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[Axiomdeveloper] [#87 solve(x + 1.1, 0.001) fails] LaTeX not working
From: 
anonymous 
Subject: 
[Axiomdeveloper] [#87 solve(x + 1.1, 0.001) fails] LaTeX not working 
Date: 
Sun, 13 Feb 2005 04:13:58 0600 
++added:
??changed:
Of course, yes. However, there is a dilemma: when you give Axiom an equation
with floating point coefficients, should Axiom "solve" this algebraically, as
if 'Float' is just like any other domain, or numerically, giving 'Float' a
special treatment? Since Axiom algorithms are categorical, rather than writing
two separate algorithms, Axiom solves, if possible, algebraically (that is,
exactly) and gives numerical answers as options when the precision parameter is
given. This choice does not work well with equations over 'Float' because
'Float' does not have some of the algebraic properties as 'Fraction Integer' or
'Fraction Complex Integer' (such as factorization or GCD), which is why there
is a warning in 'solve(x^21.234)'. The package is numsolve.spad and you see
that these restrictions are well documented. So the above signature is really
not meant to be used at the moment. A similar situation occurs, for example
'factor(1.23)' is legal, but is really useless. Axiom does n!
ot use a mechanism to exclude specific domains from a category. It adopts an
"include" philosophy but let things fail with warning or error. If you look
into numsolve.spad, you will find that the 'innerSolve1' algorithm {\it
implementation} is restricted. (So if later someone finds a way to implement a
'solve' algorithm over 'Float', that would be just fine).

So a lot of Axiom failures are not bugs, but by design. One way to improve the
user interface would seem to be to automatically lifting a polynomial over
'Float' to one over 'Fraction Integer'. A moment's reflection would convince
you this is not always possible (for example, 'sqrt(2)' or '%pi' are
technically both belong to 'Float' (model for real numbers), but of course, in
reality, every floating point number is a rational number. Such a lifting
package would have to take into consideration the precision to convert some
symbolic constants to a decimal approximation and then convert that to an exact
rational number. However, even this would not create satisfactory results
because we know the sensitivity of solutions of polynomial equations to small
changes of its coefficients. Wilkinson has this example


<center>
f(x) = (x+1)(x+2) ... (x+20) = x<SUP>20</SUP> + 210 x<SUP>19</SUP> + ... + 20!
= 0
</center>


where a change of the coefficient 210 by 2<SUP>23</SUP> (approximately 1.2
× 10<SUP>7</SUP>) would turn the root 20 to 20.8 and five pairs of
zeros to complex roots. (This perturbed equation will take a *very long* time
in Axiom, will not be solved exactly by Mathematica, but is trivially solved
*numerically* in Mathematica).



So if we want numerically accurate solutions, we should use a robust numerical
library. I believe this is not yet available in Axiom (the NAG version allowed
interface with its Fortran libraries, at extra costs).

If we are really (no pun intended) only using truely floating point
coefficients, then it can easily be converted to Fraction Integer, but one has
to beware that the algorithm would take a very long time because exact
arithmetic with large integer coefficients are expensive.

{\tt [[Kostas Oikonomou wrote:]]}
[1 more lines...]
Of course, yes. However, there is a dilemma: when you give Axiom an equation
with floating point coefficients, should Axiom "solve" this symbolically, as if
'Float' is just like any other domain, or numerically, giving 'Float' a special
treatment? Since Axiom algorithms are categorical, rather than writing two
separate algorithms, Axiom solves, if possible, exactly and gives numerical
answers as options when the precision parameter is given. This choice does not
work well with equations over 'Float' because 'Float' does not have some of the
algebraic properties as 'Fraction Integer' or 'Fraction Complex Integer' (such
as factorization or GCD), which is why there is a warning in
'solve(x^21.234)'.
++added:
The package is numsolve.spad and you see that these restrictions are well
documented. So the above signature is really not meant to be used at the
moment. A similar situation occurs, for example 'factor(1.23)' is legal, but is
really useless. Axiom does not use a mechanism to exclude specific domains from
a category. It adopts an "include" philosophy but let things fail with warning
or error. If you look into numsolve.spad, you will find that the 'innerSolve1'
algorithm *implementation* is restricted. (So if later someone finds a way to
implement a 'solve' algorithm over 'Float', that would be just fine).
So a lot of Axiom failures are not bugs, but by design. One way to improve the
user interface would seem to be to automatically lifting a polynomial over
'Float' to one over 'Fraction Integer'. A moment's reflection would convince
you this is not always possible (for example, 'sqrt(2)' or '%pi' technically
both belong to 'Float' (model for real numbers)), but of course, in reality,
every (finite precision) floating point number is a rational number. Such a
lifting package would have to take into consideration the precision to convert
some symbolic constants to their decimal approximations and then convert them
to exact rational numbers. However, even this would not create satisfactory
results because we know the sensitivity of solutions of polynomial equations to
small changes of its coefficients. Wilkinson has this example
<center>
*f*(*x*) = (*x*+1)(*x*+2) ... (*x*+20) = *x*<SUP>20</SUP> + 210
*x*<SUP>19</SUP> + ... + 20! = 0
</center>
where a change of the coefficient 210 by 2<SUP>−23</SUP> (approximately
1.2 × 10<SUP>−7</SUP>) would turn the root −20 to −20.8
and five pairs of zeros to complex roots. (This perturbed equation will take a
*very long* time in Axiom, will not be solved exactly by Mathematica, but is
easily solved *numerically* in Mathematica).
So if we want numerically accurate solutions, we should use a robust numerical
library. I believe this is not yet available in Axiom (the NAG version allowed
interface with its Fortran libraries, at extra costs).
If we are really (no pun intended) only using truely floating point
coefficients, then it can easily be converted to Fraction Integer, but one has
to beware that the algorithm would take a very long time because exact
arithmetic with large integer coefficients are expensive.
{\tt [[Kostas Oikonomou wrote:]]}
Also, while 'solve(x+1.1)' "works", so to speak, 'solve(x^2  1.234)' returns a
warning.
??changed:
{\tt [[Martin Rubey Sat Feb 12 13:07:55 0600 2005 <address@hidden>
wrote:]]}

I think the idea is to convert your Polynomial Float into a Polynomial Fraction
Integer and then use solve (POLY FRAC INT, FLOAT) > result.
{\tt [[Martin Rubey Sat Feb 12 13:07:55 0600 2005 <address@hidden> wrote:]]}
I think the idea is to convert your 'Polynomial Float' into a 'Polynomial
Fraction
Integer' and then use 'solve (POLY FRAC INT, FLOAT) > ... ' to get results.
??changed:
Yes, as explained above. Algebraic methods do not work well with Float.


Yes, as explained above. Symbolic methods do not work well with 'Float'.

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