Re: [Axiom-developer] "empty list" results, inverse trig substitutions, "Floats" in "solve"

[William Sit]

Hi James:

When you ask Axiom to solve 

solve([a=3+x,b=1-x,x=2],[a,b])

I believe (note, this is an educated guess, but simple 
problems in Axiom can often have deep reasons, so much so 
it is difficult to understand whether things are bugs or 
features) that x is treated as a parameter, or as lying in 
the coefficient domain (usually an integral domain). All 
polynomial equations are converted into polynomial ideals 
over the coefficient domain before using the solver. Thus 
an equation like x=2 is converted to x-2 (=0), but x-2 is 
a non-zero element of the coefficient domain when x is an 
indeterminate. To solve the equation in a,b, the 
coefficient domain is extended to its field of fractions, 
and x-2 is therefore a unit in the polynomial ring with a, 
b as indeterminates. The polynomial ideal is then a unit 
ideal, hence there is no solution.

Of course, this is not what the user wants. The user wants 
to solve for a,b,x but just wants the projected part a, b 
of the solution. However, the "equation" x=2 does not 
involve a or b. If it is replaced by one that does, 
provided it forms a consistent system in the variables a, 
b with the rest (thus no single equation in x alone should 
be resulted through elimination), then the system will 
solved.

solve([a=b+x,b=a-x,x=a-3],[a,b])
[[a= x + 3,b= 3]]

If x is intended to be solved (not a parameter), then it 
should be considered as an indeterminate like a, b. 
Otherwise, if it is intended to substitute x with a value 
like 2, then it should be done after the system in a, b 
has been solved.

By distinguishing the two cases when the variables to be 
solved is [a,b] or [a,b,x], Axiom allows the user to 
indicate his intention.

I can't do much with the tan problem, since we have:

solve([tan(b) = a],[b])
> > Error detected within library code:
No identity element for reduce of empty list using 
operation
append

(You mileage may differ, since I am using a very old 
version of Axiom).

This is clearly a bug, for the old version.

William



On Fri, 02 Oct 2009 13:29:29 -0600
 James <address@hidden> wrote:
> I'm new to axiom, using it for calculations.  There seems 
> to be odd behaviour
> when using "solve" on a list of equations.  This is with 
> the most recent axiom
> binary package, Version Axiom (May 2009), on Ubuntu 8.10 
> "intrepid".  For
> instance:
>
>  (8) -> solve([a=3+x,b=1-x,x=2],[a,b])
>  ...
>  (8)  []
>                          Type: List List Equation 
> Fraction Polynomial Integer
>
> So, the answer is "the empty list"?!  That's not very 
> useful, and seems not
> correct either.  What am I missing here?
>
> Then instead:
>
>  (9) -> solve([a=3+x,b=1-x,x=2],[a,b,x])
>  (9) ->
>    (9)  [[a= 5,b= - 1,x= 2]]
>                          Type: List List Equation 
> Fraction Polynomial Integer
>
> That works.  Why?
>
> Similarly, axiom seems confused about finding 
> substitutions.  For instance:
>
>  (12) -> solve([ tan(bt)=(a*r)/(s*(r-d)), x=100*cos(bt), 
> y=d*sin(bt)],[x,y,bt])
>  (12) ->
>     (12)  [[]]
>                                    Type: List List 
> Equation Expression Integer
>
> Here, "the empty list" again - why?
>
> If, instead, I "spell it out" for axiom, by taking the 
> arctan instead, then
>
>  (13) -> 
> solve([bt=atan((a*r)/(s*(r-d))),x=100*cos(bt),y=d*sin(bt)],[x,y,bt])
>  (13) ->
>     (13)
>     [
>                  (100r - 100d)s                        a 
> d r
>       [x= ----------------------------, y= 
> ----------------------------,
>            +-------------------------+ 
>     +-------------------------+
>            |  2           2  2    2 2       |  2 
>          2  2    2 2
>           \|(r  - 2d r + d )s  + a r       \|(r  - 2d r 
> + d )s  + a r
>                    a r
>        bt= atan(--------)]
>                 (r - d)s
>       ]
>                                    Type: List List 
> Equation Expression Integer
>
> Axiom doesn't know about substituting inverse trig 
> functions by itself?
>
> But then, again,
>
>  (14) -> 
> solve([bt=atan((a*r)/(s*(r-d))),x=100*cos(bt),y=d*sin(bt)],[x,y])
>  (14) ->
>     (14)  [[]]
>                                    Type: List List 
> Equation Expression Integer
>
> Arrrrgh!  Ok, why is that again, returning "the empty 
> list" when using "solve"
> with the truncated list of variables?
>
>
> On anther topic, "Floats" in "solve", where this works, 
> mixing integers and
> floats:
>
>  (16) -> solve([a=3+x,b=1-x,x=2.0],[a,b,x])
>     ...
>     (16)  [[a= 5.0,b= - 1.0,x= 2.0]]
>                             Type: List List Equation 
> Fraction Polynomial Float
>
> and this works:
>
>  (17) -> solve([a=3+x,b=1-x,x=2],0.001)
>    ...
>    (17)  [[x= 2.0,a= 5.0,b= - 1.0]]
>                                     Type: List List 
> Equation Polynomial Float
>
>
> doing this, using "x=2.0" instead of "x=2":
>
>  (18) -> solve([a=3+x,b=1-x,x=2.0],0.001)
>    ...
>    There are 20 exposed and 3 unexposed library 
> operations named solve
>       having 2 argument(s) but none was determined to be 
> applicable.
>       ...
>    Cannot find a definition or applicable library 
> operation named solve
>       with argument type(s)
>                        List Equation Polynomial Float
>                                     Float
>
>
> Arrrrgh! - the dreaded "none was determined to be 
> applicable"!
> Does that make sense, that?
>
>
> Thanks in advance for any clues!  Are these bugs?
>
>
> James
>
>
>
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William Sit, Professor Emeritus
Mathematics, City College of New York                  
Office: R6/202C Tel: 212-650-5179
Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/