for the 0.0 case, the rightmost base-10 digit of @var{mant} is
+always nonzero. (If the rightmost digit is zero, the number is
+rearranged by dividing @var{mant} by ten and incrementing @var{exp}.)
+
+Rectangular complex numbers are stored in the form @samp{(cplx @var{re}
address@hidden)}, where @var{re} and @var{im} are each real numbers, either
+integers, fractions, or floats. The value is @address@hidden + @var{im}i}.
+The @var{im} part is nonzero; complex numbers with zero imaginary
+components are converted to real numbers automatically.
+
+Polar complex numbers are stored in the form @samp{(polar @var{r}
address@hidden)}, where @var{r} is a positive real value and @var{theta}
+is a real value or HMS form representing an angle. This angle is
+usually normalized to lie in the interval @samp{(-180 ..@: 180)} degrees,
+or @samp{(-pi ..@: pi)} radians, according to the current angular mode.
+If the angle is 0 the value is converted to a real number automatically.
+(If the angle is 180 degrees, the value is usually also converted to a
+negative real number.)
+
+Hours-minutes-seconds forms are stored as @samp{(hms @var{h} @var{m}
address@hidden)}, where @var{h} is an integer or an integer-valued float (i.e.,
+a float with @address@hidden >= 0}), @var{m} is an integer or integer-valued
+float in the range @address@hidden ..@: 60)}}, and @var{s} is any real number
+in the range @samp{[0 ..@: 60)}.
+
+Date forms are stored as @samp{(date @var{n})}, where @var{n} is
+a real number that counts days since midnight on the morning of
+January 1, 1 AD. If @var{n} is an integer, this is a pure date
+form. If @var{n} is a fraction or float, this is a date/time form.
+
+Modulo forms are stored as @samp{(mod @var{n} @var{m})}, where @var{m} is a
+positive real number or HMS form, and @var{n} is a real number or HMS
+form in the range @samp{[0 ..@: @var{m})}.
+
+Error forms are stored as @samp{(sdev @var{x} @var{sigma})}, where @var{x}
+is the mean value and @var{sigma} is the standard deviation. Each
+component is either a number, an HMS form, or a symbolic object
+(a variable or function call). If @var{sigma} is zero, the value is
+converted to a plain real number. If @var{sigma} is negative or
+complex, it is automatically normalized to be a positive real.
+
+Interval forms are stored as @samp{(intv @var{mask} @var{lo} @var{hi})},
+where @var{mask} is one of the integers 0, 1, 2, or 3, and @var{lo} and
address@hidden are real numbers, HMS forms, or symbolic objects. The @var{mask}
+is a binary integer where 1 represents the fact that the interval is
+closed on the high end, and 2 represents the fact that it is closed on
+the low end. (Thus 3 represents a fully closed interval.) The interval
address@hidden@samp{(intv 3 @var{x} @var{x})}} is converted to the plain number
@var{x};
+intervals @samp{(intv @var{mask} @var{x} @var{x})} for any other @var{mask}
+represent empty intervals. If @var{hi} is less than @var{lo}, the interval
+is converted to a standard empty interval by replacing @var{hi} with @var{lo}.
+
+Vectors are stored as @samp{(vec @var{v1} @var{v2} @dots{})}, where @var{v1}
+is the first element of the vector, @var{v2} is the second, and so on.
+An empty vector is stored as @samp{(vec)}. A matrix is simply a vector
+where all @var{v}'s are themselves vectors of equal lengths. Note that
+Calc vectors are unrelated to the Emacs Lisp ``vector'' type, which is
+generally unused by Calc data structures.
+
+Variables are stored as @samp{(var @var{name} @var{sym})}, where
address@hidden is a Lisp symbol whose print name is used as the visible name
+of the variable, and @var{sym} is a Lisp symbol in which the variable's
+value is actually stored. Thus, @samp{(var pi var-pi)} represents the
+special constant @samp{pi}. Almost always, the form is @samp{(var
address@hidden address@hidden)}. If the variable name was entered with @code{#}
+signs (which are converted to hyphens internally), the form is
address@hidden(var @var{u} @var{v})}, where @var{u} is a symbol whose name
+contains @code{#} characters, and @var{v} is a symbol that contains
address@hidden characters instead. The value of a variable is the Calc
+object stored in its @var{sym} symbol's value cell. If the symbol's
+value cell is void or if it contains @code{nil}, the variable has no
+value. Special constants have the form @samp{(special-const
address@hidden)} stored in their value cell, where @var{value} is a formula
+which is evaluated when the constant's value is requested. Variables
+which represent units are not stored in any special way; they are units
+only because their names appear in the units table. If the value
+cell contains a string, it is parsed to get the variable's value when
+the variable is used.
+
+A Lisp list with any other symbol as the first element is a function call.
+The symbols @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, @code{^},
+and @code{|} represent special binary operators; these lists are always
+of the form @samp{(@var{op} @var{lhs} @var{rhs})} where @var{lhs} is the
+sub-formula on the lefthand side and @var{rhs} is the sub-formula on the
+right. The symbol @code{neg} represents unary negation; this list is always
+of the form @samp{(neg @var{arg})}. Any other symbol @var{func} represents a
+function that would be displayed in function-call notation; the symbol
address@hidden is in general always of the form @address@hidden
+The function cell of the symbol @var{func} should contain a Lisp function
+for evaluating a call to @var{func}. This function is passed the remaining
+elements of the list (themselves already evaluated) as arguments; such
+functions should return @code{nil} or call @code{reject-arg} to signify
+that they should be left in symbolic form, or they should return a Calc
+object which represents their value, or a list of such objects if they
+wish to return multiple values. (The latter case is allowed only for
+functions which are the outer-level call in an expression whose value is
+about to be pushed on the stack; this feature is considered obsolete
+and is not used by any built-in Calc functions.)
+
address@hidden Interactive Lisp Functions, Stack Lisp Functions, Data Type
Formats, Internals
address@hidden Interactive Functions
+
address@hidden
+The functions described here are used in implementing interactive Calc
+commands. Note that this list is not exhaustive! If there is an
+existing command that behaves similarly to the one you want to define,
+you may find helpful tricks by checking the source code for that command.
+
address@hidden calc-set-command-flag flag
+Set the command flag @var{flag}. This is generally a Lisp symbol, but
+may in fact be anything. The effect is to add @var{flag} to the list
+stored in the variable @code{calc-command-flags}, unless it is already
+there. @xref{Defining Simple Commands}.
address@hidden defun
+
address@hidden calc-clear-command-flag flag
+If @var{flag} appears among the list of currently-set command flags,
+remove it from that list.
address@hidden defun
+
address@hidden calc-record-undo rec
+Add the ``undo record'' @var{rec} to the list of steps to take if the
+current operation should need to be undone. Stack push and pop functions
+automatically call @code{calc-record-undo}, so the kinds of undo records
+you might need to create take the form @samp{(set @var{sym} @var{value})},
+which says that the Lisp variable @var{sym} was changed and had previously
+contained @var{value}; @samp{(store @var{var} @var{value})} which says that
+the Calc variable @var{var} (a string which is the name of the symbol that
+contains the variable's value) was stored and its previous value was
address@hidden (either a Calc data object, or @code{nil} if the variable was
+previously void); or @samp{(eval @var{undo} @var{redo} @var{args} @dots{})},
+which means that to undo requires calling the function @samp{(@var{undo}
address@hidden @dots{})} and, if the undo is later redone, calling
address@hidden(@var{redo} @var{args} @dots{})}.
address@hidden defun
+
address@hidden calc-record-why msg args
+Record the error or warning message @var{msg}, which is normally a string.
+This message will be replayed if the user types @kbd{w} (@code{calc-why});
+if the message string begins with a @samp{*}, it is considered important
+enough to display even if the user doesn't type @kbd{w}. If one or more
address@hidden are present, the displayed message will be of the form,
address@hidden@var{msg}: @var{arg1}, @var{arg2}, @dots{}}, where the arguments
are
+formatted on the assumption that they are either strings or Calc objects of
+some sort. If @var{msg} is a symbol, it is the name of a Calc predicate
+(such as @code{integerp} or @code{numvecp}) which the arguments did not
+satisfy; it is expanded to a suitable string such as ``Expected an
+integer.'' The @code{reject-arg} function calls @code{calc-record-why}
+automatically; @pxref{Predicates}.
address@hidden defun
+
address@hidden calc-is-inverse
+This predicate returns true if the current command is inverse,
+i.e., if the Inverse (@kbd{I} key) flag was set.
address@hidden defun
+
address@hidden calc-is-hyperbolic
+This predicate is the analogous function for the @kbd{H} key.
address@hidden defun
+
address@hidden Stack Lisp Functions, Predicates, Interactive Lisp Functions,
Internals
address@hidden Stack-Oriented Functions
+
address@hidden
+The functions described here perform various operations on the Calc
+stack and trail. They are to be used in interactive Calc commands.
+
address@hidden calc-push-list vals n
+Push the Calc objects in list @var{vals} onto the stack at stack level
address@hidden If @var{n} is omitted it defaults to 1, so that the elements
+are pushed at the top of the stack. If @var{n} is greater than 1, the
+elements will be inserted into the stack so that the last element will
+end up at level @var{n}, the next-to-last at level @var{n}+1, etc.
+The elements of @var{vals} are assumed to be valid Calc objects, and
+are not evaluated, rounded, or renormalized in any way. If @var{vals}
+is an empty list, nothing happens.
+
+The stack elements are pushed without any sub-formula selections.
+You can give an optional third argument to this function, which must
+be a list the same size as @var{vals} of selections. Each selection
+must be @code{eq} to some sub-formula of the corresponding formula
+in @var{vals}, or @code{nil} if that formula should have no selection.
address@hidden defun
+
address@hidden calc-top-list n m
+Return a list of the @var{n} objects starting at level @var{m} of the
+stack. If @var{m} is omitted it defaults to 1, so that the elements are
+taken from the top of the stack. If @var{n} is omitted, it also
+defaults to 1, so that the top stack element (in the form of a
+one-element list) is returned. If @var{m} is greater than 1, the
address@hidden stack element will be at the end of the list, the @var{m}+1st
+element will be next-to-last, etc. If @var{n} or @var{m} are out of
+range, the command is aborted with a suitable error message. If @var{n}
+is zero, the function returns an empty list. The stack elements are not
+evaluated, rounded, or renormalized.
+
+If any stack elements contain selections, and selections have not
+been disabled by the @kbd{j e} (@code{calc-enable-selections}) command,
+this function returns the selected portions rather than the entire
+stack elements. It can be given a third ``selection-mode'' argument
+which selects other behaviors. If it is the symbol @code{t}, then
+a selection in any of the requested stack elements produces an
+``invalid operation on selections'' error. If it is the symbol @code{full},
+the whole stack entry is always returned regardless of selections.
+If it is the symbol @code{sel}, the selected portion is always returned,
+or @code{nil} if there is no selection. (This mode ignores the @kbd{j e}
+command.) If the symbol is @code{entry}, the complete stack entry in
+list form is returned; the first element of this list will be the whole
+formula, and the third element will be the selection (or @code{nil}).
address@hidden defun
+
address@hidden calc-pop-stack n m
+Remove the specified elements from the stack. The parameters @var{n}
+and @var{m} are defined the same as for @code{calc-top-list}. The return
+value of @code{calc-pop-stack} is uninteresting.
+
+If there are any selected sub-formulas among the popped elements, and
address@hidden e} has not been used to disable selections, this produces an
+error without changing the stack. If you supply an optional third
+argument of @code{t}, the stack elements are popped even if they
+contain selections.
address@hidden defun
+
address@hidden calc-record-list vals tag
+This function records one or more results in the trail. The @var{vals}
+are a list of strings or Calc objects. The @var{tag} is the four-character
+tag string to identify the values. If @var{tag} is omitted, a blank tag
+will be used.
address@hidden defun
+
address@hidden calc-normalize n
+This function takes a Calc object and ``normalizes'' it. At the very
+least this involves re-rounding floating-point values according to the
+current precision and other similar jobs. Also, unless the user has
+selected No-Simplify mode (@pxref{Simplification Modes}), this involves
+actually evaluating a formula object by executing the function calls
+it contains, and possibly also doing algebraic simplification, etc.
address@hidden defun
+
address@hidden calc-top-list-n n m
+This function is identical to @code{calc-top-list}, except that it calls
address@hidden on the values that it takes from the stack. They
+are also passed through @code{check-complete}, so that incomplete
+objects will be rejected with an error message. All computational
+commands should use this in preference to @code{calc-top-list}; the only
+standard Calc commands that operate on the stack without normalizing
+are stack management commands like @code{calc-enter} and @code{calc-roll-up}.
+This function accepts the same optional selection-mode argument as
address@hidden
address@hidden defun
+
address@hidden calc-top-n m
+This function is a convenient form of @code{calc-top-list-n} in which only
+a single element of the stack is taken and returned, rather than a list
+of elements. This also accepts an optional selection-mode argument.
address@hidden defun
+
address@hidden calc-enter-result n tag vals
+This function is a convenient interface to most of the above functions.
+The @var{vals} argument should be either a single Calc object, or a list
+of Calc objects; the object or objects are normalized, and the top @var{n}
+stack entries are replaced by the normalized objects. If @var{tag} is
address@hidden, the normalized objects are also recorded in the trail.
+A typical stack-based computational command would take the form,
+
address@hidden
+(calc-enter-result @var{n} @var{tag} (cons 'address@hidden
+ (calc-top-list-n @var{n})))
address@hidden smallexample
+
+If any of the @var{n} stack elements replaced contain sub-formula
+selections, and selections have not been disabled by @kbd{j e},
+this function takes one of two courses of action. If @var{n} is
+equal to the number of elements in @var{vals}, then each element of
address@hidden is spliced into the corresponding selection; this is what
+happens when you use the @key{TAB} key, or when you use a unary
+arithmetic operation like @code{sqrt}. If @var{vals} has only one
+element but @var{n} is greater than one, there must be only one
+selection among the top @var{n} stack elements; the element from
address@hidden is spliced into that selection. This is what happens when
+you use a binary arithmetic operation like @kbd{+}. Any other
+combination of @var{n} and @var{vals} is an error when selections
+are present.
address@hidden defun
+
address@hidden calc-unary-op tag func arg
+This function implements a unary operator that allows a numeric prefix
+argument to apply the operator over many stack entries. If the prefix
+argument @var{arg} is @code{nil}, this uses @code{calc-enter-result}
+as outlined above. Otherwise, it maps the function over several stack
+elements; @pxref{Prefix Arguments}. For example,
+
address@hidden
+(defun calc-zeta (arg)
+ (interactive "P")
+ (calc-unary-op "zeta" 'calcFunc-zeta arg))
address@hidden smallexample
address@hidden defun
+
address@hidden calc-binary-op tag func arg ident unary
+This function implements a binary operator, analogously to
address@hidden The optional @var{ident} and @var{unary}
+arguments specify the behavior when the prefix argument is zero or
+one, respectively. If the prefix is zero, the value @var{ident}
+is pushed onto the stack, if specified, otherwise an error message
+is displayed. If the prefix is one, the unary function @var{unary}
+is applied to the top stack element, or, if @var{unary} is not
+specified, nothing happens. When the argument is two or more,
+the binary function @var{func} is reduced across the top @var{arg}
+stack elements; when the argument is negative, the function is
+mapped between the next-to-top @address@hidden stack elements and the
+top element.
address@hidden defun
+
address@hidden calc-stack-size
+Return the number of elements on the stack as an integer. This count
+does not include elements that have been temporarily hidden by stack
+truncation; @pxref{Truncating the Stack}.
address@hidden defun
+
address@hidden calc-cursor-stack-index n
+Move the point to the @var{n}th stack entry. If @var{n} is zero, this
+will be the @samp{.} line. If @var{n} is from 1 to the current stack size,
+this will be the beginning of the first line of that stack entry's display.
+If line numbers are enabled, this will move to the first character of the
+line number, not the stack entry itself.
address@hidden defun
+
address@hidden calc-substack-height n
+Return the number of lines between the beginning of the @var{n}th stack
+entry and the bottom of the buffer. If @var{n} is zero, this
+will be one (assuming no stack truncation). If all stack entries are
+one line long (i.e., no matrices are displayed), the return value will
+be equal @var{n}+1 as long as @var{n} is in range. (Note that in Big
+mode, the return value includes the blank lines that separate stack
+entries.)
address@hidden defun
+
address@hidden calc-refresh
+Erase the @code{*Calculator*} buffer and reformat its contents from memory.
+This must be called after changing any parameter, such as the current
+display radix, which might change the appearance of existing stack
+entries. (During a keyboard macro invoked by the @kbd{X} key, refreshing
+is suppressed, but a flag is set so that the entire stack will be refreshed
+rather than just the top few elements when the macro finishes.)
address@hidden defun
+
address@hidden Predicates, Computational Lisp Functions, Stack Lisp Functions,
Internals
address@hidden Predicates
+
address@hidden
+The functions described here are predicates, that is, they return a
+true/false value where @code{nil} means false and anything else means
+true. These predicates are expanded by @code{defmath}, for example,
+from @code{zerop} to @code{math-zerop}. In many cases they correspond
+to native Lisp functions by the same name, but are extended to cover
+the full range of Calc data types.
+
address@hidden zerop x
+Returns true if @var{x} is numerically zero, in any of the Calc data
+types. (Note that for some types, such as error forms and intervals,
+it never makes sense to return true.) In @code{defmath}, the expression
address@hidden(= x 0)} will automatically be converted to @samp{(math-zerop x)},
+and @samp{(/= x 0)} will be converted to @samp{(not (math-zerop x))}.
address@hidden defun
+
address@hidden negp x
+Returns true if @var{x} is negative. This accepts negative real numbers
+of various types, negative HMS and date forms, and intervals in which
+all included values are negative. In @code{defmath}, the expression
address@hidden(< x 0)} will automatically be converted to @samp{(math-negp x)},
+and @samp{(>= x 0)} will be converted to @samp{(not (math-negp x))}.
address@hidden defun
+
address@hidden posp x
+Returns true if @var{x} is positive (and non-zero). For complex
+numbers, none of these three predicates will return true.
address@hidden defun
+
address@hidden looks-negp x
+Returns true if @var{x} is ``negative-looking.'' This returns true if
address@hidden is a negative number, or a formula with a leading minus sign
+such as @samp{-a/b}. In other words, this is an object which can be
+made simpler by calling @code{(- @var{x})}.
address@hidden defun
+
address@hidden integerp x
+Returns true if @var{x} is an integer of any size.
address@hidden defun
+
address@hidden fixnump x
+Returns true if @var{x} is a native Lisp integer.
address@hidden defun
+
address@hidden natnump x
+Returns true if @var{x} is a nonnegative integer of any size.
address@hidden defun
+
address@hidden fixnatnump x
+Returns true if @var{x} is a nonnegative Lisp integer.
address@hidden defun
+
address@hidden num-integerp x
+Returns true if @var{x} is numerically an integer, i.e., either a
+true integer or a float with no significant digits to the right of
+the decimal point.
address@hidden defun
+
address@hidden messy-integerp x
+Returns true if @var{x} is numerically, but not literally, an integer.
+A value is @code{num-integerp} if it is @code{integerp} or
address@hidden (but it is never both at once).
address@hidden defun
+
address@hidden num-natnump x
+Returns true if @var{x} is numerically a nonnegative integer.
address@hidden defun
+
address@hidden evenp x
+Returns true if @var{x} is an even integer.
address@hidden defun
+
address@hidden looks-evenp x
+Returns true if @var{x} is an even integer, or a formula with a leading
+multiplicative coefficient which is an even integer.
address@hidden defun
+
address@hidden oddp x
+Returns true if @var{x} is an odd integer.
address@hidden defun
+
address@hidden ratp x
+Returns true if @var{x} is a rational number, i.e., an integer or a
+fraction.
address@hidden defun
+
address@hidden realp x
+Returns true if @var{x} is a real number, i.e., an integer, fraction,
+or floating-point number.
address@hidden defun
+
address@hidden anglep x
+Returns true if @var{x} is a real number or HMS form.
address@hidden defun
+
address@hidden floatp x
+Returns true if @var{x} is a float, or a complex number, error form,
+interval, date form, or modulo form in which at least one component
+is a float.
address@hidden defun
+
address@hidden complexp x
+Returns true if @var{x} is a rectangular or polar complex number
+(but not a real number).
address@hidden defun
+
address@hidden rect-complexp x
+Returns true if @var{x} is a rectangular complex number.
address@hidden defun
+
address@hidden polar-complexp x
+Returns true if @var{x} is a polar complex number.
address@hidden defun
+
address@hidden numberp x
+Returns true if @var{x} is a real number or a complex number.
address@hidden defun
+
address@hidden scalarp x
+Returns true if @var{x} is a real or complex number or an HMS form.
address@hidden defun
+
address@hidden vectorp x
+Returns true if @var{x} is a vector (this simply checks if its argument
+is a list whose first element is the symbol @code{vec}).
address@hidden defun
+
address@hidden numvecp x
+Returns true if @var{x} is a number or vector.
address@hidden defun
+
address@hidden matrixp x
+Returns true if @var{x} is a matrix, i.e., a vector of one or more vectors,
+all of the same size.
address@hidden defun
+
address@hidden square-matrixp x
+Returns true if @var{x} is a square matrix.
address@hidden defun
+
address@hidden objectp x
+Returns true if @var{x} is any numeric Calc object, including real and
+complex numbers, HMS forms, date forms, error forms, intervals, and
+modulo forms. (Note that error forms and intervals may include formulas
+as their components; see @code{constp} below.)
address@hidden defun
+
address@hidden objvecp x
+Returns true if @var{x} is an object or a vector. This also accepts
+incomplete objects, but it rejects variables and formulas (except as
+mentioned above for @code{objectp}).
address@hidden defun
+
address@hidden primp x
+Returns true if @var{x} is a ``primitive'' or ``atomic'' Calc object,
+i.e., one whose components cannot be regarded as sub-formulas. This
+includes variables, and all @code{objectp} types except error forms
+and intervals.
address@hidden defun
+
address@hidden constp x
+Returns true if @var{x} is constant, i.e., a real or complex number,
+HMS form, date form, or error form, interval, or vector all of whose
+components are @code{constp}.
address@hidden defun
+
address@hidden lessp x y
+Returns true if @var{x} is numerically less than @var{y}. Returns false
+if @var{x} is greater than or equal to @var{y}, or if the order is
+undefined or cannot be determined. Generally speaking, this works
+by checking whether @address@hidden - @var{y}} is @code{negp}. In
address@hidden, the expression @samp{(< x y)} will automatically be
+converted to @samp{(lessp x y)}; expressions involving @code{>}, @code{<=},
+and @code{>=} are similarly converted in terms of @code{lessp}.
address@hidden defun
+
address@hidden beforep x y
+Returns true if @var{x} comes before @var{y} in a canonical ordering
+of Calc objects. If @var{x} and @var{y} are both real numbers, this
+will be the same as @code{lessp}. But whereas @code{lessp} considers
+other types of objects to be unordered, @code{beforep} puts any two
+objects into a definite, consistent order. The @code{beforep}
+function is used by the @kbd{V S} vector-sorting command, and also
+by @kbd{a s} to put the terms of a product into canonical order:
+This allows @samp{x y + y x} to be simplified easily to @samp{2 x y}.
address@hidden defun
+
address@hidden equal x y
+This is the standard Lisp @code{equal} predicate; it returns true if
address@hidden and @var{y} are structurally identical. This is the usual way
+to compare numbers for equality, but note that @code{equal} will treat
+0 and 0.0 as different.
address@hidden defun
+
address@hidden math-equal x y
+Returns true if @var{x} and @var{y} are numerically equal, either because
+they are @code{equal}, or because their difference is @code{zerop}. In
address@hidden, the expression @samp{(= x y)} will automatically be
+converted to @samp{(math-equal x y)}.
address@hidden defun
+
address@hidden equal-int x n
+Returns true if @var{x} and @var{n} are numerically equal, where @var{n}
+is a fixnum which is not a multiple of 10. This will automatically be
+used by @code{defmath} in place of the more general @code{math-equal}
+whenever possible.
address@hidden defun
+
address@hidden nearly-equal x y
+Returns true if @var{x} and @var{y}, as floating-point numbers, are
+equal except possibly in the last decimal place. For example,
+314.159 and 314.166 are considered nearly equal if the current
+precision is 6 (since they differ by 7 units), but not if the current
+precision is 7 (since they differ by 70 units). Most functions which
+use series expansions use @code{with-extra-prec} to evaluate the
+series with 2 extra digits of precision, then use @code{nearly-equal}
+to decide when the series has converged; this guards against cumulative
+error in the series evaluation without doing extra work which would be
+lost when the result is rounded back down to the current precision.
+In @code{defmath}, this can be written @samp{(~= @var{x} @var{y})}.
+The @var{x} and @var{y} can be numbers of any kind, including complex.
address@hidden defun
+
address@hidden nearly-zerop x y
+Returns true if @var{x} is nearly zero, compared to @var{y}. This
+checks whether @var{x} plus @var{y} would by be @code{nearly-equal}
+to @var{y} itself, to within the current precision, in other words,
+if adding @var{x} to @var{y} would have a negligible effect on @var{y}
+due to roundoff error. @var{X} may be a real or complex number, but
address@hidden must be real.
address@hidden defun
+
address@hidden is-true x
+Return true if the formula @var{x} represents a true value in
+Calc, not Lisp, terms. It tests if @var{x} is a non-zero number
+or a provably non-zero formula.
address@hidden defun
+
address@hidden reject-arg val pred
+Abort the current function evaluation due to unacceptable argument values.
+This calls @samp{(calc-record-why @var{pred} @var{val})}, then signals a
+Lisp error which @code{normalize} will trap. The net effect is that the
+function call which led here will be left in symbolic form.
address@hidden defun
+
address@hidden inexact-value
+If Symbolic mode is enabled, this will signal an error that causes
address@hidden to leave the formula in symbolic form, with the message
+``Inexact result.'' (This function has no effect when not in Symbolic mode.)
+Note that if your function calls @samp{(sin 5)} in Symbolic mode, the
address@hidden function will call @code{inexact-value}, which will cause your
+function to be left unsimplified. You may instead wish to call
address@hidden(normalize (list 'calcFunc-sin 5))}, which in Symbolic mode will
+return the formula @samp{sin(5)} to your function.
address@hidden defun
+
address@hidden overflow
+This signals an error that will be reported as a floating-point overflow.
address@hidden defun
+
address@hidden underflow
+This signals a floating-point underflow.
address@hidden defun
+
address@hidden Computational Lisp Functions, Vector Lisp Functions, Predicates,
Internals
address@hidden Computational Functions
+
address@hidden
+The functions described here do the actual computational work of the
+Calculator. In addition to these, note that any function described in
+the main body of this manual may be called from Lisp; for example, if
+the documentation refers to the @code{calc-sqrt} address@hidden command,
+this means @code{calc-sqrt} is an interactive stack-based square-root
+command and @code{sqrt} (which @code{defmath} expands to @code{calcFunc-sqrt})
+is the actual Lisp function for taking square roots.
+
+The functions @code{math-add}, @code{math-sub}, @code{math-mul},
address@hidden, @code{math-mod}, and @code{math-neg} are not included
+in this list, since @code{defmath} allows you to write native Lisp
address@hidden, @code{-}, @code{*}, @code{/}, @code{%}, and unary @code{-},
+respectively, instead.
+
address@hidden normalize val
+(Full form: @code{math-normalize}.)
+Reduce the value @var{val} to standard form. For example, if @var{val}
+is a fixnum, it will be converted to a bignum if it is too large, and
+if @var{val} is a bignum it will be normalized by clipping off trailing
+(i.e., most-significant) zero digits and converting to a fixnum if it is
+small. All the various data types are similarly converted to their standard
+forms. Variables are left alone, but function calls are actually evaluated
+in formulas. For example, normalizing @samp{(+ 2 (calcFunc-abs -4))} will
+return 6.
+
+If a function call fails, because the function is void or has the wrong
+number of parameters, or because it returns @code{nil} or calls
address@hidden or @code{inexact-result}, @code{normalize} returns
+the formula still in symbolic form.
+
+If the current simplification mode is ``none'' or ``numeric arguments
+only,'' @code{normalize} will act appropriately. However, the more
+powerful simplification modes (like Algebraic Simplification) are
+not handled by @code{normalize}. They are handled by @code{calc-normalize},
+which calls @code{normalize} and possibly some other routines, such
+as @code{simplify} or @code{simplify-units}. Programs generally will
+never call @code{calc-normalize} except when popping or pushing values
+on the stack.
address@hidden defun
+
address@hidden evaluate-expr expr
+Replace all variables in @var{expr} that have values with their values,
+then use @code{normalize} to simplify the result. This is what happens
+when you press the @kbd{=} key interactively.
address@hidden defun
+
address@hidden with-extra-prec n body
+Evaluate the Lisp forms in @var{body} with precision increased by @var{n}
+digits. This is a macro which expands to
+
address@hidden
+(math-normalize
+ (let ((calc-internal-prec (+ calc-internal-prec @var{n})))
+ @var{body}))
address@hidden smallexample
+
+The surrounding call to @code{math-normalize} causes a floating-point
+result to be rounded down to the original precision afterwards. This
+is important because some arithmetic operations assume a number's
+mantissa contains no more digits than the current precision allows.
address@hidden defmac
+
address@hidden make-frac n d
+Build a fraction @address@hidden:@var{d}}. This is equivalent to calling
address@hidden(normalize (list 'frac @var{n} @var{d}))}, but more efficient.
address@hidden defun
+
address@hidden make-float mant exp
+Build a floating-point value out of @var{mant} and @var{exp}, both
+of which are arbitrary integers. This function will return a
+properly normalized float value, or signal an overflow or underflow
+if @var{exp} is out of range.
address@hidden defun
+
address@hidden make-sdev x sigma
+Build an error form out of @var{x} and the absolute value of @var{sigma}.
+If @var{sigma} is zero, the result is the number @var{x} directly.
+If @var{sigma} is negative or complex, its absolute value is used.
+If @var{x} or @var{sigma} is not a valid type of object for use in
+error forms, this calls @code{reject-arg}.
address@hidden defun
+
address@hidden make-intv mask lo hi
+Build an interval form out of @var{mask} (which is assumed to be an
+integer from 0 to 3), and the limits @var{lo} and @var{hi}. If
address@hidden is greater than @var{hi}, an empty interval form is returned.
+This calls @code{reject-arg} if @var{lo} or @var{hi} is unsuitable.
address@hidden defun
+
address@hidden sort-intv mask lo hi
+Build an interval form, similar to @code{make-intv}, except that if
address@hidden is less than @var{hi} they are simply exchanged, and the
+bits of @var{mask} are swapped accordingly.
address@hidden defun
+
address@hidden make-mod n m
+Build a modulo form out of @var{n} and the modulus @var{m}. Since modulo
+forms do not allow formulas as their components, if @var{n} or @var{m}
+is not a real number or HMS form the result will be a formula which
+is a call to @code{makemod}, the algebraic version of this function.
address@hidden defun
+
address@hidden float x
+Convert @var{x} to floating-point form. Integers and fractions are
+converted to numerically equivalent floats; components of complex
+numbers, vectors, HMS forms, date forms, error forms, intervals, and
+modulo forms are recursively floated. If the argument is a variable
+or formula, this calls @code{reject-arg}.
address@hidden defun
+
address@hidden compare x y
+Compare the numbers @var{x} and @var{y}, and return @mathit{-1} if
address@hidden(lessp @var{x} @var{y})}, 1 if @samp{(lessp @var{y} @var{x})},
+0 if @samp{(math-equal @var{x} @var{y})}, or 2 if the order is
+undefined or cannot be determined.
address@hidden defun
+
address@hidden numdigs n
+Return the number of digits of integer @var{n}, effectively
address@hidden(log10(@var{n}))}, but much more efficient. Zero is
+considered to have zero digits.
address@hidden defun
+
address@hidden scale-int x n
+Shift integer @var{x} left @var{n} decimal digits, or right @address@hidden
+digits with truncation toward zero.
address@hidden defun
+
address@hidden scale-rounding x n
+Like @code{scale-int}, except that a right shift rounds to the nearest
+integer rather than truncating.
address@hidden defun
+
address@hidden fixnum n
+Return the integer @var{n} as a fixnum, i.e., a native Lisp integer.
+If @var{n} is outside the permissible range for Lisp integers (usually
+24 binary bits) the result is undefined.
address@hidden defun
+
address@hidden sqr x
+Compute the square of @var{x}; short for @samp{(* @var{x} @var{x})}.
address@hidden defun
+
address@hidden quotient x y
+Divide integer @var{x} by integer @var{y}; return an integer quotient
+and discard the remainder. If @var{x} or @var{y} is negative, the
+direction of rounding is undefined.
address@hidden defun
+
address@hidden idiv x y
+Perform an integer division; if @var{x} and @var{y} are both nonnegative
+integers, this uses the @code{quotient} function, otherwise it computes
address@hidden(@var{x}/@var{y})}. Thus the result is well-defined but
+slower than for @code{quotient}.
address@hidden defun
+
address@hidden imod x y
+Divide integer @var{x} by integer @var{y}; return the integer remainder
+and discard the quotient. Like @code{quotient}, this works only for
+integer arguments and is not well-defined for negative arguments.
+For a more well-defined result, use @samp{(% @var{x} @var{y})}.
address@hidden defun
+
address@hidden idivmod x y
+Divide integer @var{x} by integer @var{y}; return a cons cell whose
address@hidden is @samp{(quotient @var{x} @var{y})} and whose @code{cdr}
+is @samp{(imod @var{x} @var{y})}.
address@hidden defun
+
address@hidden pow x y
+Compute @var{x} to the power @var{y}. In @code{defmath} code, this can
+also be written @samp{(^ @var{x} @var{y})} or
address@hidden@samp{(expt @var{x} @var{y})}}.
address@hidden defun
+
address@hidden abs-approx x
+Compute a fast approximation to the absolute value of @var{x}. For
+example, for a rectangular complex number the result is the sum of
+the absolute values of the components.
address@hidden defun
+
address@hidden e
address@hidden gamma-const
address@hidden ln-2
address@hidden ln-10
address@hidden phi
address@hidden pi-over-2
address@hidden pi-over-4
address@hidden pi-over-180
address@hidden sqrt-two-pi
address@hidden sqrt-e
address@hidden two-pi
address@hidden pi
+The function @samp{(pi)} computes @samp{pi} to the current precision.
+Other related constant-generating functions are @code{two-pi},
address@hidden, @code{pi-over-4}, @code{pi-over-180}, @code{sqrt-two-pi},
address@hidden, @code{sqrt-e}, @code{ln-2}, @code{ln-10}, @code{phi} and
address@hidden Each function returns a floating-point value in the
+current precision, and each uses caching so that all calls after the
+first are essentially free.
address@hidden defun
+
address@hidden math-defcache @var{func} @var{initial} @var{form}
+This macro, usually used as a top-level call like @code{defun} or
address@hidden, defines a new cached constant analogous to @code{pi}, etc.
+It defines a function @code{func} which returns the requested value;
+if @var{initial} is address@hidden it must be a @samp{(float @dots{})}
+form which serves as an initial value for the cache. If @var{func}
+is called when the cache is empty or does not have enough digits to
+satisfy the current precision, the Lisp expression @var{form} is evaluated
+with the current precision increased by four, and the result minus its
+two least significant digits is stored in the cache. For example,
+calling @samp{(pi)} with a precision of 30 computes @samp{pi} to 34
+digits, rounds it down to 32 digits for future use, then rounds it
+again to 30 digits for use in the present request.
address@hidden defmac
+
address@hidden half-circle
address@hidden quarter-circle
address@hidden full-circle symb
+If the current angular mode is Degrees or HMS, this function returns the
+integer 360. In Radians mode, this function returns either the
+corresponding value in radians to the current precision, or the formula
address@hidden, depending on the Symbolic mode. There are also similar
+function @code{half-circle} and @code{quarter-circle}.
address@hidden defun
+
address@hidden power-of-2 n
+Compute two to the integer power @var{n}, as a (potentially very large)
+integer. Powers of two are cached, so only the first call for a
+particular @var{n} is expensive.
address@hidden defun
+
address@hidden integer-log2 n
+Compute the base-2 logarithm of @var{n}, which must be an integer which
+is a power of two. If @var{n} is not a power of two, this function will
+return @code{nil}.
address@hidden defun
+
address@hidden div-mod a b m
+Divide @var{a} by @var{b}, modulo @var{m}. This returns @code{nil} if
+there is no solution, or if any of the arguments are not integers.
address@hidden defun
+
address@hidden pow-mod a b m
+Compute @var{a} to the power @var{b}, modulo @var{m}. If @var{a},
address@hidden, and @var{m} are integers, this uses an especially efficient
+algorithm. Otherwise, it simply computes @samp{(% (^ a b) m)}.
address@hidden defun
+
address@hidden isqrt n
+Compute the integer square root of @var{n}. This is the square root
+of @var{n} rounded down toward zero, i.e., @samp{floor(sqrt(@var{n}))}.
+If @var{n} is itself an integer, the computation is especially efficient.
address@hidden defun
+
address@hidden to-hms a ang
+Convert the argument @var{a} into an HMS form. If @var{ang} is specified,
+it is the angular mode in which to interpret @var{a}, either @code{deg}
+or @code{rad}. Otherwise, the current angular mode is used. If @var{a}
+is already an HMS form it is returned as-is.
address@hidden defun
+
address@hidden from-hms a ang
+Convert the HMS form @var{a} into a real number. If @var{ang} is specified,
+it is the angular mode in which to express the result, otherwise the
+current angular mode is used. If @var{a} is already a real number, it
+is returned as-is.
address@hidden defun
+
address@hidden to-radians a
+Convert the number or HMS form @var{a} to radians from the current
+angular mode.
address@hidden defun
+
address@hidden from-radians a
+Convert the number @var{a} from radians to the current angular mode.
+If @var{a} is a formula, this returns the formula @samp{deg(@var{a})}.
address@hidden defun
+
address@hidden to-radians-2 a
+Like @code{to-radians}, except that in Symbolic mode a degrees to
+radians conversion yields a formula like @address@hidden/180}.
address@hidden defun
+
address@hidden from-radians-2 a
+Like @code{from-radians}, except that in Symbolic mode a radians to
+degrees conversion yields a formula like @address@hidden/pi}.
address@hidden defun
+
address@hidden random-digit
+Produce a random base-1000 digit in the range 0 to 999.
address@hidden defun
+
address@hidden random-digits n
+Produce a random @var{n}-digit integer; this will be an integer
+in the interval @samp{[0, address@hidden)}.
address@hidden defun
+
address@hidden random-float
+Produce a random float in the interval @samp{[0, 1)}.
address@hidden defun
+
address@hidden prime-test n iters
+Determine whether the integer @var{n} is prime. Return a list which has
+one of these forms: @samp{(nil @var{f})} means the number is non-prime
+because it was found to be divisible by @var{f}; @samp{(nil)} means it
+was found to be non-prime by table look-up (so no factors are known);
address@hidden(nil unknown)} means it is definitely non-prime but no factors
+are known because @var{n} was large enough that Fermat's probabilistic
+test had to be used; @samp{(t)} means the number is definitely prime;
+and @samp{(maybe @var{i} @var{p})} means that Fermat's test, after @var{i}
+iterations, is @var{p} percent sure that the number is prime. The
address@hidden parameter is the number of Fermat iterations to use, in the
+case that this is necessary. If @code{prime-test} returns ``maybe,''
+you can call it again with the same @var{n} to get a greater certainty;
address@hidden remembers where it left off.
address@hidden defun
+
address@hidden to-simple-fraction f
+If @var{f} is a floating-point number which can be represented exactly
+as a small rational number. return that number, else return @var{f}.
+For example, 0.75 would be converted to 3:4. This function is very
+fast.
address@hidden defun
+
address@hidden to-fraction f tol
+Find a rational approximation to floating-point number @var{f} to within
+a specified tolerance @var{tol}; this corresponds to the algebraic
+function @code{frac}, and can be rather slow.
address@hidden defun
+
address@hidden quarter-integer n
+If @var{n} is an integer or integer-valued float, this function
+returns zero. If @var{n} is a half-integer (i.e., an integer plus
address@hidden:2} or 0.5), it returns 2. If @var{n} is a quarter-integer,
+it returns 1 or 3. If @var{n} is anything else, this function
+returns @code{nil}.
address@hidden defun
+
address@hidden Vector Lisp Functions, Symbolic Lisp Functions, Computational
Lisp Functions, Internals
address@hidden Vector Functions
+
address@hidden
+The functions described here perform various operations on vectors and
+matrices.
+
address@hidden math-concat x y
+Do a vector concatenation; this operation is written @address@hidden | @var{y}}
+in a symbolic formula. @xref{Building Vectors}.
address@hidden defun
+
address@hidden vec-length v
+Return the length of vector @var{v}. If @var{v} is not a vector, the
+result is zero. If @var{v} is a matrix, this returns the number of
+rows in the matrix.
address@hidden defun
+
address@hidden mat-dimens m
+Determine the dimensions of vector or matrix @var{m}. If @var{m} is not
+a vector, the result is an empty list. If @var{m} is a plain vector
+but not a matrix, the result is a one-element list containing the length
+of the vector. If @var{m} is a matrix with @var{r} rows and @var{c} columns,
+the result is the list @samp{(@var{r} @var{c})}. Higher-order tensors
+produce lists of more than two dimensions. Note that the object
address@hidden, 2, 3], [4, 5]]} is a vector of vectors not all the same size,
+and is treated by this and other Calc routines as a plain vector of two
+elements.
address@hidden defun
+
address@hidden dimension-error
+Abort the current function with a message of ``Dimension error.''
+The Calculator will leave the function being evaluated in symbolic
+form; this is really just a special case of @code{reject-arg}.
address@hidden defun
+
address@hidden build-vector args
+Return a Calc vector with @var{args} as elements.
+For example, @samp{(build-vector 1 2 3)} returns the Calc vector
address@hidden, 2, 3]}, stored internally as the list @samp{(vec 1 2 3)}.
address@hidden defun
+
address@hidden make-vec obj dims
+Return a Calc vector or matrix all of whose elements are equal to
address@hidden For example, @samp{(make-vec 27 3 4)} returns a 3x4 matrix
+filled with 27's.
address@hidden defun
+
address@hidden row-matrix v
+If @var{v} is a plain vector, convert it into a row matrix, i.e.,
+a matrix whose single row is @var{v}. If @var{v} is already a matrix,
+leave it alone.
address@hidden defun
+
address@hidden col-matrix v
+If @var{v} is a plain vector, convert it into a column matrix, i.e., a
+matrix with each element of @var{v} as a separate row. If @var{v} is
+already a matrix, leave it alone.
address@hidden defun
+
address@hidden map-vec f v
+Map the Lisp function @var{f} over the Calc vector @var{v}. For example,
address@hidden(map-vec 'math-floor v)} returns a vector of the floored
components
+of vector @var{v}.
address@hidden defun
+
address@hidden map-vec-2 f a b
+Map the Lisp function @var{f} over the two vectors @var{a} and @var{b}.
+If @var{a} and @var{b} are vectors of equal length, the result is a
+vector of the results of calling @samp{(@var{f} @var{ai} @var{bi})}
+for each pair of elements @var{ai} and @var{bi}. If either @var{a} or
address@hidden is a scalar, it is matched with each value of the other vector.
+For example, @samp{(map-vec-2 'math-add v 1)} returns the vector @var{v}
+with each element increased by one. Note that using @samp{'+} would not
+work here, since @code{defmath} does not expand function names everywhere,
+just where they are in the function position of a Lisp expression.
address@hidden defun
+
address@hidden reduce-vec f v
+Reduce the function @var{f} over the vector @var{v}. For example, if
address@hidden is @samp{[10, 20, 30, 40]}, this calls @samp{(f (f (f 10 20) 30)
40)}.
+If @var{v} is a matrix, this reduces over the rows of @var{v}.
address@hidden defun
+
address@hidden reduce-cols f m
+Reduce the function @var{f} over the columns of matrix @var{m}. For
+example, if @var{m} is @samp{[[1, 2], [3, 4], [5, 6]]}, the result
+is a vector of the two elements @samp{(f (f 1 3) 5)} and @samp{(f (f 2 4) 6)}.
address@hidden defun
+
address@hidden mat-row m n
+Return the @var{n}th row of matrix @var{m}. This is equivalent to
address@hidden(elt m n)}. For a slower but safer version, use @code{mrow}.
+(@xref{Extracting Elements}.)
address@hidden defun
+
address@hidden mat-col m n
+Return the @var{n}th column of matrix @var{m}, in the form of a vector.
+The arguments are not checked for correctness.
address@hidden defun
+
address@hidden mat-less-row m n
+Return a copy of matrix @var{m} with its @var{n}th row deleted. The
+number @var{n} must be in range from 1 to the number of rows in @var{m}.
address@hidden defun
+
address@hidden mat-less-col m n
+Return a copy of matrix @var{m} with its @var{n}th column deleted.
address@hidden defun
+
address@hidden transpose m
+Return the transpose of matrix @var{m}.
address@hidden defun
+
address@hidden flatten-vector v
+Flatten nested vector @var{v} into a vector of scalars. For example,
+if @var{v} is @samp{[[1, 2, 3], [4, 5]]} the result is @samp{[1, 2, 3, 4, 5]}.
address@hidden defun
+
address@hidden copy-matrix m
+If @var{m} is a matrix, return a copy of @var{m}. This maps
address@hidden over the rows of @var{m}; in Lisp terms, each
+element of the result matrix will be @code{eq} to the corresponding
+element of @var{m}, but none of the @code{cons} cells that make up
+the structure of the matrix will be @code{eq}. If @var{m} is a plain
+vector, this is the same as @code{copy-sequence}.
address@hidden defun
+
address@hidden swap-rows m r1 r2
+Exchange rows @var{r1} and @var{r2} of matrix @var{m} in-place. In
+other words, unlike most of the other functions described here, this
+function changes @var{m} itself rather than building up a new result
+matrix. The return value is @var{m}, i.e., @samp{(eq (swap-rows m 1 2) m)}
+is true, with the side effect of exchanging the first two rows of
address@hidden
address@hidden defun
+
address@hidden Symbolic Lisp Functions, Formatting Lisp Functions, Vector Lisp
Functions, Internals
address@hidden Symbolic Functions
+
address@hidden
+The functions described here operate on symbolic formulas in the
+Calculator.
+
address@hidden calc-prepare-selection num
+Prepare a stack entry for selection operations. If @var{num} is
+omitted, the stack entry containing the cursor is used; otherwise,
+it is the number of the stack entry to use. This function stores
+useful information about the current stack entry into a set of
+variables. @code{calc-selection-cache-num} contains the number of
+the stack entry involved (equal to @var{num} if you specified it);
address@hidden contains the stack entry as a
+list (such as @code{calc-top-list} would return with @code{entry}
+as the selection mode); and @code{calc-selection-cache-comp} contains
+a special ``tagged'' composition (@pxref{Formatting Lisp Functions})
+which allows Calc to relate cursor positions in the buffer with
+their corresponding sub-formulas.
+
+A slight complication arises in the selection mechanism because
+formulas may contain small integers. For example, in the vector
address@hidden, 2, 1]} the first and last elements are @code{eq} to each
+other; selections are recorded as the actual Lisp object that
+appears somewhere in the tree of the whole formula, but storing
address@hidden would falsely select both @code{1}'s in the vector. So
address@hidden also checks the stack entry and
+replaces any plain integers with ``complex number'' lists of the form
address@hidden(cplx @var{n} 0)}. This list will be displayed the same as a
+plain @var{n} and the change will be completely invisible to the
+user, but it will guarantee that no two sub-formulas of the stack
+entry will be @code{eq} to each other. Next time the stack entry
+is involved in a computation, @code{calc-normalize} will replace
+these lists with plain numbers again, again invisibly to the user.
address@hidden defun
+
address@hidden calc-encase-atoms x
+This modifies the formula @var{x} to ensure that each part of the
+formula is a unique atom, using the @samp{(cplx @var{n} 0)} trick
+described above. This function may use @code{setcar} to modify
+the formula in-place.
address@hidden defun
+
address@hidden calc-find-selected-part
+Find the smallest sub-formula of the current formula that contains
+the cursor. This assumes @code{calc-prepare-selection} has been
+called already. If the cursor is not actually on any part of the
+formula, this returns @code{nil}.
address@hidden defun
+
address@hidden calc-change-current-selection selection
+Change the currently prepared stack element's selection to
address@hidden, which should be @code{eq} to some sub-formula
+of the stack element, or @code{nil} to unselect the formula.
+The stack element's appearance in the Calc buffer is adjusted
+to reflect the new selection.
address@hidden defun
+
address@hidden calc-find-nth-part expr n
+Return the @var{n}th sub-formula of @var{expr}. This function is used
+by the selection commands, and (unless @kbd{j b} has been used) treats
+sums and products as flat many-element formulas. Thus if @var{expr}
+is @samp{((a + b) - c) + d}, calling @code{calc-find-nth-part} with
address@hidden equal to four will return @samp{d}.
address@hidden defun
+
address@hidden calc-find-parent-formula expr part
+Return the sub-formula of @var{expr} which immediately contains
address@hidden If @var{expr} is @samp{a*b + (c+1)*d} and @var{part}
+is @code{eq} to the @samp{c+1} term of @var{expr}, then this function
+will return @samp{(c+1)*d}. If @var{part} turns out not to be a
+sub-formula of @var{expr}, the function returns @code{nil}. If
address@hidden is @code{eq} to @var{expr}, the function returns @code{t}.
+This function does not take associativity into account.
address@hidden defun
+
address@hidden calc-find-assoc-parent-formula expr part
+This is the same as @code{calc-find-parent-formula}, except that
+(unless @kbd{j b} has been used) it continues widening the selection
+to contain a complete level of the formula. Given @samp{a} from
address@hidden((a + b) - c) + d}, @code{calc-find-parent-formula} will
+return @samp{a + b} but @code{calc-find-assoc-parent-formula} will
+return the whole expression.
address@hidden defun
+
address@hidden calc-grow-assoc-formula expr part
+This expands sub-formula @var{part} of @var{expr} to encompass a
+complete level of the formula. If @var{part} and its immediate
+parent are not compatible associative operators, or if @kbd{j b}
+has been used, this simply returns @var{part}.
address@hidden defun
+
address@hidden calc-find-sub-formula expr part
+This finds the immediate sub-formula of @var{expr} which contains
address@hidden It returns an index @var{n} such that
address@hidden(calc-find-nth-part @var{expr} @var{n})} would return @var{part}.
+If @var{part} is not a sub-formula of @var{expr}, it returns @code{nil}.
+If @var{part} is @code{eq} to @var{expr}, it returns @code{t}. This
+function does not take associativity into account.
address@hidden defun
+
address@hidden calc-replace-sub-formula expr old new
+This function returns a copy of formula @var{expr}, with the
+sub-formula that is @code{eq} to @var{old} replaced by @var{new}.
address@hidden defun
+
address@hidden simplify expr
+Simplify the expression @var{expr} by applying various algebraic rules.
+This is what the @address@hidden s}} (@code{calc-simplify}) command uses. This
+always returns a copy of the expression; the structure @var{expr} points
+to remains unchanged in memory.
+
+More precisely, here is what @code{simplify} does: The expression is
+first normalized and evaluated by calling @code{normalize}. If any
address@hidden have been defined, they are then applied. Then
+the expression is traversed in a depth-first, bottom-up fashion; at
+each level, any simplifications that can be made are made until no
+further changes are possible. Once the entire formula has been
+traversed in this way, it is compared with the original formula (from
+before the call to @code{normalize}) and, if it has changed,
+the entire procedure is repeated (starting with @code{normalize})
+until no further changes occur. Usually only two iterations are
+needed:@: one to simplify the formula, and another to verify that no
+further simplifications were possible.
address@hidden defun
+
address@hidden simplify-extended expr
+Simplify the expression @var{expr}, with additional rules enabled that
+help do a more thorough job, while not being entirely ``safe'' in all
+circumstances. (For example, this mode will simplify @samp{sqrt(x^2)}
+to @samp{x}, which is only valid when @var{x} is positive.) This is
+implemented by temporarily binding the variable @code{math-living-dangerously}
+to @code{t} (using a @code{let} form) and calling @code{simplify}.
+Dangerous simplification rules are written to check this variable
+before taking any action.
address@hidden defun
+
address@hidden simplify-units expr
+Simplify the expression @var{expr}, treating variable names as units
+whenever possible. This works by binding the variable
address@hidden to @code{t} while calling @code{simplify}.
address@hidden defun
+
address@hidden math-defsimplify funcs body
+Register a new simplification rule; this is normally called as a top-level
+form, like @code{defun} or @code{defmath}. If @var{funcs} is a symbol
+(like @code{+} or @code{calcFunc-sqrt}), this simplification rule is
+applied to the formulas which are calls to the specified function. Or,
address@hidden can be a list of such symbols; the rule applies to all
+functions on the list. The @var{body} is written like the body of a
+function with a single argument called @code{expr}. The body will be
+executed with @code{expr} bound to a formula which is a call to one of
+the functions @var{funcs}. If the function body returns @code{nil}, or
+if it returns a result @code{equal} to the original @code{expr}, it is
+ignored and Calc goes on to try the next simplification rule that applies.
+If the function body returns something different, that new formula is
+substituted for @var{expr} in the original formula.
+
+At each point in the formula, rules are tried in the order of the
+original calls to @code{math-defsimplify}; the search stops after the
+first rule that makes a change. Thus later rules for that same
+function will not have a chance to trigger until the next iteration
+of the main @code{simplify} loop.
+
+Note that, since @code{defmath} is not being used here, @var{body} must
+be written in true Lisp code without the conveniences that @code{defmath}
+provides. If you prefer, you can have @var{body} simply call another
+function (defined with @code{defmath}) which does the real work.
+
+The arguments of a function call will already have been simplified
+before any rules for the call itself are invoked. Since a new argument
+list is consed up when this happens, this means that the rule's body is
+allowed to rearrange the function's arguments destructively if that is
+convenient. Here is a typical example of a simplification rule:
+
address@hidden
+(math-defsimplify calcFunc-arcsinh
+ (or (and (math-looks-negp (nth 1 expr))
+ (math-neg (list 'calcFunc-arcsinh
+ (math-neg (nth 1 expr)))))
+ (and (eq (car-safe (nth 1 expr)) 'calcFunc-sinh)
+ (or math-living-dangerously
+ (math-known-realp (nth 1 (nth 1 expr))))
+ (nth 1 (nth 1 expr)))))
address@hidden smallexample
+
+This is really a pair of rules written with one @code{math-defsimplify}
+for convenience; the first replaces @samp{arcsinh(-x)} with
address@hidden(x)}, and the second, which is safe only for real @samp{x},
+replaces @samp{arcsinh(sinh(x))} with @samp{x}.
address@hidden defmac
+
address@hidden common-constant-factor expr
+Check @var{expr} to see if it is a sum of terms all multiplied by the
+same rational value. If so, return this value. If not, return @code{nil}.
+For example, if called on @samp{6x + 9y + 12z}, it would return 3, since
+3 is a common factor of all the terms.
address@hidden defun
+
address@hidden cancel-common-factor expr factor
+Assuming @var{expr} is a sum with @var{factor} as a common factor,
+divide each term of the sum by @var{factor}. This is done by
+destructively modifying parts of @var{expr}, on the assumption that
+it is being used by a simplification rule (where such things are
+allowed; see above). For example, consider this built-in rule for
+square roots:
+
address@hidden
+(math-defsimplify calcFunc-sqrt
+ (let ((fac (math-common-constant-factor (nth 1 expr))))
+ (and fac (not (eq fac 1))
+ (math-mul (math-normalize (list 'calcFunc-sqrt fac))
+ (math-normalize
+ (list 'calcFunc-sqrt
+ (math-cancel-common-factor
+ (nth 1 expr) fac)))))))
address@hidden smallexample
address@hidden defun
+
address@hidden frac-gcd a b
+Compute a ``rational GCD'' of @var{a} and @var{b}, which must both be
+rational numbers. This is the fraction composed of the GCD of the
+numerators of @var{a} and @var{b}, over the GCD of the denominators.
+It is used by @code{common-constant-factor}. Note that the standard
address@hidden function uses the LCM to combine the denominators.
address@hidden defun
+
address@hidden map-tree func expr many
+Try applying Lisp function @var{func} to various sub-expressions of
address@hidden Initially, call @var{func} with @var{expr} itself as an
+argument. If this returns an expression which is not @code{equal} to
address@hidden, apply @var{func} again until eventually it does return
address@hidden with no changes. Then, if @var{expr} is a function call,
+recursively apply @var{func} to each of the arguments. This keeps going
+until no changes occur anywhere in the expression; this final expression
+is returned by @code{map-tree}. Note that, unlike simplification rules,
address@hidden functions may @emph{not} make destructive changes to
address@hidden If a third argument @var{many} is provided, it is an
+integer which says how many times @var{func} may be applied; the
+default, as described above, is infinitely many times.
address@hidden defun
+
address@hidden compile-rewrites rules
+Compile the rewrite rule set specified by @var{rules}, which should
+be a formula that is either a vector or a variable name. If the latter,
+the compiled rules are saved so that later @code{compile-rules} calls
+for that same variable can return immediately. If there are problems
+with the rules, this function calls @code{error} with a suitable
+message.
address@hidden defun
+
address@hidden apply-rewrites expr crules heads
+Apply the compiled rewrite rule set @var{crules} to the expression
address@hidden This will make only one rewrite and only checks at the
+top level of the expression. The result @code{nil} if no rules
+matched, or if the only rules that matched did not actually change
+the expression. The @var{heads} argument is optional; if is given,
+it should be a list of all function names that (may) appear in
address@hidden The rewrite compiler tags each rule with the
+rarest-looking function name in the rule; if you specify @var{heads},
address@hidden can use this information to narrow its search
+down to just a few rules in the rule set.
address@hidden defun
+
address@hidden rewrite-heads expr
+Compute a @var{heads} list for @var{expr} suitable for use with
address@hidden, as discussed above.
address@hidden defun
+
address@hidden rewrite expr rules many
+This is an all-in-one rewrite function. It compiles the rule set
+specified by @var{rules}, then uses @code{map-tree} to apply the
+rules throughout @var{expr} up to @var{many} (default infinity)
+times.
address@hidden defun
+
address@hidden match-patterns pat vec not-flag
+Given a Calc vector @var{vec} and an uncompiled pattern set or
+pattern set variable @var{pat}, this function returns a new vector
+of all elements of @var{vec} which do (or don't, if @var{not-flag} is
address@hidden) match any of the patterns in @var{pat}.
address@hidden defun
+
address@hidden deriv expr var value symb
+Compute the derivative of @var{expr} with respect to variable @var{var}
+(which may actually be any sub-expression). If @var{value} is specified,
+the derivative is evaluated at the value of @var{var}; otherwise, the
+derivative is left in terms of @var{var}. If the expression contains
+functions for which no derivative formula is known, new derivative
+functions are invented by adding primes to the names; @pxref{Calculus}.
+However, if @var{symb} is address@hidden, the presence of undifferentiable
+functions in @var{expr} instead cancels the whole differentiation, and
address@hidden returns @code{nil} instead.
+
+Derivatives of an @var{n}-argument function can be defined by
+adding a @address@hidden property to the property list
+of the symbol for the function's derivative, which will be the
+function name followed by an apostrophe. The value of the property
+should be a Lisp function; it is called with the same arguments as the
+original function call that is being differentiated. It should return
+a formula for the derivative. For example, the derivative of @code{ln}
+is defined by
+
address@hidden
+(put 'calcFunc-ln\' 'math-derivative-1
+ (function (lambda (u) (math-div 1 u))))
address@hidden smallexample
+
+The two-argument @code{log} function has two derivatives,
address@hidden
+(put 'calcFunc-log\' 'math-derivative-2 ; d(log(x,b)) / dx
+ (function (lambda (x b) ... )))
+(put 'calcFunc-log\'2 'math-derivative-2 ; d(log(x,b)) / db
+ (function (lambda (x b) ... )))
address@hidden smallexample
address@hidden defun
+
address@hidden tderiv expr var value symb
+Compute the total derivative of @var{expr}. This is the same as
address@hidden, except that variables other than @var{var} are not
+assumed to be constant with respect to @var{var}.
address@hidden defun
+
address@hidden integ expr var low high
+Compute the integral of @var{expr} with respect to @var{var}.
address@hidden, for further details.
address@hidden defun
+
address@hidden math-defintegral funcs body
+Define a rule for integrating a function or functions of one argument;
+this macro is very similar in format to @code{math-defsimplify}.
+The main difference is that here @var{body} is the body of a function
+with a single argument @code{u} which is bound to the argument to the
+function being integrated, not the function call itself. Also, the
+variable of integration is available as @code{math-integ-var}. If
+evaluation of the integral requires doing further integrals, the body
+should call @samp{(math-integral @var{x})} to find the integral of
address@hidden with respect to @code{math-integ-var}; this function returns
address@hidden if the integral could not be done. Some examples:
+
address@hidden
+(math-defintegral calcFunc-conj
+ (let ((int (math-integral u)))
+ (and int
+ (list 'calcFunc-conj int))))
+
+(math-defintegral calcFunc-cos
+ (and (equal u math-integ-var)
+ (math-from-radians-2 (list 'calcFunc-sin u))))
address@hidden smallexample
+
+In the @code{cos} example, we define only the integral of @samp{cos(x) dx},
+relying on the general integration-by-substitution facility to handle
+cosines of more complicated arguments. An integration rule should return
address@hidden if it can't do the integral; if several rules are defined for
+the same function, they are tried in order until one returns a address@hidden
+result.
address@hidden defmac
+
address@hidden math-defintegral-2 funcs body
+Define a rule for integrating a function or functions of two arguments.
+This is exactly analogous to @code{math-defintegral}, except that @var{body}
+is written as the body of a function with two arguments, @var{u} and
address@hidden
address@hidden defmac
+
address@hidden solve-for lhs rhs var full
+Attempt to solve the equation @address@hidden = @var{rhs}} by isolating
+the variable @var{var} on the lefthand side; return the resulting righthand
+side, or @code{nil} if the equation cannot be solved. The variable
address@hidden must appear at least once in @var{lhs} or @var{rhs}. Note that
+the return value is a formula which does not contain @var{var}; this is
+different from the user-level @code{solve} and @code{finv} functions,
+which return a rearranged equation or a functional inverse, respectively.
+If @var{full} is address@hidden, a full solution including dummy signs
+and dummy integers will be produced. User-defined inverses are provided
+as properties in a manner similar to derivatives:
+
address@hidden
+(put 'calcFunc-ln 'math-inverse
+ (function (lambda (x) (list 'calcFunc-exp x))))
address@hidden smallexample
+
+This function can call @samp{(math-solve-get-sign @var{x})} to create
+a new arbitrary sign variable, returning @var{x} times that sign, and
address@hidden(math-solve-get-int @var{x})} to create a new arbitrary integer
+variable multiplied by @var{x}. These functions simply return @var{x}
+if the caller requested a non-``full'' solution.
address@hidden defun
+
address@hidden solve-eqn expr var full
+This version of @code{solve-for} takes an expression which will
+typically be an equation or inequality. (If it is not, it will be
+interpreted as the equation @address@hidden = 0}.) It returns an
+equation or inequality, or @code{nil} if no solution could be found.
address@hidden defun
+
address@hidden solve-system exprs vars full
+This function solves a system of equations. Generally, @var{exprs}
+and @var{vars} will be vectors of equal length.
address@hidden Systems of Equations}, for other options.
address@hidden defun
+
address@hidden expr-contains expr var
+Returns a address@hidden value if @var{var} occurs as a subexpression
+of @var{expr}.
+
+This function might seem at first to be identical to
address@hidden The key difference is that
address@hidden uses @code{equal} to test for matches, whereas
address@hidden uses @code{eq}. In the formula
address@hidden(a, a)}, the two @samp{a}s will be @code{equal} but not
address@hidden to each other.
address@hidden defun
+
address@hidden expr-contains-count expr var
+Returns the number of occurrences of @var{var} as a subexpression
+of @var{expr}, or @code{nil} if there are no occurrences.
address@hidden defun
+
address@hidden expr-depends expr var
+Returns true if @var{expr} refers to any variable the occurs in @var{var}.
+In other words, it checks if @var{expr} and @var{var} have any variables
+in common.
address@hidden defun
+
address@hidden expr-contains-vars expr
+Return true if @var{expr} contains any variables, or @code{nil} if @var{expr}
+contains only constants and functions with constant arguments.
address@hidden defun
+
address@hidden expr-subst expr old new
+Returns a copy of @var{expr}, with all occurrences of @var{old} replaced
+by @var{new}. This treats @code{lambda} forms specially with respect
+to the dummy argument variables, so that the effect is always to return
address@hidden evaluated at @var{old} = @var{new}.
address@hidden defun
+
address@hidden multi-subst expr old new
+This is like @code{expr-subst}, except that @var{old} and @var{new}
+are lists of expressions to be substituted simultaneously. If one
+list is shorter than the other, trailing elements of the longer list
+are ignored.
address@hidden defun
+
address@hidden expr-weight expr
+Returns the ``weight'' of @var{expr}, basically a count of the total
+number of objects and function calls that appear in @var{expr}. For
+``primitive'' objects, this will be one.
address@hidden defun
+
address@hidden expr-height expr
+Returns the ``height'' of @var{expr}, which is the deepest level to
+which function calls are nested. (Note that @address@hidden + @var{b}}
+counts as a function call.) For primitive objects, this returns zero.
address@hidden defun
+
address@hidden polynomial-p expr var
+Check if @var{expr} is a polynomial in variable (or sub-expression)
address@hidden If so, return the degree of the polynomial, that is, the
+highest power of @var{var} that appears in @var{expr}. For example,
+for @samp{(x^2 + 3)^3 + 4} this would return 6. This function returns
address@hidden unless @var{expr}, when expanded out by @kbd{a x}
+(@code{calc-expand}), would consist of a sum of terms in which @var{var}
+appears only raised to nonnegative integer powers. Note that if
address@hidden does not occur in @var{expr}, then @var{expr} is considered
+a polynomial of degree 0.
address@hidden defun
+
address@hidden is-polynomial expr var degree loose
+Check if @var{expr} is a polynomial in variable or sub-expression
address@hidden, and, if so, return a list representation of the polynomial
+where the elements of the list are coefficients of successive powers of
address@hidden: @address@hidden + @var{b} x + @var{c} x^3} would produce the
+list @samp{(@var{a} @var{b} 0 @var{c})}, and @samp{(x + 1)^2} would
+produce the list @samp{(1 2 1)}. The highest element of the list will
+be non-zero, with the special exception that if @var{expr} is the
+constant zero, the returned value will be @samp{(0)}. Return @code{nil}
+if @var{expr} is not a polynomial in @var{var}. If @var{degree} is
+specified, this will not consider polynomials of degree higher than that
+value. This is a good precaution because otherwise an input of
address@hidden(x+1)^1000} will cause a huge coefficient list to be built. If
address@hidden is address@hidden, then a looser definition of a polynomial
+is used in which coefficients are no longer required not to depend on
address@hidden, but are only required not to take the form of polynomials
+themselves. For example, @samp{sin(x) x^2 + cos(x)} is a loose
+polynomial with coefficients @samp{((calcFunc-cos x) 0 (calcFunc-sin
+x))}. The result will never be @code{nil} in loose mode, since any
+expression can be interpreted as a ``constant'' loose polynomial.
address@hidden defun
+
address@hidden polynomial-base expr pred
+Check if @var{expr} is a polynomial in any variable that occurs in it;
+if so, return that variable. (If @var{expr} is a multivariate polynomial,
+this chooses one variable arbitrarily.) If @var{pred} is specified, it should
+be a Lisp function which is called as @samp{(@var{pred} @var{subexpr})},
+and which should return true if @code{mpb-top-expr} (a global name for
+the original @var{expr}) is a suitable polynomial in @var{subexpr}.
+The default predicate uses @samp{(polynomial-p mpb-top-expr @var{subexpr})};
+you can use @var{pred} to specify additional conditions. Or, you could
+have @var{pred} build up a list of every suitable @var{subexpr} that
+is found.
address@hidden defun
+
address@hidden poly-simplify poly
+Simplify polynomial coefficient list @var{poly} by (destructively)
+clipping off trailing zeros.
address@hidden defun
+
address@hidden poly-mix a ac b bc
+Mix two polynomial lists @var{a} and @var{b} (in the form returned by
address@hidden) in a linear combination with coefficient expressions
address@hidden and @var{bc}. The result is a (not necessarily simplified)
+polynomial list representing @address@hidden @var{a} + @var{bc} @var{b}}.
address@hidden defun
+
address@hidden poly-mul a b
+Multiply two polynomial coefficient lists @var{a} and @var{b}. The
+result will be in simplified form if the inputs were simplified.
address@hidden defun
+
address@hidden build-polynomial-expr poly var
+Construct a Calc formula which represents the polynomial coefficient
+list @var{poly} applied to variable @var{var}. The @kbd{a c}
+(@code{calc-collect}) command uses @code{is-polynomial} to turn an
+expression into a coefficient list, then @code{build-polynomial-expr}
+to turn the list back into an expression in regular form.
address@hidden defun
+
address@hidden check-unit-name var
+Check if @var{var} is a variable which can be interpreted as a unit
+name. If so, return the units table entry for that unit. This
+will be a list whose first element is the unit name (not counting
+prefix characters) as a symbol and whose second element is the
+Calc expression which defines the unit. (Refer to the Calc sources
+for details on the remaining elements of this list.) If @var{var}
+is not a variable or is not a unit name, return @code{nil}.
address@hidden defun
+
address@hidden units-in-expr-p expr sub-exprs
+Return true if @var{expr} contains any variables which can be
+interpreted as units. If @var{sub-exprs} is @code{t}, the entire
+expression is searched. If @var{sub-exprs} is @code{nil}, this
+checks whether @var{expr} is directly a units expression.
address@hidden defun
+
address@hidden single-units-in-expr-p expr
+Check whether @var{expr} contains exactly one units variable. If so,
+return the units table entry for the variable. If @var{expr} does
+not contain any units, return @code{nil}. If @var{expr} contains
+two or more units, return the symbol @code{wrong}.
address@hidden defun
+
address@hidden to-standard-units expr which
+Convert units expression @var{expr} to base units. If @var{which}
+is @code{nil}, use Calc's native base units. Otherwise, @var{which}
+can specify a units system, which is a list of two-element lists,
+where the first element is a Calc base symbol name and the second
+is an expression to substitute for it.
address@hidden defun
+
address@hidden remove-units expr
+Return a copy of @var{expr} with all units variables replaced by ones.
+This expression is generally normalized before use.
address@hidden defun
+
address@hidden extract-units expr
+Return a copy of @var{expr} with everything but units variables replaced
+by ones.
address@hidden defun
+
address@hidden Formatting Lisp Functions, Hooks, Symbolic Lisp Functions,
Internals
address@hidden I/O and Formatting Functions
+
address@hidden
+The functions described here are responsible for parsing and formatting
+Calc numbers and formulas.
+
address@hidden calc-eval str sep arg1 arg2 @dots{}
+This is the simplest interface to the Calculator from another Lisp program.
address@hidden Calc from Your Programs}.
address@hidden defun
+
address@hidden read-number str
+If string @var{str} contains a valid Calc number, either integer,
+fraction, float, or HMS form, this function parses and returns that
+number. Otherwise, it returns @code{nil}.
address@hidden defun
+
address@hidden read-expr str
+Read an algebraic expression from string @var{str}. If @var{str} does
+not have the form of a valid expression, return a list of the form
address@hidden(error @var{pos} @var{msg})} where @var{pos} is an integer index
+into @var{str} of the general location of the error, and @var{msg} is
+a string describing the problem.
address@hidden defun
+
address@hidden read-exprs str
+Read a list of expressions separated by commas, and return it as a
+Lisp list. If an error occurs in any expressions, an error list as
+shown above is returned instead.
address@hidden defun
+
address@hidden calc-do-alg-entry initial prompt no-norm
+Read an algebraic formula or formulas using the minibuffer. All
+conventions of regular algebraic entry are observed. The return value
+is a list of Calc formulas; there will be more than one if the user
+entered a list of values separated by commas. The result is @code{nil}
+if the user presses Return with a blank line. If @var{initial} is
+given, it is a string which the minibuffer will initially contain.
+If @var{prompt} is given, it is the prompt string to use; the default
+is ``Algebraic:''. If @var{no-norm} is @code{t}, the formulas will
+be returned exactly as parsed; otherwise, they will be passed through
address@hidden first.
+
+To support the use of @kbd{$} characters in the algebraic entry, use
address@hidden to bind @code{calc-dollar-values} to a list of the values
+to be substituted for @kbd{$}, @kbd{$$}, and so on, and bind
address@hidden to 0. Upon return, @code{calc-dollar-used}
+will have been changed to the highest number of consecutive @kbd{$}s
+that actually appeared in the input.
address@hidden defun
+
address@hidden format-number a
+Convert the real or complex number or HMS form @var{a} to string form.
address@hidden defun
+
address@hidden format-flat-expr a prec
+Convert the arbitrary Calc number or formula @var{a} to string form,
+in the style used by the trail buffer and the @code{calc-edit} command.
+This is a simple format designed
+mostly to guarantee the string is of a form that can be re-parsed by
address@hidden Most formatting modes, such as digit grouping,
+complex number format, and point character, are ignored to ensure the
+result will be re-readable. The @var{prec} parameter is normally 0; if
+you pass a large integer like 1000 instead, the expression will be
+surrounded by parentheses unless it is a plain number or variable name.
address@hidden defun
+
address@hidden format-nice-expr a width
+This is like @code{format-flat-expr} (with @var{prec} equal to 0),
+except that newlines will be inserted to keep lines down to the
+specified @var{width}, and vectors that look like matrices or rewrite
+rules are written in a pseudo-matrix format. The @code{calc-edit}
+command uses this when only one stack entry is being edited.
address@hidden defun
+
address@hidden format-value a width
+Convert the Calc number or formula @var{a} to string form, using the
+format seen in the stack buffer. Beware the string returned may
+not be re-readable by @code{read-expr}, for example, because of digit
+grouping. Multi-line objects like matrices produce strings that
+contain newline characters to separate the lines. The @var{w}
+parameter, if given, is the target window size for which to format
+the expressions. If @var{w} is omitted, the width of the Calculator
+window is used.
address@hidden defun
+
address@hidden compose-expr a prec
+Format the Calc number or formula @var{a} according to the current
+language mode, returning a ``composition.'' To learn about the
+structure of compositions, see the comments in the Calc source code.
+You can specify the format of a given type of function call by putting
+a @address@hidden property on the function's symbol,
+whose value is a Lisp function that takes @var{a} and @var{prec} as
+arguments and returns a composition. Here @var{lang} is a language
+mode name, one of @code{normal}, @code{big}, @code{c}, @code{pascal},
address@hidden, @code{tex}, @code{eqn}, @code{math}, or @code{maple}.
+In Big mode, Calc actually tries @code{math-compose-big} first, then
+tries @code{math-compose-normal}. If this property does not exist,
+or if the function returns @code{nil}, the function is written in the
+normal function-call notation for that language.
address@hidden defun
+
address@hidden composition-to-string c w
+Convert a composition structure returned by @code{compose-expr} into
+a string. Multi-line compositions convert to strings containing
+newline characters. The target window size is given by @var{w}.
+The @code{format-value} function basically calls @code{compose-expr}
+followed by @code{composition-to-string}.
address@hidden defun
+
address@hidden comp-width c
+Compute the width in characters of composition @var{c}.
address@hidden defun
+
address@hidden comp-height c
+Compute the height in lines of composition @var{c}.
address@hidden defun
+
address@hidden comp-ascent c
+Compute the portion of the height of composition @var{c} which is on or
+above the baseline. For a one-line composition, this will be one.
address@hidden defun
+
address@hidden comp-descent c
+Compute the portion of the height of composition @var{c} which is below
+the baseline. For a one-line composition, this will be zero.
address@hidden defun
+
address@hidden comp-first-char c
+If composition @var{c} is a ``flat'' composition, return the first
+(leftmost) character of the composition as an integer. Otherwise,
+return @code{nil}.
address@hidden defun
+
address@hidden comp-last-char c
+If composition @var{c} is a ``flat'' composition, return the last
+(rightmost) character, otherwise return @code{nil}.
address@hidden defun
+
address@hidden @node Lisp Variables, Hooks, Formatting Lisp Functions, Internals
address@hidden @subsubsection Lisp Variables
address@hidden
address@hidden @noindent
address@hidden (This section is currently unfinished.)
+
address@hidden Hooks, , Formatting Lisp Functions, Internals
address@hidden Hooks
+
address@hidden
+Hooks are variables which contain Lisp functions (or lists of functions)
+which are called at various times. Calc defines a number of hooks
+that help you to customize it in various ways. Calc uses the Lisp
+function @code{run-hooks} to invoke the hooks shown below. Several
+other customization-related variables are also described here.
+
address@hidden calc-load-hook
+This hook is called at the end of @file{calc.el}, after the file has
+been loaded, before any functions in it have been called, but after
address@hidden and similar variables have been set up.
address@hidden defvar
+
address@hidden calc-ext-load-hook
+This hook is called at the end of @file{calc-ext.el}.
address@hidden defvar
+
address@hidden calc-start-hook
+This hook is called as the last step in a @kbd{M-x calc} command.
+At this point, the Calc buffer has been created and initialized if
+necessary, the Calc window and trail window have been created,
+and the ``Welcome to Calc'' message has been displayed.
address@hidden defvar
+
address@hidden calc-mode-hook
+This hook is called when the Calc buffer is being created. Usually
+this will only happen once per Emacs session. The hook is called
+after Emacs has switched to the new buffer, the mode-settings file
+has been read if necessary, and all other buffer-local variables
+have been set up. After this hook returns, Calc will perform a
address@hidden operation, set up the mode line display, then
+evaluate any deferred @code{calc-define} properties that have not
+been evaluated yet.
address@hidden defvar
+
address@hidden calc-trail-mode-hook
+This hook is called when the Calc Trail buffer is being created.
+It is called as the very last step of setting up the Trail buffer.
+Like @code{calc-mode-hook}, this will normally happen only once
+per Emacs session.
address@hidden defvar
+
address@hidden calc-end-hook
+This hook is called by @code{calc-quit}, generally because the user
+presses @kbd{q} or @kbd{C-x * c} while in Calc. The Calc buffer will
+be the current buffer. The hook is called as the very first
+step, before the Calc window is destroyed.
address@hidden defvar
+
address@hidden calc-window-hook
+If this hook is address@hidden, it is called to create the Calc window.
+Upon return, this new Calc window should be the current window.
+(The Calc buffer will already be the current buffer when the
+hook is called.) If the hook is not defined, Calc will
+generally use @code{split-window}, @code{set-window-buffer},
+and @code{select-window} to create the Calc window.
address@hidden defvar
+
address@hidden calc-trail-window-hook
+If this hook is address@hidden, it is called to create the Calc Trail
+window. The variable @code{calc-trail-buffer} will contain the buffer
+which the window should use. Unlike @code{calc-window-hook}, this hook
+must @emph{not} switch into the new window.
address@hidden defvar
+
address@hidden calc-embedded-mode-hook
+This hook is called the first time that Embedded mode is entered.
address@hidden defvar
+
address@hidden calc-embedded-new-buffer-hook
+This hook is called each time that Embedded mode is entered in a
+new buffer.
address@hidden defvar
+
address@hidden calc-embedded-new-formula-hook
+This hook is called each time that Embedded mode is enabled for a
+new formula.
address@hidden defvar
+
address@hidden calc-edit-mode-hook
+This hook is called by @code{calc-edit} (and the other ``edit''
+commands) when the temporary editing buffer is being created.
+The buffer will have been selected and set up to be in
address@hidden, but will not yet have been filled with
+text. (In fact it may still have leftover text from a previous
address@hidden command.)
address@hidden defvar
+
address@hidden calc-mode-save-hook
+This hook is called by the @code{calc-save-modes} command,
+after Calc's own mode features have been inserted into the
+Calc init file and just before the ``End of mode settings''
+message is inserted.
address@hidden defvar
+
address@hidden calc-reset-hook
+This hook is called after @kbd{C-x * 0} (@code{calc-reset}) has
+reset all modes. The Calc buffer will be the current buffer.
address@hidden defvar
+
address@hidden calc-other-modes
+This variable contains a list of strings. The strings are
+concatenated at the end of the modes portion of the Calc
+mode line (after standard modes such as ``Deg'', ``Inv'' and
+``Hyp''). Each string should be a short, single word followed
+by a space. The variable is @code{nil} by default.
address@hidden defvar
+
address@hidden calc-mode-map
+This is the keymap that is used by Calc mode. The best time
+to adjust it is probably in a @code{calc-mode-hook}. If the
+Calc extensions package (@file{calc-ext.el}) has not yet been
+loaded, many of these keys will be bound to @code{calc-missing-key},
+which is a command that loads the extensions package and
+``retypes'' the key. If your @code{calc-mode-hook} rebinds
+one of these keys, it will probably be overridden when the
+extensions are loaded.
address@hidden defvar
+
address@hidden calc-digit-map
+This is the keymap that is used during numeric entry. Numeric
+entry uses the minibuffer, but this map binds every non-numeric
+key to @code{calcDigit-nondigit} which generally calls
address@hidden and ``retypes'' the key.
address@hidden defvar
+
address@hidden calc-alg-ent-map
+This is the keymap that is used during algebraic entry. This is
+mostly a copy of @code{minibuffer-local-map}.
address@hidden defvar
+
address@hidden calc-store-var-map
+This is the keymap that is used during entry of variable names for
+commands like @code{calc-store} and @code{calc-recall}. This is
+mostly a copy of @code{minibuffer-local-completion-map}.
address@hidden defvar
+
address@hidden calc-edit-mode-map
+This is the (sparse) keymap used by @code{calc-edit} and other
+temporary editing commands. It binds @key{RET}, @key{LFD},
+and @kbd{C-c C-c} to @code{calc-edit-finish}.
address@hidden defvar
+
address@hidden calc-mode-var-list
+This is a list of variables which are saved by @code{calc-save-modes}.
+Each entry is a list of two items, the variable (as a Lisp symbol)
+and its default value. When modes are being saved, each variable
+is compared with its default value (using @code{equal}) and any
+non-default variables are written out.
address@hidden defvar
+
address@hidden calc-local-var-list
+This is a list of variables which should be buffer-local to the
+Calc buffer. Each entry is a variable name (as a Lisp symbol).
+These variables also have their default values manipulated by
+the @code{calc} and @code{calc-quit} commands; @pxref{Multiple Calculators}.
+Since @code{calc-mode-hook} is called after this list has been
+used the first time, your hook should add a variable to the
+list and also call @code{make-local-variable} itself.
address@hidden defvar
+
address@hidden Copying, GNU Free Documentation License, Programming, Top
address@hidden GNU GENERAL PUBLIC LICENSE
address@hidden gpl.texi
+
address@hidden GNU Free Documentation License, Customizing Calc, Copying, Top
address@hidden GNU Free Documentation License
address@hidden doclicense.texi
+
address@hidden Customizing Calc, Reporting Bugs, GNU Free Documentation
License, Top
address@hidden Customizing Calc
+
+The usual prefix for Calc is the key sequence @kbd{C-x *}. If you wish
+to use a different prefix, you can put
+
address@hidden
+(global-set-key "NEWPREFIX" 'calc-dispatch)
address@hidden example
+
address@hidden
+in your .emacs file.
+(@xref{Key Bindings,,Customizing Key Bindings,emacs,
+The GNU Emacs Manual}, for more information on binding keys.)
+A convenient way to start Calc is with @kbd{C-x * *}; to make it equally
+convenient for users who use a different prefix, the prefix can be
+followed by @kbd{=}, @kbd{&}, @kbd{#}, @kbd{\}, @kbd{/}, @kbd{+} or
address@hidden as well as @kbd{*} to start Calc, and so in many cases the last
+character of the prefix can simply be typed twice.
+
+Calc is controlled by many variables, most of which can be reset
+from within Calc. Some variables are less involved with actual
+calculation, and can be set outside of Calc using Emacs's
+customization facilities. These variables are listed below.
+Typing @kbd{M-x customize-variable RET @var{variable-name} RET}
+will bring up a buffer in which the variable's value can be redefined.
+Typing @kbd{M-x customize-group RET calc RET} will bring up a buffer which
+contains all of Calc's customizable variables. (These variables can
+also be reset by putting the appropriate lines in your .emacs file;
address@hidden File, ,Init File, emacs, The GNU Emacs Manual}.)
+
+Some of the customizable variables are regular expressions. A regular
+expression is basically a pattern that Calc can search for.
+See @ref{Regexp Search,, Regular Expression Search, emacs, The GNU Emacs
Manual}
+to see how regular expressions work.
+
address@hidden calc-settings-file
+The variable @code{calc-settings-file} holds the file name in
+which commands like @kbd{m m} and @kbd{Z P} store ``permanent''
+definitions.
+If @code{calc-settings-file} is not your user init file (typically
address@hidden/.emacs}) and if the variable @code{calc-loaded-settings-file} is
address@hidden, then Calc will automatically load your settings file (if it
+exists) the first time Calc is invoked.
+
+The default value for this variable is @code{"~/.calc.el"}.
address@hidden defvar
+
address@hidden calc-gnuplot-name
+See @address@hidden
+The variable @code{calc-gnuplot-name} should be the name of the
+GNUPLOT program (a string). If you have GNUPLOT installed on your
+system but Calc is unable to find it, you may need to set this
+variable. You may also need to set some Lisp variables to show Calc how
+to run GNUPLOT on your system, see @ref{Devices, ,Graphical Devices} .
+The default value of @code{calc-gnuplot-name} is @code{"gnuplot"}.
address@hidden defvar
+
address@hidden calc-gnuplot-plot-command
address@hidden calc-gnuplot-print-command
+See @ref{Devices, ,Graphical address@hidden
+The variables @code{calc-gnuplot-plot-command} and
address@hidden represent system commands to
+display and print the output of GNUPLOT, respectively. These may be
address@hidden if no command is necessary, or strings which can include
address@hidden to signify the name of the file to be displayed or printed.
+Or, these variables may contain Lisp expressions which are evaluated
+to display or print the output.
+
+The default value of @code{calc-gnuplot-plot-command} is @code{nil},
+and the default value of @code{calc-gnuplot-print-command} is
address@hidden"lp %s"}.
address@hidden defvar
+
address@hidden calc-language-alist
+See @ref{Basic Embedded address@hidden
+The variable @code{calc-language-alist} controls the languages that
+Calc will associate with major modes. When Calc embedded mode is
+enabled, it will try to use the current major mode to
+determine what language should be used. (This can be overridden using
+Calc's mode changing commands, @xref{Mode Settings in Embedded Mode}.)
+The variable @code{calc-language-alist} consists of a list of pairs of
+the form @code{(@var{MAJOR-MODE} . @var{LANGUAGE})}; for example,
address@hidden(latex-mode . latex)} is one such pair. If Calc embedded is
+activated in a buffer whose major mode is @var{MAJOR-MODE}, it will set itself
+to use the language @var{LANGUAGE}.
+
+The default value of @code{calc-language-alist} is
address@hidden
+ ((latex-mode . latex)
+ (tex-mode . tex)
+ (plain-tex-mode . tex)
+ (context-mode . tex)
+ (nroff-mode . eqn)
+ (pascal-mode . pascal)
+ (c-mode . c)
+ (c++-mode . c)
+ (fortran-mode . fortran)
+ (f90-mode . fortran))
address@hidden example
address@hidden defvar
+
address@hidden calc-embedded-announce-formula
address@hidden calc-embedded-announce-formula-alist
+See @ref{Customizing Embedded address@hidden
+The variable @code{calc-embedded-announce-formula} helps determine
+what formulas @kbd{C-x * a} will activate in a buffer. It is a
+regular expression, and when activating embedded formulas with
address@hidden * a}, it will tell Calc that what follows is a formula to be
+activated. (Calc also uses other patterns to find formulas, such as
address@hidden>} and @samp{:=}.)
+
+The default pattern is @code{"%Embed\n\\(% .*\n\\)*"}, which checks
+for @samp{%Embed} followed by any number of lines beginning with
address@hidden and a space.
+
+The variable @code{calc-embedded-announce-formula-alist} is used to
+set @code{calc-embedded-announce-formula} to different regular
+expressions depending on the major mode of the editing buffer.
+It consists of a list of pairs of the form @code{(@var{MAJOR-MODE} .
address@hidden)}, and its default value is
address@hidden
+ ((c++-mode . "//Embed\n\\(// .*\n\\)*")
+ (c-mode . "/\\*Embed\\*/\n\\(/\\* .*\\*/\n\\)*")
+ (f90-mode . "!Embed\n\\(! .*\n\\)*")
+ (fortran-mode . "C Embed\n\\(C .*\n\\)*")
+ (html-helper-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
+ (html-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
+ (nroff-mode . "\\\\\"Embed\n\\(\\\\\" .*\n\\)*")
+ (pascal-mode . "@address@hidden(@address@hidden)*")
+ (sgml-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
+ (xml-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
+ (texinfo-mode . "@@c Embed\n\\(@@c .*\n\\)*"))
address@hidden example
+Any major modes added to @code{calc-embedded-announce-formula-alist}
+should also be added to @code{calc-embedded-open-close-plain-alist}
+and @code{calc-embedded-open-close-mode-alist}.
address@hidden defvar
+
address@hidden calc-embedded-open-formula
address@hidden calc-embedded-close-formula
address@hidden calc-embedded-open-close-formula-alist
+See @ref{Customizing Embedded address@hidden
+The variables @code{calc-embedded-open-formula} and
address@hidden control the region that Calc will
+activate as a formula when Embedded mode is entered with @kbd{C-x * e}.
+They are regular expressions;
+Calc normally scans backward and forward in the buffer for the
+nearest text matching these regular expressions to be the ``formula
+delimiters''.
+
+The simplest delimiters are blank lines. Other delimiters that
+Embedded mode understands by default are:
address@hidden
address@hidden
+The @TeX{} and address@hidden math delimiters @samp{$ $}, @samp{$$ $$},
address@hidden \]}, and @samp{\( \)};
address@hidden
+Lines beginning with @samp{\begin} and @samp{\end} (except matrix delimiters);
address@hidden
+Lines beginning with @samp{@@} (Texinfo delimiters).
address@hidden
+Lines beginning with @samp{.EQ} and @samp{.EN} (@dfn{eqn} delimiters);
address@hidden
+Lines containing a single @samp{%} or @samp{.\"} symbol and nothing else.
address@hidden enumerate
+
+The variable @code{calc-embedded-open-close-formula-alist} is used to
+set @code{calc-embedded-open-formula} and
address@hidden to different regular
+expressions depending on the major mode of the editing buffer.
+It consists of a list of lists of the form
address@hidden(@var{MAJOR-MODE} @var{OPEN-FORMULA-REGEXP}
address@hidden)}, and its default value is
address@hidden
address@hidden defvar
+
address@hidden calc-embedded-open-word
address@hidden calc-embedded-close-word
address@hidden calc-embedded-open-close-word-alist
+See @ref{Customizing Embedded address@hidden
+The variables @code{calc-embedded-open-word} and
address@hidden control the region that Calc will
+activate when Embedded mode is entered with @kbd{C-x * w}. They are
+regular expressions.
+
+The default values of @code{calc-embedded-open-word} and
address@hidden are @code{"^\\|[^-+0-9.eE]"} and
address@hidden"$\\|[^-+0-9.eE]"} respectively.
+
+The variable @code{calc-embedded-open-close-word-alist} is used to
+set @code{calc-embedded-open-word} and
address@hidden to different regular
+expressions depending on the major mode of the editing buffer.
+It consists of a list of lists of the form
address@hidden(@var{MAJOR-MODE} @var{OPEN-WORD-REGEXP}
address@hidden)}, and its default value is
address@hidden
address@hidden defvar
+
address@hidden calc-embedded-open-plain
address@hidden calc-embedded-close-plain
address@hidden calc-embedded-open-close-plain-alist
+See @ref{Customizing Embedded address@hidden
+The variables @code{calc-embedded-open-plain} and
address@hidden are used to delimit ``plain''
+formulas. Note that these are actual strings, not regular
+expressions, because Calc must be able to write these string into a
+buffer as well as to recognize them.
+
+The default string for @code{calc-embedded-open-plain} is
address@hidden"%%% "}, note the trailing space. The default string for
address@hidden is @code{" %%%\n"}, without
+the trailing newline here, the first line of a Big mode formula
+that followed might be shifted over with respect to the other lines.
+
+The variable @code{calc-embedded-open-close-plain-alist} is used to
+set @code{calc-embedded-open-plain} and
address@hidden to different strings
+depending on the major mode of the editing buffer.
+It consists of a list of lists of the form
address@hidden(@var{MAJOR-MODE} @var{OPEN-PLAIN-STRING}
address@hidden)}, and its default value is
address@hidden
+ ((c++-mode "// %% " " %%\n")
+ (c-mode "/* %% " " %% */\n")
+ (f90-mode "! %% " " %%\n")
+ (fortran-mode "C %% " " %%\n")
+ (html-helper-mode "<!-- %% " " %% -->\n")
+ (html-mode "<!-- %% " " %% -->\n")
+ (nroff-mode "\\\" %% " " %%\n")
+ (pascal-mode "@{%% " " address@hidden")
+ (sgml-mode "<!-- %% " " %% -->\n")
+ (xml-mode "<!-- %% " " %% -->\n")
+ (texinfo-mode "@@c %% " " %%\n"))
address@hidden example
+Any major modes added to @code{calc-embedded-open-close-plain-alist}
+should also be added to @code{calc-embedded-announce-formula-alist}
+and @code{calc-embedded-open-close-mode-alist}.
address@hidden defvar
+
address@hidden calc-embedded-open-new-formula
address@hidden calc-embedded-close-new-formula
address@hidden calc-embedded-open-close-new-formula-alist
+See @ref{Customizing Embedded address@hidden
+The variables @code{calc-embedded-open-new-formula} and
address@hidden are strings which are
+inserted before and after a new formula when you type @kbd{C-x * f}.
+
+The default value of @code{calc-embedded-open-new-formula} is
address@hidden"\n\n"}. If this string begins with a newline character and the
address@hidden * f} is typed at the beginning of a line, @kbd{C-x * f} will skip
+this first newline to avoid introducing unnecessary blank lines in the
+file. The default value of @code{calc-embedded-close-new-formula} is
+also @code{"\n\n"}. The final newline is omitted by @address@hidden * f}}
+if typed at the end of a line. (It follows that if @kbd{C-x * f} is
+typed on a blank line, both a leading opening newline and a trailing
+closing newline are omitted.)
+
+The variable @code{calc-embedded-open-close-new-formula-alist} is used to
+set @code{calc-embedded-open-new-formula} and
address@hidden to different strings
+depending on the major mode of the editing buffer.
+It consists of a list of lists of the form
address@hidden(@var{MAJOR-MODE} @var{OPEN-NEW-FORMULA-STRING}
address@hidden)}, and its default value is
address@hidden
address@hidden defvar
+
address@hidden calc-embedded-open-mode
address@hidden calc-embedded-close-mode
address@hidden calc-embedded-open-close-mode-alist
+See @ref{Customizing Embedded address@hidden
+The variables @code{calc-embedded-open-mode} and
address@hidden are strings which Calc will place before
+and after any mode annotations that it inserts. Calc never scans for
+these strings; Calc always looks for the annotation itself, so it is not
+necessary to add them to user-written annotations.
+
+The default value of @code{calc-embedded-open-mode} is @code{"% "}
+and the default value of @code{calc-embedded-close-mode} is
address@hidden"\n"}.
+If you change the value of @code{calc-embedded-close-mode}, it is a good
+idea still to end with a newline so that mode annotations will appear on
+lines by themselves.
+
+The variable @code{calc-embedded-open-close-mode-alist} is used to
+set @code{calc-embedded-open-mode} and
address@hidden to different strings
+expressions depending on the major mode of the editing buffer.
+It consists of a list of lists of the form
address@hidden(@var{MAJOR-MODE} @var{OPEN-MODE-STRING}
address@hidden)}, and its default value is
address@hidden
+ ((c++-mode "// " "\n")
+ (c-mode "/* " " */\n")
+ (f90-mode "! " "\n")
+ (fortran-mode "C " "\n")
+ (html-helper-mode "<!-- " " -->\n")
+ (html-mode "<!-- " " -->\n")
+ (nroff-mode "\\\" " "\n")
+ (pascal-mode "@{ " " @}\n")
+ (sgml-mode "<!-- " " -->\n")
+ (xml-mode "<!-- " " -->\n")
+ (texinfo-mode "@@c " "\n"))
address@hidden example
+Any major modes added to @code{calc-embedded-open-close-mode-alist}
+should also be added to @code{calc-embedded-announce-formula-alist}
+and @code{calc-embedded-open-close-plain-alist}.
address@hidden defvar
+
address@hidden calc-multiplication-has-precedence
+The variable @code{calc-multiplication-has-precedence} determines
+whether multiplication has precedence over division in algebraic formulas
+in normal language modes. If @code{calc-multiplication-has-precedence}
+is address@hidden, then multiplication has precedence, and so for
+example @samp{a/b*c} will be interpreted as @samp{a/(b*c)}. If
address@hidden is @code{nil}, then
+multiplication has the same precedence as division, and so for example
address@hidden/b*c} will be interpreted as @samp{(a/b)*c}. The default value
+of @code{calc-multiplication-has-precedence} is @code{t}.
address@hidden defvar
+
address@hidden Reporting Bugs, Summary, Customizing Calc, Top
address@hidden Reporting Bugs
+
address@hidden
+If you find a bug in Calc, send e-mail to Jay Belanger,
+
address@hidden
+jay.p.belanger@@gmail.com
address@hidden example
+
address@hidden
+There is an automatic command @kbd{M-x report-calc-bug} which helps
+you to report bugs. This command prompts you for a brief subject
+line, then leaves you in a mail editing buffer. Type @kbd{C-c C-c} to
+send your mail. Make sure your subject line indicates that you are
+reporting a Calc bug; this command sends mail to the maintainer's
+regular mailbox.
+
+If you have suggestions for additional features for Calc, please send
+them. Some have dared to suggest that Calc is already top-heavy with
+features; this obviously cannot be the case, so if you have ideas, send
+them right in.
+
+At the front of the source file, @file{calc.el}, is a list of ideas for
+future work. If any enthusiastic souls wish to take it upon themselves
+to work on these, please send a message (using @kbd{M-x report-calc-bug})
+so any efforts can be coordinated.
+
+The latest version of Calc is available from Savannah, in the Emacs
+CVS tree. See @uref{http://savannah.gnu.org/projects/emacs}.
+
address@hidden [summary]
address@hidden Summary, Key Index, Reporting Bugs, Top
address@hidden Calc Summary
+
address@hidden
+This section includes a complete list of Calc 2.1 keystroke commands.
+Each line lists the stack entries used by the command (top-of-stack
+last), the keystrokes themselves, the prompts asked by the command,
+and the result of the command (also with top-of-stack last).
+The result is expressed using the equivalent algebraic function.
+Commands which put no results on the stack show the full @kbd{M-x}
+command name in that position. Numbers preceding the result or
+command name refer to notes at the end.
+
+Algebraic functions and @kbd{M-x} commands that don't have corresponding
+keystrokes are not listed in this summary.
address@hidden Index}. @xref{Function Index}.
+
address@hidden
address@hidden
address@hidden
+\vskip-2\baselineskip \null
+\gdef\sumrow#1{\sumrowx#1\relax}%
+\gdef\sumrowx#1\:#2\:#3\:#4\:#5\:#6\relax{%
+\leavevmode%
+{\smallfonts
+\hbox to5em{\sl\hss#1}%
+\hbox to5em{\tt#2\hss}%
+\hbox to4em{\sl#3\hss}%
+\hbox to5em{\rm\hss#4}%
+\thinspace%
+{\tt#5}%
+{\sl#6}%
+}}%
+\gdef\sumlpar{{\rm(}}%
+\gdef\sumrpar{{\rm)}}%
+\gdef\sumcomma{{\rm,\thinspace}}%
+\gdef\sumexcl{{\rm!}}%
+\gdef\sumbreak{\vskip-2.5\baselineskip\goodbreak}%
+\gdef\minus#1{{\tt-}}%
address@hidden tex
address@hidden@:address@hidden
address@hidden@address@hidden
address@hidden@(address@hidden @let(address@hidden
address@hidden@)address@hidden @let)address@hidden
address@hidden@,address@hidden @let,address@hidden
address@hidden@address@hidden @address@hidden
address@hidden iftex
address@hidden
address@hidden
address@hidden@baselineskip-2.5pt
address@hidden@address@hidden
address@hidden iftex
address@hidden @: C-x * a @: @: 33
@:calc-embedded-activate@:}
address@hidden @: C-x * b @: @:
@:calc-big-or-small@:}
address@hidden @: C-x * c @: @: @:calc@:}
address@hidden @: C-x * d @: @:
@:calc-embedded-duplicate@:}
address@hidden @: C-x * e @: @: 34 @:calc-embedded@:}
address@hidden @: C-x * f @:formula @:
@:calc-embedded-new-formula@:}
address@hidden @: C-x * g @: @: 35
@:calc-grab-region@:}
address@hidden @: C-x * i @: @: @:calc-info@:}
address@hidden @: C-x * j @: @:
@:calc-embedded-select@:}
address@hidden @: C-x * k @: @: @:calc-keypad@:}
address@hidden @: C-x * l @: @:
@:calc-load-everything@:}
address@hidden @: C-x * m @: @:
@:read-kbd-macro@:}
address@hidden @: C-x * n @: @: 4
@:calc-embedded-next@:}
address@hidden @: C-x * o @: @:
@:calc-other-window@:}
address@hidden @: C-x * p @: @: 4
@:calc-embedded-previous@:}
address@hidden @: C-x * q @:formula @: @:quick-calc@:}
address@hidden @: C-x * r @: @: 36
@:calc-grab-rectangle@:}
address@hidden @: C-x * s @: @:
@:calc-info-summary@:}
address@hidden @: C-x * t @: @: @:calc-tutorial@:}
address@hidden @: C-x * u @: @:
@:calc-embedded-update-formula@:}
address@hidden @: C-x * w @: @:
@:calc-embedded-word@:}
address@hidden @: C-x * x @: @: @:calc-quit@:}
address@hidden @: C-x * y @: @:1,28,49
@:calc-copy-to-buffer@:}
address@hidden @: C-x * z @: @:
@:calc-user-invocation@:}
address@hidden @: C-x * : @: @: 36
@:calc-grab-sum-down@:}
address@hidden @: C-x * _ @: @: 36
@:calc-grab-sum-across@:}
address@hidden @: C-x * ` @:editing @: 30
@:calc-embedded-edit@:}
address@hidden @: C-x * 0 @:(zero) @: @:calc-reset@:}
+
address@hidden
address@hidden @: 0-9 @:number @: @:@:number}
address@hidden @: . @:number @: @:@:0.number}
address@hidden @: _ @:number @: @:-@:number}
address@hidden @: e @:number @: @:@:1e number}
address@hidden @: # @:number @: @:@:address@hidden
address@hidden @: P @:(in number) @: @:+/-@:}
address@hidden @: M @:(in number) @: @:mod@:}
address@hidden @: @@ ' " @: (in number)@: @:@:HMS form}
address@hidden @: h m s @: (in number)@: @:@:HMS form}
+
address@hidden
address@hidden @: ' @:formula @: 37,46 @:@:formula}
address@hidden @: $ @:formula @: 37,46 @:$@:formula}
address@hidden @: " @:string @: 37,46 @:@:string}
+
address@hidden
address@hidden a b@: + @: @: 2 @:add@:(a,b) a+b}
address@hidden a b@: - @: @: 2 @:sub@:(a,b)
address@hidden
address@hidden a b@: * @: @: 2 @:mul@:(a,b) a b,
a*b}
address@hidden a b@: / @: @: 2 @:div@:(a,b) a/b}
address@hidden a b@: ^ @: @: 2 @:pow@:(a,b) a^b}
address@hidden a b@: I ^ @: @: 2 @:nroot@:(a,b)
a^(1/b)}
address@hidden a b@: % @: @: 2 @:mod@:(a,b) a%b}
address@hidden a b@: \ @: @: 2 @:idiv@:(a,b) a\b}
address@hidden a b@: : @: @: 2 @:fdiv@:(a,b)}
address@hidden a b@: | @: @: 2 @:vconcat@:(a,b)
a|b}
address@hidden a b@: I | @: @: @:vconcat@:(b,a)
b|a}
address@hidden a b@: H | @: @: 2 @:append@:(a,b)}
address@hidden a b@: I H | @: @: @:append@:(b,a)}
address@hidden a@: & @: @: 1 @:inv@:(a) 1/a}
address@hidden a@: ! @: @: 1 @:fact@:(a) a!}
address@hidden a@: = @: @: 1 @:evalv@:(a)}
address@hidden a@: M-% @: @: @:percent@:(a) a%}
+
address@hidden
address@hidden ... a@: @key{RET} @: @: 1 @:@:... a a}
address@hidden ... a@: @key{SPC} @: @: 1 @:@:... a a}
address@hidden a b@: @key{TAB} @: @: 3 @:@:... b a}
address@hidden a b c@: address@hidden @: @: 3 @:@:... b c
a}
address@hidden a b@: @key{LFD} @: @: 1 @:@:... a b a}
address@hidden ... a@: @key{DEL} @: @: 1 @:@:...}
address@hidden a b@: address@hidden @: @: 1 @:@:... b}
address@hidden @: address@hidden @: @: 4
@:calc-last-args@:}
address@hidden a@: ` @:editing @: 1,30 @:calc-edit@:}
+
address@hidden
address@hidden ... a@: C-d @: @: 1 @:@:...}
address@hidden @: C-k @: @: 27 @:calc-kill@:}
address@hidden @: C-w @: @: 27
@:calc-kill-region@:}
address@hidden @: C-y @: @: @:calc-yank@:}
address@hidden @: C-_ @: @: 4 @:calc-undo@:}
address@hidden @: M-k @: @: 27
@:calc-copy-as-kill@:}
address@hidden @: M-w @: @: 27
@:calc-copy-region-as-kill@:}
+
address@hidden
address@hidden @: [ @: @: @:@:[...}
address@hidden a b@: ] @: @: @:@:[a,b]}
address@hidden @: ( @: @: @:@:(...}
address@hidden(.. a b@: ) @: @: @:@:(a,b)}
address@hidden @: , @: @: @:@:vector or rect
complex}
address@hidden @: ; @: @: @:@:matrix or polar
complex}
address@hidden @: .. @: @: @:@:interval}
+
address@hidden
address@hidden @: ~ @: @: @:calc-num-prefix@:}
address@hidden @: < @: @: 4
@:calc-scroll-left@:}
address@hidden @: > @: @: 4
@:calc-scroll-right@:}
address@hidden @: @{ @: @: 4
@:calc-scroll-down@:}
address@hidden @: @} @: @: 4 @:calc-scroll-up@:}
address@hidden @: ? @: @: @:calc-help@:}
+
address@hidden
address@hidden a@: n @: @: 1 @:neg@:(a)
@minus{}a}
address@hidden @: o @: @: 4 @:calc-realign@:}
address@hidden @: p @:precision @: 31 @:calc-precision@:}
address@hidden @: q @: @: @:calc-quit@:}
address@hidden @: w @: @: @:calc-why@:}
address@hidden @: x @:command @: @:M-x
calc-@:command}
address@hidden a@: y @: @:1,28,49
@:calc-copy-to-buffer@:}
+
address@hidden
address@hidden a@: A @: @: 1 @:abs@:(a)}
address@hidden a b@: B @: @: 2 @:log@:(a,b)}
address@hidden a b@: I B @: @: 2 @:alog@:(a,b) b^a}
address@hidden a@: C @: @: 1 @:cos@:(a)}
address@hidden a@: I C @: @: 1 @:arccos@:(a)}
address@hidden a@: H C @: @: 1 @:cosh@:(a)}
address@hidden a@: I H C @: @: 1 @:arccosh@:(a)}
address@hidden @: D @: @: 4 @:calc-redo@:}
address@hidden a@: E @: @: 1 @:exp@:(a)}
address@hidden a@: H E @: @: 1 @:exp10@:(a) 10.^a}
address@hidden a@: F @: @: 1,11 @:floor@:(a,d)}
address@hidden a@: I F @: @: 1,11 @:ceil@:(a,d)}
address@hidden a@: H F @: @: 1,11 @:ffloor@:(a,d)}
address@hidden a@: I H F @: @: 1,11 @:fceil@:(a,d)}
address@hidden a@: G @: @: 1 @:arg@:(a)}
address@hidden @: H @:command @: 32 @:@:Hyperbolic}
address@hidden @: I @:command @: 32 @:@:Inverse}
address@hidden a@: J @: @: 1 @:conj@:(a)}
address@hidden @: K @:command @: 32 @:@:Keep-args}
address@hidden a@: L @: @: 1 @:ln@:(a)}
address@hidden a@: H L @: @: 1 @:log10@:(a)}
address@hidden @: M @: @:
@:calc-more-recursion-depth@:}
address@hidden @: I M @: @:
@:calc-less-recursion-depth@:}
address@hidden a@: N @: @: 5 @:evalvn@:(a)}
address@hidden @: P @: @: @:@:pi}
address@hidden @: I P @: @: @:@:gamma}
address@hidden @: H P @: @: @:@:e}
address@hidden @: I H P @: @: @:@:phi}
address@hidden a@: Q @: @: 1 @:sqrt@:(a)}
address@hidden a@: I Q @: @: 1 @:sqr@:(a) a^2}
address@hidden a@: R @: @: 1,11 @:round@:(a,d)}
address@hidden a@: I R @: @: 1,11 @:trunc@:(a,d)}
address@hidden a@: H R @: @: 1,11 @:fround@:(a,d)}
address@hidden a@: I H R @: @: 1,11 @:ftrunc@:(a,d)}
address@hidden a@: S @: @: 1 @:sin@:(a)}
address@hidden a@: I S @: @: 1 @:arcsin@:(a)}
address@hidden a@: H S @: @: 1 @:sinh@:(a)}
address@hidden a@: I H S @: @: 1 @:arcsinh@:(a)}
address@hidden a@: T @: @: 1 @:tan@:(a)}
address@hidden a@: I T @: @: 1 @:arctan@:(a)}
address@hidden a@: H T @: @: 1 @:tanh@:(a)}
address@hidden a@: I H T @: @: 1 @:arctanh@:(a)}
address@hidden @: U @: @: 4 @:calc-undo@:}
address@hidden @: X @: @: 4
@:calc-call-last-kbd-macro@:}
+
address@hidden
address@hidden a b@: a = @: @: 2 @:eq@:(a,b) a=b}
address@hidden a b@: a # @: @: 2 @:neq@:(a,b) a!=b}
address@hidden a b@: a < @: @: 2 @:lt@:(a,b) a<b}
address@hidden a b@: a > @: @: 2 @:gt@:(a,b) a>b}
address@hidden a b@: a [ @: @: 2 @:leq@:(a,b) a<=b}
address@hidden a b@: a ] @: @: 2 @:geq@:(a,b) a>=b}
address@hidden a b@: a @{ @: @: 2 @:in@:(a,b)}
address@hidden a b@: a & @: @: 2,45 @:land@:(a,b) a&&b}
address@hidden a b@: a | @: @: 2,45 @:lor@:(a,b) a||b}
address@hidden a@: a ! @: @: 1,45 @:lnot@:(a) !a}
address@hidden a b c@: a : @: @: 45 @:if@:(a,b,c)
a?b:c}
address@hidden a@: a . @: @: 1 @:rmeq@:(a)}
address@hidden a@: a " @: @: 7,8
@:calc-expand-formula@:}
+
address@hidden
address@hidden a@: a + @:i, l, h @: 6,38 @:sum@:(a,i,l,h)}
address@hidden a@: a - @:i, l, h @: 6,38 @:asum@:(a,i,l,h)}
address@hidden a@: a * @:i, l, h @: 6,38 @:prod@:(a,i,l,h)}
address@hidden a b@: a _ @: @: 2 @:subscr@:(a,b)
a_b}
+
address@hidden
address@hidden a b@: a \ @: @: 2 @:pdiv@:(a,b)}
address@hidden a b@: a % @: @: 2 @:prem@:(a,b)}
address@hidden a b@: a / @: @: 2 @:pdivrem@:(a,b)
[q,r]}
address@hidden a b@: H a / @: @: 2 @:pdivide@:(a,b)
q+r/b}
+
address@hidden
address@hidden a@: a a @: @: 1 @:apart@:(a)}
address@hidden a@: a b @:old, new @: 38
@:subst@:(a,old,new)}
address@hidden a@: a c @:v @: 38 @:collect@:(a,v)}
address@hidden a@: a d @:v @: 4,38 @:deriv@:(a,v)}
address@hidden a@: H a d @:v @: 4,38 @:tderiv@:(a,v)}
address@hidden a@: a e @: @: @:esimplify@:(a)}
address@hidden a@: a f @: @: 1 @:factor@:(a)}
address@hidden a@: H a f @: @: 1 @:factors@:(a)}
address@hidden a b@: a g @: @: 2 @:pgcd@:(a,b)}
address@hidden a@: a i @:v @: 38 @:integ@:(a,v)}
address@hidden a@: a m @:pats @: 38 @:match@:(a,pats)}
address@hidden a@: I a m @:pats @: 38
@:matchnot@:(a,pats)}
address@hidden data x@: a p @: @: 28 @:polint@:(data,x)}
address@hidden data x@: H a p @: @: 28 @:ratint@:(data,x)}
address@hidden a@: a n @: @: 1 @:nrat@:(a)}
address@hidden a@: a r @:rules @:4,8,38
@:rewrite@:(a,rules,n)}
address@hidden a@: a s @: @: @:simplify@:(a)}
address@hidden a@: a t @:v, n @: 31,39 @:taylor@:(a,v,n)}
address@hidden a@: a v @: @: 7,8
@:calc-alg-evaluate@:}
address@hidden a@: a x @: @: 4,8 @:expand@:(a)}
+
address@hidden
address@hidden data@: a F @:model, vars @: 48
@:fit@:(m,iv,pv,data)}
address@hidden data@: I a F @:model, vars @: 48
@:xfit@:(m,iv,pv,data)}
address@hidden data@: H a F @:model, vars @: 48
@:efit@:(m,iv,pv,data)}
address@hidden a@: a I @:v, l, h @: 38 @:ninteg@:(a,v,l,h)}
address@hidden a b@: a M @:op @: 22 @:mapeq@:(op,a,b)}
address@hidden a b@: I a M @:op @: 22 @:mapeqr@:(op,a,b)}
address@hidden a b@: H a M @:op @: 22 @:mapeqp@:(op,a,b)}
address@hidden a g@: a N @:v @: 38 @:minimize@:(a,v,g)}
address@hidden a g@: H a N @:v @: 38
@:wminimize@:(a,v,g)}
address@hidden a@: a P @:v @: 38 @:roots@:(a,v)}
address@hidden a g@: a R @:v @: 38 @:root@:(a,v,g)}
address@hidden a g@: H a R @:v @: 38 @:wroot@:(a,v,g)}
address@hidden a@: a S @:v @: 38 @:solve@:(a,v)}
address@hidden a@: I a S @:v @: 38 @:finv@:(a,v)}
address@hidden a@: H a S @:v @: 38 @:fsolve@:(a,v)}
address@hidden a@: I H a S @:v @: 38 @:ffinv@:(a,v)}
address@hidden a@: a T @:i, l, h @: 6,38 @:table@:(a,i,l,h)}
address@hidden a g@: a X @:v @: 38 @:maximize@:(a,v,g)}
address@hidden a g@: H a X @:v @: 38
@:wmaximize@:(a,v,g)}
+
address@hidden
address@hidden a b@: b a @: @: 9 @:and@:(a,b,w)}
address@hidden a@: b c @: @: 9 @:clip@:(a,w)}
address@hidden a b@: b d @: @: 9 @:diff@:(a,b,w)}
address@hidden a@: b l @: @: 10 @:lsh@:(a,n,w)}
address@hidden a n@: H b l @: @: 9 @:lsh@:(a,n,w)}
address@hidden a@: b n @: @: 9 @:not@:(a,w)}
address@hidden a b@: b o @: @: 9 @:or@:(a,b,w)}
address@hidden v@: b p @: @: 1 @:vpack@:(v)}
address@hidden a@: b r @: @: 10 @:rsh@:(a,n,w)}
address@hidden a n@: H b r @: @: 9 @:rsh@:(a,n,w)}
address@hidden a@: b t @: @: 10 @:rot@:(a,n,w)}
address@hidden a n@: H b t @: @: 9 @:rot@:(a,n,w)}
address@hidden a@: b u @: @: 1 @:vunpack@:(a)}
address@hidden @: b w @:w @: 9,50 @:calc-word-size@:}
address@hidden a b@: b x @: @: 9 @:xor@:(a,b,w)}
+
address@hidden
address@hidden s l p@: b D @: @: @:ddb@:(c,s,l,p)}
address@hidden r n p@: b F @: @: @:fv@:(r,n,p)}
address@hidden r n p@: I b F @: @: @:fvb@:(r,n,p)}
address@hidden r n p@: H b F @: @: @:fvl@:(r,n,p)}
address@hidden v@: b I @: @: 19 @:irr@:(v)}
address@hidden v@: I b I @: @: 19 @:irrb@:(v)}
address@hidden a@: b L @: @: 10 @:ash@:(a,n,w)}
address@hidden a n@: H b L @: @: 9 @:ash@:(a,n,w)}
address@hidden r n a@: b M @: @: @:pmt@:(r,n,a)}
address@hidden r n a@: I b M @: @: @:pmtb@:(r,n,a)}
address@hidden r n a@: H b M @: @: @:pmtl@:(r,n,a)}
address@hidden r v@: b N @: @: 19 @:npv@:(r,v)}
address@hidden r v@: I b N @: @: 19 @:npvb@:(r,v)}
address@hidden r n p@: b P @: @: @:pv@:(r,n,p)}
address@hidden r n p@: I b P @: @: @:pvb@:(r,n,p)}
address@hidden r n p@: H b P @: @: @:pvl@:(r,n,p)}
address@hidden a@: b R @: @: 10 @:rash@:(a,n,w)}
address@hidden a n@: H b R @: @: 9 @:rash@:(a,n,w)}
address@hidden c s l@: b S @: @: @:sln@:(c,s,l)}
address@hidden n p a@: b T @: @: @:rate@:(n,p,a)}
address@hidden n p a@: I b T @: @: @:rateb@:(n,p,a)}
address@hidden n p a@: H b T @: @: @:ratel@:(n,p,a)}
address@hidden s l p@: b Y @: @: @:syd@:(c,s,l,p)}
+
address@hidden r p a@: b # @: @: @:nper@:(r,p,a)}
address@hidden r p a@: I b # @: @: @:nperb@:(r,p,a)}
address@hidden r p a@: H b # @: @: @:nperl@:(r,p,a)}
address@hidden a b@: b % @: @: @:relch@:(a,b)}
+
address@hidden
address@hidden a@: c c @: @: 5 @:pclean@:(a,p)}
address@hidden a@: c 0-9 @: @: @:pclean@:(a,p)}
address@hidden a@: H c c @: @: 5 @:clean@:(a,p)}
address@hidden a@: H c 0-9 @: @: @:clean@:(a,p)}
address@hidden a@: c d @: @: 1 @:deg@:(a)}
address@hidden a@: c f @: @: 1 @:pfloat@:(a)}
address@hidden a@: H c f @: @: 1 @:float@:(a)}
address@hidden a@: c h @: @: 1 @:hms@:(a)}
address@hidden a@: c p @: @: @:polar@:(a)}
address@hidden a@: I c p @: @: @:rect@:(a)}
address@hidden a@: c r @: @: 1 @:rad@:(a)}
+
address@hidden
address@hidden a@: c F @: @: 5 @:pfrac@:(a,p)}
address@hidden a@: H c F @: @: 5 @:frac@:(a,p)}
+
address@hidden
address@hidden a@: c % @: @: @:percent@:(a*100)}
+
address@hidden
address@hidden @: d . @:char @: 50 @:calc-point-char@:}
address@hidden @: d , @:char @: 50 @:calc-group-char@:}
address@hidden @: d < @: @: 13,50
@:calc-left-justify@:}
address@hidden @: d = @: @: 13,50
@:calc-center-justify@:}
address@hidden @: d > @: @: 13,50
@:calc-right-justify@:}
address@hidden @: d @{ @:label @: 50
@:calc-left-label@:}
address@hidden @: d @} @:label @: 50
@:calc-right-label@:}
address@hidden @: d [ @: @: 4
@:calc-truncate-up@:}
address@hidden @: d ] @: @: 4
@:calc-truncate-down@:}
address@hidden @: d " @: @: 12,50
@:calc-display-strings@:}
address@hidden @: d @key{SPC} @: @:
@:calc-refresh@:}
address@hidden @: d @key{RET} @: @: 1
@:calc-refresh-top@:}
+
address@hidden
address@hidden @: d 0 @: @: 50
@:calc-decimal-radix@:}
address@hidden @: d 2 @: @: 50
@:calc-binary-radix@:}
address@hidden @: d 6 @: @: 50 @:calc-hex-radix@:}
address@hidden @: d 8 @: @: 50
@:calc-octal-radix@:}
+
address@hidden
address@hidden @: d b @: @:12,13,50
@:calc-line-breaking@:}
address@hidden @: d c @: @: 50
@:calc-complex-notation@:}
address@hidden @: d d @:format @: 50
@:calc-date-notation@:}
address@hidden @: d e @: @: 5,50
@:calc-eng-notation@:}
address@hidden @: d f @:num @: 31,50
@:calc-fix-notation@:}
address@hidden @: d g @: @:12,13,50
@:calc-group-digits@:}
address@hidden @: d h @:format @: 50
@:calc-hms-notation@:}
address@hidden @: d i @: @: 50 @:calc-i-notation@:}
address@hidden @: d j @: @: 50 @:calc-j-notation@:}
address@hidden @: d l @: @: 12,50
@:calc-line-numbering@:}
address@hidden @: d n @: @: 5,50
@:calc-normal-notation@:}
address@hidden @: d o @:format @: 50
@:calc-over-notation@:}
address@hidden @: d p @: @: 12,50 @:calc-show-plain@:}
address@hidden @: d r @:radix @: 31,50 @:calc-radix@:}
address@hidden @: d s @: @: 5,50
@:calc-sci-notation@:}
address@hidden @: d t @: @: 27
@:calc-truncate-stack@:}
address@hidden @: d w @: @: 12,13 @:calc-auto-why@:}
address@hidden @: d z @: @: 12,50
@:calc-leading-zeros@:}
+
address@hidden
address@hidden @: d B @: @: 50
@:calc-big-language@:}
address@hidden @: d C @: @: 50 @:calc-c-language@:}
address@hidden @: d E @: @: 50
@:calc-eqn-language@:}
address@hidden @: d F @: @: 50
@:calc-fortran-language@:}
address@hidden @: d M @: @: 50
@:calc-mathematica-language@:}
address@hidden @: d N @: @: 50
@:calc-normal-language@:}
address@hidden @: d O @: @: 50
@:calc-flat-language@:}
address@hidden @: d P @: @: 50
@:calc-pascal-language@:}
address@hidden @: d T @: @: 50
@:calc-tex-language@:}
address@hidden @: d L @: @: 50
@:calc-latex-language@:}
address@hidden @: d U @: @: 50
@:calc-unformatted-language@:}
address@hidden @: d W @: @: 50
@:calc-maple-language@:}
+
address@hidden
address@hidden a@: f [ @: @: 4 @:decr@:(a,n)}
address@hidden a@: f ] @: @: 4 @:incr@:(a,n)}
+
address@hidden
address@hidden a b@: f b @: @: 2 @:beta@:(a,b)}
address@hidden a@: f e @: @: 1 @:erf@:(a)}
address@hidden a@: I f e @: @: 1 @:erfc@:(a)}
address@hidden a@: f g @: @: 1 @:gamma@:(a)}
address@hidden a b@: f h @: @: 2 @:hypot@:(a,b)}
address@hidden a@: f i @: @: 1 @:im@:(a)}
address@hidden n a@: f j @: @: 2 @:besJ@:(n,a)}
address@hidden a b@: f n @: @: 2 @:min@:(a,b)}
address@hidden a@: f r @: @: 1 @:re@:(a)}
address@hidden a@: f s @: @: 1 @:sign@:(a)}
address@hidden a b@: f x @: @: 2 @:max@:(a,b)}
address@hidden n a@: f y @: @: 2 @:besY@:(n,a)}
+
address@hidden
address@hidden a@: f A @: @: 1 @:abssqr@:(a)}
address@hidden x a b@: f B @: @: @:betaI@:(x,a,b)}
address@hidden x a b@: H f B @: @: @:betaB@:(x,a,b)}
address@hidden a@: f E @: @: 1 @:expm1@:(a)}
address@hidden a x@: f G @: @: 2 @:gammaP@:(a,x)}
address@hidden a x@: I f G @: @: 2 @:gammaQ@:(a,x)}
address@hidden a x@: H f G @: @: 2 @:gammag@:(a,x)}
address@hidden a x@: I H f G @: @: 2 @:gammaG@:(a,x)}
address@hidden a b@: f I @: @: 2 @:ilog@:(a,b)}
address@hidden a b@: I f I @: @: 2 @:alog@:(a,b) b^a}
address@hidden a@: f L @: @: 1 @:lnp1@:(a)}
address@hidden a@: f M @: @: 1 @:mant@:(a)}
address@hidden a@: f Q @: @: 1 @:isqrt@:(a)}
address@hidden a@: I f Q @: @: 1 @:sqr@:(a) a^2}
address@hidden a n@: f S @: @: 2 @:scf@:(a,n)}
address@hidden y x@: f T @: @: @:arctan2@:(y,x)}
address@hidden a@: f X @: @: 1 @:xpon@:(a)}
+
address@hidden
address@hidden x y@: g a @: @: 28,40 @:calc-graph-add@:}
address@hidden @: g b @: @: 12
@:calc-graph-border@:}
address@hidden @: g c @: @:
@:calc-graph-clear@:}
address@hidden @: g d @: @: 41
@:calc-graph-delete@:}
address@hidden x y@: g f @: @: 28,40 @:calc-graph-fast@:}
address@hidden @: g g @: @: 12 @:calc-graph-grid@:}
address@hidden @: g h @:title @:
@:calc-graph-header@:}
address@hidden @: g j @: @: 4
@:calc-graph-juggle@:}
address@hidden @: g k @: @: 12 @:calc-graph-key@:}
address@hidden @: g l @: @: 12
@:calc-graph-log-x@:}
address@hidden @: g n @:name @: @:calc-graph-name@:}
address@hidden @: g p @: @: 42 @:calc-graph-plot@:}
address@hidden @: g q @: @: @:calc-graph-quit@:}
address@hidden @: g r @:range @:
@:calc-graph-range-x@:}
address@hidden @: g s @: @: 12,13
@:calc-graph-line-style@:}
address@hidden @: g t @:title @:
@:calc-graph-title-x@:}
address@hidden @: g v @: @:
@:calc-graph-view-commands@:}
address@hidden @: g x @:display @:
@:calc-graph-display@:}
address@hidden @: g z @: @: 12
@:calc-graph-zero-x@:}
+
address@hidden
address@hidden x y z@: g A @: @: 28,40
@:calc-graph-add-3d@:}
address@hidden @: g C @:command @:
@:calc-graph-command@:}
address@hidden @: g D @:device @: 43,44
@:calc-graph-device@:}
address@hidden x y z@: g F @: @: 28,40
@:calc-graph-fast-3d@:}
address@hidden @: g H @: @: 12 @:calc-graph-hide@:}
address@hidden @: g K @: @: @:calc-graph-kill@:}
address@hidden @: g L @: @: 12
@:calc-graph-log-y@:}
address@hidden @: g N @:number @: 43,51
@:calc-graph-num-points@:}
address@hidden @: g O @:filename @: 43,44
@:calc-graph-output@:}
address@hidden @: g P @: @: 42
@:calc-graph-print@:}
address@hidden @: g R @:range @:
@:calc-graph-range-y@:}
address@hidden @: g S @: @: 12,13
@:calc-graph-point-style@:}
address@hidden @: g T @:title @:
@:calc-graph-title-y@:}
address@hidden @: g V @: @:
@:calc-graph-view-trail@:}
address@hidden @: g X @:format @:
@:calc-graph-geometry@:}
address@hidden @: g Z @: @: 12
@:calc-graph-zero-y@:}
+
address@hidden
address@hidden @: g C-l @: @: 12
@:calc-graph-log-z@:}
address@hidden @: g C-r @:range @:
@:calc-graph-range-z@:}
address@hidden @: g C-t @:title @:
@:calc-graph-title-z@:}
+
address@hidden
address@hidden @: h b @: @:
@:calc-describe-bindings@:}
address@hidden @: h c @:key @:
@:calc-describe-key-briefly@:}
address@hidden @: h f @:function @:
@:calc-describe-function@:}
address@hidden @: h h @: @: @:calc-full-help@:}
address@hidden @: h i @: @: @:calc-info@:}
address@hidden @: h k @:key @:
@:calc-describe-key@:}
address@hidden @: h n @: @: @:calc-view-news@:}
address@hidden @: h s @: @:
@:calc-info-summary@:}
address@hidden @: h t @: @: @:calc-tutorial@:}
address@hidden @: h v @:var @:
@:calc-describe-variable@:}
+
address@hidden
address@hidden @: j 1-9 @: @:
@:calc-select-part@:}
address@hidden @: j @key{RET} @: @: 27
@:calc-copy-selection@:}
address@hidden @: j @key{DEL} @: @: 27
@:calc-del-selection@:}
address@hidden @: j ' @:formula @: 27
@:calc-enter-selection@:}
address@hidden @: j ` @:editing @: 27,30
@:calc-edit-selection@:}
address@hidden @: j " @: @: 7,27
@:calc-sel-expand-formula@:}
+
address@hidden
address@hidden @: j + @:formula @: 27
@:calc-sel-add-both-sides@:}
address@hidden @: j - @:formula @: 27
@:calc-sel-sub-both-sides@:}
address@hidden @: j * @:formula @: 27
@:calc-sel-mul-both-sides@:}
address@hidden @: j / @:formula @: 27
@:calc-sel-div-both-sides@:}
address@hidden @: j & @: @: 27 @:calc-sel-invert@:}
+
address@hidden
address@hidden @: j a @: @: 27
@:calc-select-additional@:}
address@hidden @: j b @: @: 12
@:calc-break-selections@:}
address@hidden @: j c @: @:
@:calc-clear-selections@:}
address@hidden @: j d @: @: 12,50
@:calc-show-selections@:}
address@hidden @: j e @: @: 12
@:calc-enable-selections@:}
address@hidden @: j l @: @: 4,27
@:calc-select-less@:}
address@hidden @: j m @: @: 4,27
@:calc-select-more@:}
address@hidden @: j n @: @: 4
@:calc-select-next@:}
address@hidden @: j o @: @: 4,27
@:calc-select-once@:}
address@hidden @: j p @: @: 4
@:calc-select-previous@:}
address@hidden @: j r @:rules @:4,8,27
@:calc-rewrite-selection@:}
address@hidden @: j s @: @: 4,27
@:calc-select-here@:}
address@hidden @: j u @: @: 27 @:calc-unselect@:}
address@hidden @: j v @: @: 7,27
@:calc-sel-evaluate@:}
+
address@hidden
address@hidden @: j C @: @: 27
@:calc-sel-commute@:}
address@hidden @: j D @: @: 4,27
@:calc-sel-distribute@:}
address@hidden @: j E @: @: 27
@:calc-sel-jump-equals@:}
address@hidden @: j I @: @: 27
@:calc-sel-isolate@:}
address@hidden @: H j I @: @: 27
@:calc-sel-isolate@: (full)}
address@hidden @: j L @: @: 4,27
@:calc-commute-left@:}
address@hidden @: j M @: @: 27 @:calc-sel-merge@:}
address@hidden @: j N @: @: 27 @:calc-sel-negate@:}
address@hidden @: j O @: @: 4,27
@:calc-select-once-maybe@:}
address@hidden @: j R @: @: 4,27
@:calc-commute-right@:}
address@hidden @: j S @: @: 4,27
@:calc-select-here-maybe@:}
address@hidden @: j U @: @: 27 @:calc-sel-unpack@:}
+
address@hidden
address@hidden @: k a @: @:
@:calc-random-again@:}
address@hidden n@: k b @: @: 1 @:bern@:(n)}
address@hidden n x@: H k b @: @: 2 @:bern@:(n,x)}
address@hidden n m@: k c @: @: 2 @:choose@:(n,m)}
address@hidden n m@: H k c @: @: 2 @:perm@:(n,m)}
address@hidden n@: k d @: @: 1 @:dfact@:(n) n!!}
address@hidden n@: k e @: @: 1 @:euler@:(n)}
address@hidden n x@: H k e @: @: 2 @:euler@:(n,x)}
address@hidden n@: k f @: @: 4 @:prfac@:(n)}
address@hidden n m@: k g @: @: 2 @:gcd@:(n,m)}
address@hidden m n@: k h @: @: 14 @:shuffle@:(n,m)}
address@hidden n m@: k l @: @: 2 @:lcm@:(n,m)}
address@hidden n@: k m @: @: 1 @:moebius@:(n)}
address@hidden n@: k n @: @: 4 @:nextprime@:(n)}
address@hidden n@: I k n @: @: 4 @:prevprime@:(n)}
address@hidden n@: k p @: @: 4,28 @:calc-prime-test@:}
address@hidden m@: k r @: @: 14 @:random@:(m)}
address@hidden n m@: k s @: @: 2 @:stir1@:(n,m)}
address@hidden n m@: H k s @: @: 2 @:stir2@:(n,m)}
address@hidden n@: k t @: @: 1 @:totient@:(n)}
+
address@hidden
address@hidden n p x@: k B @: @: @:utpb@:(x,n,p)}
address@hidden n p x@: I k B @: @: @:ltpb@:(x,n,p)}
address@hidden v x@: k C @: @: @:utpc@:(x,v)}
address@hidden v x@: I k C @: @: @:ltpc@:(x,v)}
address@hidden n m@: k E @: @: @:egcd@:(n,m)}
address@hidden v2 x@: k F @: @: @:utpf@:(x,v1,v2)}
address@hidden v2 x@: I k F @: @: @:ltpf@:(x,v1,v2)}
address@hidden m s x@: k N @: @: @:utpn@:(x,m,s)}
address@hidden m s x@: I k N @: @: @:ltpn@:(x,m,s)}
address@hidden m x@: k P @: @: @:utpp@:(x,m)}
address@hidden m x@: I k P @: @: @:ltpp@:(x,m)}
address@hidden v x@: k T @: @: @:utpt@:(x,v)}
address@hidden v x@: I k T @: @: @:ltpt@:(x,v)}
+
address@hidden
address@hidden @: m a @: @: 12,13
@:calc-algebraic-mode@:}
address@hidden @: m d @: @:
@:calc-degrees-mode@:}
address@hidden @: m e @: @:
@:calc-embedded-preserve-modes@:}
address@hidden @: m f @: @: 12 @:calc-frac-mode@:}
address@hidden @: m g @: @: 52 @:calc-get-modes@:}
address@hidden @: m h @: @: @:calc-hms-mode@:}
address@hidden @: m i @: @: 12,13
@:calc-infinite-mode@:}
address@hidden @: m m @: @: @:calc-save-modes@:}
address@hidden @: m p @: @: 12 @:calc-polar-mode@:}
address@hidden @: m r @: @:
@:calc-radians-mode@:}
address@hidden @: m s @: @: 12
@:calc-symbolic-mode@:}
address@hidden @: m t @: @: 12
@:calc-total-algebraic-mode@:}
address@hidden @: m v @: @: 12,13
@:calc-matrix-mode@:}
address@hidden @: m w @: @: 13 @:calc-working@:}
address@hidden @: m x @: @:
@:calc-always-load-extensions@:}
+
address@hidden
address@hidden @: m A @: @: 12
@:calc-alg-simplify-mode@:}
address@hidden @: m B @: @: 12
@:calc-bin-simplify-mode@:}
address@hidden @: m C @: @: 12
@:calc-auto-recompute@:}
address@hidden @: m D @: @:
@:calc-default-simplify-mode@:}
address@hidden @: m E @: @: 12
@:calc-ext-simplify-mode@:}
address@hidden @: m F @:filename @: 13
@:calc-settings-file-name@:}
address@hidden @: m N @: @: 12
@:calc-num-simplify-mode@:}
address@hidden @: m O @: @: 12
@:calc-no-simplify-mode@:}
address@hidden @: m R @: @: 12,13
@:calc-mode-record-mode@:}
address@hidden @: m S @: @: 12
@:calc-shift-prefix@:}
address@hidden @: m U @: @: 12
@:calc-units-simplify-mode@:}
+
address@hidden
address@hidden @: s c @:var1, var2 @: 29
@:calc-copy-variable@:}
address@hidden @: s d @:var, decl @:
@:calc-declare-variable@:}
address@hidden @: s e @:var, editing @: 29,30
@:calc-edit-variable@:}
address@hidden @: s i @:buffer @:
@:calc-insert-variables@:}
address@hidden @: s k @:const, var @: 29
@:calc-copy-special-constant@:}
address@hidden a b@: s l @:var @: 29 @:@:a (letting
var=b)}
address@hidden a ...@: s m @:op, var @: 22,29 @:calc-store-map@:}
address@hidden @: s n @:var @: 29,47 @:calc-store-neg@:
(v/-1)}
address@hidden @: s p @:var @: 29
@:calc-permanent-variable@:}
address@hidden @: s r @:var @: 29 @:@:v (recalled
value)}
address@hidden @: r 0-9 @: @:
@:calc-recall-quick@:}
address@hidden a@: s s @:var @: 28,29 @:calc-store@:}
address@hidden a@: s 0-9 @: @:
@:calc-store-quick@:}
address@hidden a@: s t @:var @: 29 @:calc-store-into@:}
address@hidden a@: t 0-9 @: @:
@:calc-store-into-quick@:}
address@hidden @: s u @:var @: 29 @:calc-unstore@:}
address@hidden a@: s x @:var @: 29
@:calc-store-exchange@:}
+
address@hidden
address@hidden @: s A @:editing @: 30
@:calc-edit-AlgSimpRules@:}
address@hidden @: s D @:editing @: 30 @:calc-edit-Decls@:}
address@hidden @: s E @:editing @: 30
@:calc-edit-EvalRules@:}
address@hidden @: s F @:editing @: 30
@:calc-edit-FitRules@:}
address@hidden @: s G @:editing @: 30
@:calc-edit-GenCount@:}
address@hidden @: s H @:editing @: 30
@:calc-edit-Holidays@:}
address@hidden @: s I @:editing @: 30
@:calc-edit-IntegLimit@:}
address@hidden @: s L @:editing @: 30
@:calc-edit-LineStyles@:}
address@hidden @: s P @:editing @: 30
@:calc-edit-PointStyles@:}
address@hidden @: s R @:editing @: 30
@:calc-edit-PlotRejects@:}
address@hidden @: s T @:editing @: 30
@:calc-edit-TimeZone@:}
address@hidden @: s U @:editing @: 30 @:calc-edit-Units@:}
address@hidden @: s X @:editing @: 30
@:calc-edit-ExtSimpRules@:}
+
address@hidden
address@hidden a@: s + @:var @: 29,47 @:calc-store-plus@:
(v+a)}
address@hidden a@: s - @:var @: 29,47
@:calc-store-minus@: (v-a)}
address@hidden a@: s * @:var @: 29,47
@:calc-store-times@: (v*a)}
address@hidden a@: s / @:var @: 29,47 @:calc-store-div@:
(v/a)}
address@hidden a@: s ^ @:var @: 29,47
@:calc-store-power@: (v^a)}
address@hidden a@: s | @:var @: 29,47
@:calc-store-concat@: (v|a)}
address@hidden @: s & @:var @: 29,47 @:calc-store-inv@:
(v^-1)}
address@hidden @: s [ @:var @: 29,47 @:calc-store-decr@:
(v-1)}
address@hidden @: s ] @:var @: 29,47 @:calc-store-incr@:
(v-(-1))}
address@hidden a b@: s : @: @: 2 @:assign@:(a,b) a
@tfn{:=} b}
address@hidden a@: s = @: @: 1 @:evalto@:(a,b) a
@tfn{=>}}
+
address@hidden
address@hidden @: t [ @: @: 4
@:calc-trail-first@:}
address@hidden @: t ] @: @: 4 @:calc-trail-last@:}
address@hidden @: t < @: @: 4
@:calc-trail-scroll-left@:}
address@hidden @: t > @: @: 4
@:calc-trail-scroll-right@:}
address@hidden @: t . @: @: 12
@:calc-full-trail-vectors@:}
+
address@hidden
address@hidden @: t b @: @: 4
@:calc-trail-backward@:}
address@hidden @: t d @: @: 12,50
@:calc-trail-display@:}
address@hidden @: t f @: @: 4
@:calc-trail-forward@:}
address@hidden @: t h @: @: @:calc-trail-here@:}
address@hidden @: t i @: @: @:calc-trail-in@:}
address@hidden @: t k @: @: 4 @:calc-trail-kill@:}
address@hidden @: t m @:string @:
@:calc-trail-marker@:}
address@hidden @: t n @: @: 4 @:calc-trail-next@:}
address@hidden @: t o @: @: @:calc-trail-out@:}
address@hidden @: t p @: @: 4
@:calc-trail-previous@:}
address@hidden @: t r @:string @:
@:calc-trail-isearch-backward@:}
address@hidden @: t s @:string @:
@:calc-trail-isearch-forward@:}
address@hidden @: t y @: @: 4 @:calc-trail-yank@:}
+
address@hidden
address@hidden d@: t C @:oz, nz @: @:tzconv@:(d,oz,nz)}
address@hidden oz nz@: t C @:$ @: @:tzconv@:(d,oz,nz)}
address@hidden d@: t D @: @: 15 @:date@:(d)}
address@hidden d@: t I @: @: 4 @:incmonth@:(d,n)}
address@hidden d@: t J @: @: 16 @:julian@:(d,z)}
address@hidden d@: t M @: @: 17 @:newmonth@:(d,n)}
address@hidden @: t N @: @: 16 @:now@:(z)}
address@hidden d@: t P @:1 @: 31 @:year@:(d)}
address@hidden d@: t P @:2 @: 31 @:month@:(d)}
address@hidden d@: t P @:3 @: 31 @:day@:(d)}
address@hidden d@: t P @:4 @: 31 @:hour@:(d)}
address@hidden d@: t P @:5 @: 31 @:minute@:(d)}
address@hidden d@: t P @:6 @: 31 @:second@:(d)}
address@hidden d@: t P @:7 @: 31 @:weekday@:(d)}
address@hidden d@: t P @:8 @: 31 @:yearday@:(d)}
address@hidden d@: t P @:9 @: 31 @:time@:(d)}
address@hidden d@: t U @: @: 16 @:unixtime@:(d,z)}
address@hidden d@: t W @: @: 17 @:newweek@:(d,w)}
address@hidden d@: t Y @: @: 17 @:newyear@:(d,n)}
+
address@hidden
address@hidden a b@: t + @: @: 2 @:badd@:(a,b)}
address@hidden a b@: t - @: @: 2 @:bsub@:(a,b)}
+
address@hidden
address@hidden @: u a @: @: 12
@:calc-autorange-units@:}
address@hidden a@: u b @: @: @:calc-base-units@:}
address@hidden a@: u c @:units @: 18
@:calc-convert-units@:}
address@hidden defn@: u d @:unit, descr @:
@:calc-define-unit@:}
address@hidden @: u e @: @:
@:calc-explain-units@:}
address@hidden @: u g @:unit @:
@:calc-get-unit-definition@:}
address@hidden @: u p @: @:
@:calc-permanent-units@:}
address@hidden a@: u r @: @:
@:calc-remove-units@:}
address@hidden a@: u s @: @: @:usimplify@:(a)}
address@hidden a@: u t @:units @: 18
@:calc-convert-temperature@:}
address@hidden @: u u @:unit @:
@:calc-undefine-unit@:}
address@hidden @: u v @: @:
@:calc-enter-units-table@:}
address@hidden a@: u x @: @:
@:calc-extract-units@:}
address@hidden a@: u 0-9 @: @:
@:calc-quick-units@:}
+
address@hidden
address@hidden v1 v2@: u C @: @: 20 @:vcov@:(v1,v2)}
address@hidden v1 v2@: I u C @: @: 20 @:vpcov@:(v1,v2)}
address@hidden v1 v2@: H u C @: @: 20 @:vcorr@:(v1,v2)}
address@hidden v@: u G @: @: 19 @:vgmean@:(v)}
address@hidden a b@: H u G @: @: 2 @:agmean@:(a,b)}
address@hidden v@: u M @: @: 19 @:vmean@:(v)}
address@hidden v@: I u M @: @: 19 @:vmeane@:(v)}
address@hidden v@: H u M @: @: 19 @:vmedian@:(v)}
address@hidden v@: I H u M @: @: 19 @:vhmean@:(v)}
address@hidden v@: u N @: @: 19 @:vmin@:(v)}
address@hidden v@: u S @: @: 19 @:vsdev@:(v)}
address@hidden v@: I u S @: @: 19 @:vpsdev@:(v)}
address@hidden v@: H u S @: @: 19 @:vvar@:(v)}
address@hidden v@: I H u S @: @: 19 @:vpvar@:(v)}
address@hidden @: u V @: @:
@:calc-view-units-table@:}
address@hidden v@: u X @: @: 19 @:vmax@:(v)}
+
address@hidden
address@hidden v@: u + @: @: 19 @:vsum@:(v)}
address@hidden v@: u * @: @: 19 @:vprod@:(v)}
address@hidden v@: u # @: @: 19 @:vcount@:(v)}
+
address@hidden
address@hidden @: V ( @: @: 50
@:calc-vector-parens@:}
address@hidden @: V @{ @: @: 50
@:calc-vector-braces@:}
address@hidden @: V [ @: @: 50
@:calc-vector-brackets@:}
address@hidden @: V ] @:ROCP @: 50
@:calc-matrix-brackets@:}
address@hidden @: V , @: @: 50
@:calc-vector-commas@:}
address@hidden @: V < @: @: 50
@:calc-matrix-left-justify@:}
address@hidden @: V = @: @: 50
@:calc-matrix-center-justify@:}
address@hidden @: V > @: @: 50
@:calc-matrix-right-justify@:}
address@hidden @: V / @: @: 12,50
@:calc-break-vectors@:}
address@hidden @: V . @: @: 12,50
@:calc-full-vectors@:}
+
address@hidden
address@hidden s t@: V ^ @: @: 2 @:vint@:(s,t)}
address@hidden s t@: V - @: @: 2 @:vdiff@:(s,t)}
address@hidden s@: V ~ @: @: 1 @:vcompl@:(s)}
address@hidden s@: V # @: @: 1 @:vcard@:(s)}
address@hidden s@: V : @: @: 1 @:vspan@:(s)}
address@hidden s@: V + @: @: 1 @:rdup@:(s)}
+
address@hidden
address@hidden m@: V & @: @: 1 @:inv@:(m) 1/m}
+
address@hidden
address@hidden v@: v a @:n @: @:arrange@:(v,n)}
address@hidden a@: v b @:n @: @:cvec@:(a,n)}
address@hidden v@: v c @:n >0 @: 21,31 @:mcol@:(v,n)}
address@hidden v@: v c @:n <0 @: 31 @:mrcol@:(v,-n)}
address@hidden m@: v c @:0 @: 31 @:getdiag@:(m)}
address@hidden v@: v d @: @: 25 @:diag@:(v,n)}
address@hidden v m@: v e @: @: 2 @:vexp@:(v,m)}
address@hidden v m f@: H v e @: @: 2 @:vexp@:(v,m,f)}
address@hidden v a@: v f @: @: 26 @:find@:(v,a,n)}
address@hidden v@: v h @: @: 1 @:head@:(v)}
address@hidden v@: I v h @: @: 1 @:tail@:(v)}
address@hidden v@: H v h @: @: 1 @:rhead@:(v)}
address@hidden v@: I H v h @: @: 1 @:rtail@:(v)}
address@hidden @: v i @:n @: 31 @:idn@:(1,n)}
address@hidden @: v i @:0 @: 31 @:idn@:(1)}
address@hidden h t@: v k @: @: 2 @:cons@:(h,t)}
address@hidden h t@: H v k @: @: 2 @:rcons@:(h,t)}
address@hidden v@: v l @: @: 1 @:vlen@:(v)}
address@hidden v@: H v l @: @: 1 @:mdims@:(v)}
address@hidden v m@: v m @: @: 2 @:vmask@:(v,m)}
address@hidden v@: v n @: @: 1 @:rnorm@:(v)}
address@hidden a b c@: v p @: @: 24 @:calc-pack@:}
address@hidden v@: v r @:n >0 @: 21,31 @:mrow@:(v,n)}
address@hidden v@: v r @:n <0 @: 31 @:mrrow@:(v,-n)}
address@hidden m@: v r @:0 @: 31 @:getdiag@:(m)}
address@hidden v i j@: v s @: @: @:subvec@:(v,i,j)}
address@hidden v i j@: I v s @: @: @:rsubvec@:(v,i,j)}
address@hidden m@: v t @: @: 1 @:trn@:(m)}
address@hidden v@: v u @: @: 24 @:calc-unpack@:}
address@hidden v@: v v @: @: 1 @:rev@:(v)}
address@hidden @: v x @:n @: 31 @:index@:(n)}
address@hidden n s i@: C-u v x @: @: @:index@:(n,s,i)}
+
address@hidden
address@hidden v@: V A @:op @: 22 @:apply@:(op,v)}
address@hidden v1 v2@: V C @: @: 2 @:cross@:(v1,v2)}
address@hidden m@: V D @: @: 1 @:det@:(m)}
address@hidden s@: V E @: @: 1 @:venum@:(s)}
address@hidden s@: V F @: @: 1 @:vfloor@:(s)}
address@hidden v@: V G @: @: @:grade@:(v)}
address@hidden v@: I V G @: @: @:rgrade@:(v)}
address@hidden v@: V H @:n @: 31 @:histogram@:(v,n)}
address@hidden v w@: H V H @:n @: 31
@:histogram@:(v,w,n)}
address@hidden v1 v2@: V I @:mop aop @: 22
@:inner@:(mop,aop,v1,v2)}
address@hidden m@: V J @: @: 1 @:ctrn@:(m)}
address@hidden m@: V L @: @: 1 @:lud@:(m)}
address@hidden v@: V M @:op @: 22,23 @:map@:(op,v)}
address@hidden v@: V N @: @: 1 @:cnorm@:(v)}
address@hidden v1 v2@: V O @:op @: 22 @:outer@:(op,v1,v2)}
address@hidden v@: V R @:op @: 22,23 @:reduce@:(op,v)}
address@hidden v@: I V R @:op @: 22,23 @:rreduce@:(op,v)}
address@hidden a n@: H V R @:op @: 22 @:nest@:(op,a,n)}
address@hidden a@: I H V R @:op @: 22 @:fixp@:(op,a)}
address@hidden v@: V S @: @: @:sort@:(v)}
address@hidden v@: I V S @: @: @:rsort@:(v)}
address@hidden m@: V T @: @: 1 @:tr@:(m)}
address@hidden v@: V U @:op @: 22 @:accum@:(op,v)}
address@hidden v@: I V U @:op @: 22 @:raccum@:(op,v)}
address@hidden a n@: H V U @:op @: 22 @:anest@:(op,a,n)}
address@hidden a@: I H V U @:op @: 22 @:afixp@:(op,a)}
address@hidden s t@: V V @: @: 2 @:vunion@:(s,t)}
address@hidden s t@: V X @: @: 2 @:vxor@:(s,t)}
+
address@hidden
address@hidden @: Y @: @: @:@:user commands}
+
address@hidden
address@hidden @: z @: @: @:@:user commands}
+
address@hidden
address@hidden c@: Z [ @: @: 45 @:calc-kbd-if@:}
address@hidden c@: Z | @: @: 45
@:calc-kbd-else-if@:}
address@hidden @: Z : @: @: @:calc-kbd-else@:}
address@hidden @: Z ] @: @: @:calc-kbd-end-if@:}
+
address@hidden
address@hidden @: Z @{ @: @: 4 @:calc-kbd-loop@:}
address@hidden c@: Z / @: @: 45 @:calc-kbd-break@:}
address@hidden @: Z @} @: @:
@:calc-kbd-end-loop@:}
address@hidden n@: Z < @: @: @:calc-kbd-repeat@:}
address@hidden @: Z > @: @:
@:calc-kbd-end-repeat@:}
address@hidden n m@: Z ( @: @: @:calc-kbd-for@:}
address@hidden s@: Z ) @: @:
@:calc-kbd-end-for@:}
+
address@hidden
address@hidden @: Z C-g @: @: @:@:cancel if/loop
command}
+
address@hidden
address@hidden @: Z ` @: @: @:calc-kbd-push@:}
address@hidden @: Z ' @: @: @:calc-kbd-pop@:}
address@hidden @: Z # @: @: @:calc-kbd-query@:}
+
address@hidden
address@hidden comp@: Z C @:func, args @: 50
@:calc-user-define-composition@:}
address@hidden @: Z D @:key, command @:
@:calc-user-define@:}
address@hidden @: Z E @:key, editing @: 30
@:calc-user-define-edit@:}
address@hidden defn@: Z F @:k, c, f, a, n@: 28
@:calc-user-define-formula@:}
address@hidden @: Z G @:key @:
@:calc-get-user-defn@:}
address@hidden @: Z I @: @:
@:calc-user-define-invocation@:}
address@hidden @: Z K @:key, command @:
@:calc-user-define-kbd-macro@:}
address@hidden @: Z P @:key @:
@:calc-user-define-permanent@:}
address@hidden @: Z S @: @: 30
@:calc-edit-user-syntax@:}
address@hidden @: Z T @: @: 12 @:calc-timing@:}
address@hidden @: Z U @:key @:
@:calc-user-undefine@:}
+
address@hidden format
+
address@hidden
+NOTES
+
address@hidden
address@hidden 1
address@hidden
+Positive prefix arguments apply to @expr{n} stack entries.
+Negative prefix arguments apply to the @expr{-n}th stack entry.
+A prefix of zero applies to the entire stack. (For @key{LFD} and
address@hidden@key{DEL}}, the meaning of the sign is reversed.)
+
address@hidden 2
address@hidden
+Positive prefix arguments apply to @expr{n} stack entries.
+Negative prefix arguments apply to the top stack entry
+and the next @expr{-n} stack entries.
+
address@hidden 3
address@hidden
+Positive prefix arguments rotate top @expr{n} stack entries by one.
+Negative prefix arguments rotate the entire stack by @expr{-n}.
+A prefix of zero reverses the entire stack.
+
address@hidden 4
address@hidden
+Prefix argument specifies a repeat count or distance.
+
address@hidden 5
address@hidden
+Positive prefix arguments specify a precision @expr{p}.
+Negative prefix arguments reduce the current precision by @expr{-p}.
+
address@hidden 6
address@hidden
+A prefix argument is interpreted as an additional step-size parameter.
+A plain @kbd{C-u} prefix means to prompt for the step size.
+
address@hidden 7
address@hidden
+A prefix argument specifies simplification level and depth.
+1=Default, 2=like @kbd{a s}, 3=like @kbd{a e}.
+
address@hidden 8
address@hidden
+A negative prefix operates only on the top level of the input formula.
+
address@hidden 9
address@hidden
+Positive prefix arguments specify a word size of @expr{w} bits, unsigned.
+Negative prefix arguments specify a word size of @expr{w} bits, signed.
+
address@hidden 10
address@hidden
+Prefix arguments specify the shift amount @expr{n}. The @expr{w} argument
+cannot be specified in the keyboard version of this command.
+
address@hidden 11
address@hidden
+From the keyboard, @expr{d} is omitted and defaults to zero.
+
address@hidden 12
address@hidden
+Mode is toggled; a positive prefix always sets the mode, and a negative
+prefix always clears the mode.
+
address@hidden 13
address@hidden
+Some prefix argument values provide special variations of the mode.
+
address@hidden 14
address@hidden
+A prefix argument, if any, is used for @expr{m} instead of taking
address@hidden from the stack. @expr{M} may take any of these values:
address@hidden
address@hidden@tableindent10pt
address@hidden iftex
address@hidden @asis
address@hidden Integer
+Random integer in the interval @expr{[0 .. m)}.
address@hidden Float
+Random floating-point number in the interval @expr{[0 .. m)}.
address@hidden 0.0
+Gaussian with mean 1 and standard deviation 0.
address@hidden Error form
+Gaussian with specified mean and standard deviation.
address@hidden Interval
+Random integer or floating-point number in that interval.
address@hidden Vector
+Random element from the vector.
address@hidden table
address@hidden
+}
address@hidden iftex
+
address@hidden 15
address@hidden
+A prefix argument from 1 to 6 specifies number of date components
+to remove from the stack. @xref{Date Conversions}.
+
address@hidden 16
address@hidden
+A prefix argument specifies a time zone; @kbd{C-u} says to take the
+time zone number or name from the top of the stack. @xref{Time Zones}.
+
address@hidden 17
address@hidden
+A prefix argument specifies a day number (0-6, 0-31, or 0-366).
+
address@hidden 18
address@hidden
+If the input has no units, you will be prompted for both the old and
+the new units.
+
address@hidden 19
address@hidden
+With a prefix argument, collect that many stack entries to form the
+input data set. Each entry may be a single value or a vector of values.
+
address@hidden 20
address@hidden
+With a prefix argument of 1, take a single
address@hidden @address@hidden
address@hidden @address@hidden
+matrix from the stack instead of two separate data vectors.
+
address@hidden 21
address@hidden
+The row or column number @expr{n} may be given as a numeric prefix
+argument instead. A plain @kbd{C-u} prefix says to take @expr{n}
+from the top of the stack. If @expr{n} is a vector or interval,
+a subvector/submatrix of the input is created.
+
address@hidden 22
address@hidden
+The @expr{op} prompt can be answered with the key sequence for the
+desired function, or with @kbd{x} or @kbd{z} followed by a function name,
+or with @kbd{$} to take a formula from the top of the stack, or with
address@hidden'} and a typed formula. In the last two cases, the formula may
+be a nameless function like @samp{<#1+#2>} or @samp{<x, y : x+y>}, or it
+may include @kbd{$}, @kbd{$$}, etc. (where @kbd{$} will correspond to the
+last argument of the created function), or otherwise you will be
+prompted for an argument list. The number of vectors popped from the
+stack by @kbd{V M} depends on the number of arguments of the function.
+
address@hidden 23
address@hidden
+One of the mapping direction keys @kbd{_} (horizontal, i.e., map
+by rows or reduce across), @kbd{:} (vertical, i.e., map by columns or
+reduce down), or @kbd{=} (map or reduce by rows) may be used before
+entering @expr{op}; these modify the function name by adding the letter
address@hidden for ``rows,'' @code{c} for ``columns,'' @code{a} for ``across,''
+or @code{d} for ``down.''
+
address@hidden 24
address@hidden
+The prefix argument specifies a packing mode. A nonnegative mode
+is the number of items (for @kbd{v p}) or the number of levels
+(for @kbd{v u}). A negative mode is as described below. With no
+prefix argument, the mode is taken from the top of the stack and
+may be an integer or a vector of integers.
address@hidden
address@hidden@tableindent-20pt
address@hidden iftex
address@hidden @cite
address@hidden -1
+(@var{2}) Rectangular complex number.
address@hidden -2
+(@var{2}) Polar complex number.
address@hidden -3
+(@var{3}) HMS form.
address@hidden -4
+(@var{2}) Error form.
address@hidden -5
+(@var{2}) Modulo form.
address@hidden -6
+(@var{2}) Closed interval.
address@hidden -7
+(@var{2}) Closed .. open interval.
address@hidden -8
+(@var{2}) Open .. closed interval.
address@hidden -9
+(@var{2}) Open interval.
address@hidden -10
+(@var{2}) Fraction.
address@hidden -11
+(@var{2}) Float with integer mantissa.
address@hidden -12
+(@var{2}) Float with mantissa in @expr{[1 .. 10)}.
address@hidden -13
+(@var{1}) Date form (using date numbers).
address@hidden -14
+(@var{3}) Date form (using year, month, day).
address@hidden -15
+(@var{6}) Date form (using year, month, day, hour, minute, second).
address@hidden table
address@hidden
+}
address@hidden iftex
+
address@hidden 25
address@hidden
+A prefix argument specifies the size @expr{n} of the matrix. With no
+prefix argument, @expr{n} is omitted and the size is inferred from
+the input vector.
+
address@hidden 26
address@hidden
+The prefix argument specifies the starting position @expr{n} (default 1).
+
address@hidden 27
address@hidden
+Cursor position within stack buffer affects this command.
+
address@hidden 28
address@hidden
+Arguments are not actually removed from the stack by this command.
+
address@hidden 29
address@hidden
+Variable name may be a single digit or a full name.
+
address@hidden 30
address@hidden
+Editing occurs in a separate buffer. Press @kbd{C-c C-c} (or
address@hidden, or in some cases @key{RET}) to finish the edit, or kill the
+buffer with @kbd{C-x k} to cancel the edit. The @key{LFD} key prevents
evaluation
+of the result of the edit.
+
address@hidden 31
address@hidden
+The number prompted for can also be provided as a prefix argument.
+
address@hidden 32
address@hidden
+Press this key a second time to cancel the prefix.
+
address@hidden 33
address@hidden
+With a negative prefix, deactivate all formulas. With a positive
+prefix, deactivate and then reactivate from scratch.
+
address@hidden 34
address@hidden
+Default is to scan for nearest formula delimiter symbols. With a
+prefix of zero, formula is delimited by mark and point. With a
+non-zero prefix, formula is delimited by scanning forward or
+backward by that many lines.
+
address@hidden 35
address@hidden
+Parse the region between point and mark as a vector. A nonzero prefix
+parses @var{n} lines before or after point as a vector. A zero prefix
+parses the current line as a vector. A @kbd{C-u} prefix parses the
+region between point and mark as a single formula.
+
address@hidden 36
address@hidden
+Parse the rectangle defined by point and mark as a matrix. A positive
+prefix @var{n} divides the rectangle into columns of width @var{n}.
+A zero or @kbd{C-u} prefix parses each line as one formula. A negative
+prefix suppresses special treatment of bracketed portions of a line.
+
address@hidden 37
address@hidden
+A numeric prefix causes the current language mode to be ignored.
+
address@hidden 38
address@hidden
+Responding to a prompt with a blank line answers that and all
+later prompts by popping additional stack entries.
+
address@hidden 39
address@hidden
+Answer for @expr{v} may also be of the form @expr{v = v_0} or
address@hidden - v_0}.
+
address@hidden 40
address@hidden
+With a positive prefix argument, stack contains many @expr{y}'s and one
+common @expr{x}. With a zero prefix, stack contains a vector of
address@hidden and a common @expr{x}. With a negative prefix, stack
+contains many @expr{[x,y]} vectors. (For 3D plots, substitute
address@hidden for @expr{y} and @expr{x,y} for @expr{x}.)
+
address@hidden 41
address@hidden
+With any prefix argument, all curves in the graph are deleted.
+
address@hidden 42
address@hidden
+With a positive prefix, refines an existing plot with more data points.
+With a negative prefix, forces recomputation of the plot data.
+
address@hidden 43
address@hidden
+With any prefix argument, set the default value instead of the
+value for this graph.
+
address@hidden 44
address@hidden
+With a negative prefix argument, set the value for the printer.
+
address@hidden 45
address@hidden
+Condition is considered ``true'' if it is a nonzero real or complex
+number, or a formula whose value is known to be nonzero; it is ``false''
+otherwise.
+
address@hidden 46
address@hidden
+Several formulas separated by commas are pushed as multiple stack
+entries. Trailing @kbd{)}, @kbd{]}, @address@hidden, @kbd{>}, and @kbd{"}
+delimiters may be omitted. The notation @kbd{$$$} refers to the value
+in stack level three, and causes the formula to replace the top three
+stack levels. The notation @kbd{$3} refers to stack level three without
+causing that value to be removed from the stack. Use @key{LFD} in place
+of @key{RET} to prevent evaluation; use @kbd{M-=} in place of @key{RET}
+to evaluate variables.
+
address@hidden 47
address@hidden
+The variable is replaced by the formula shown on the right. The
+Inverse flag reverses the order of the operands, e.g., @kbd{I s - x}
+assigns
address@hidden @math{x \coloneq a-x}.
address@hidden @expr{x := a-x}.
+
address@hidden 48
address@hidden
+Press @kbd{?} repeatedly to see how to choose a model. Answer the
+variables prompt with @expr{iv} or @expr{iv;pv} to specify
+independent and parameter variables. A positive prefix argument
+takes @address@hidden vectors from the stack; a zero prefix takes a matrix
+and a vector from the stack.
+
address@hidden 49
address@hidden
+With a plain @kbd{C-u} prefix, replace the current region of the
+destination buffer with the yanked text instead of inserting.
+
address@hidden 50
address@hidden
+All stack entries are reformatted; the @kbd{H} prefix inhibits this.
+The @kbd{I} prefix sets the mode temporarily, redraws the top stack
+entry, then restores the original setting of the mode.
+
address@hidden 51
address@hidden
+A negative prefix sets the default 3D resolution instead of the
+default 2D resolution.
+
address@hidden 52
address@hidden
+This grabs a vector of the form address@hidden, @var{wsize}, @var{ssize},
address@hidden, @var{flfmt}, @var{ang}, @var{frac}, @var{symb}, @var{polar},
address@hidden, @var{simp}, @var{inf}]. A prefix argument from 1 to 12
+grabs the @var{n}th mode value only.
address@hidden enumerate
+
address@hidden
+(Space is provided below for you to keep your own written notes.)
address@hidden
address@hidden
address@hidden iftex
+
+
address@hidden [end-summary]
+
address@hidden Key Index, Command Index, Summary, Top
address@hidden Index of Key Sequences
+
address@hidden ky
+
address@hidden Command Index, Function Index, Key Index, Top
address@hidden Index of Calculator Commands
+
+Since all Calculator commands begin with the prefix @samp{calc-}, the
address@hidden key has been provided as a variant of @kbd{M-x} which
automatically
+types @samp{calc-} for you. Thus, @kbd{x last-args} is short for
address@hidden calc-last-args}.
+
address@hidden pg
+
address@hidden Function Index, Concept Index, Command Index, Top
address@hidden Index of Algebraic Functions
+
+This is a list of built-in functions and operators usable in algebraic
+expressions. Their full Lisp names are derived by adding the prefix
address@hidden, as in @code{calcFunc-sqrt}.
address@hidden
+All functions except those noted with ``*'' have corresponding
+Calc keystrokes and can also be found in the Calc Summary.
address@hidden iftex
+
address@hidden tp
+
address@hidden Concept Index, Variable Index, Function Index, Top
address@hidden Concept Index
+
address@hidden cp
+
address@hidden Variable Index, Lisp Function Index, Concept Index, Top
address@hidden Index of Variables
+
+The variables in this list that do not contain dashes are accessible
+as Calc variables. Add a @samp{var-} prefix to get the name of the
+corresponding Lisp variable.
+
+The remaining variables are Lisp variables suitable for @code{setq}ing
+in your Calc init file or @file{.emacs} file.
+
address@hidden vr
+
address@hidden Lisp Function Index, , Variable Index, Top
address@hidden Index of Lisp Math Functions
+
+The following functions are meant to be used with @code{defmath}, not
address@hidden definitions. For names that do not start with @samp{calc-},
+the corresponding full Lisp name is derived by adding a prefix of
address@hidden
+
address@hidden fn
+
address@hidden
+
+
address@hidden
+ arch-tag: 77a71809-fa4d-40be-b2cc-da3e8fb137c0
address@hidden ignore