Thanks for the comments.
Many of you may know about these things already, but in case that
isn't so, I thought I'd add to the comments on this list FYI.
Searching in Google on "solving quintics and galois groups" produces some
interesting links. It is the subject of fairly recent research.
Given a quintic rational polynomial $p(x)$, D.S. Dummit produced a
sextic polynomial $f(x)$ which has a rational root if and only if
$p(x)$ can be solved by radicals. The polynomial $f(x)$ is complicated, but
not so bad for CAS. For $x^5 + a*x +b$ it is pretty simple.
Subsequent work and references can be found in the paper "Solving
Solvable Quintics in Radicals" by Piezas who also has a type of
"cookbook" method for pasting into CAS.
There are references to Mathematica and sites at wolfram.com, so these
methods may be already implemented in Mathematica. I don't know.
In any event, it may be a worthwhile exercise for students of lisp,
CAS, and Algebra to implement these methods in maxima.
-sen
>
>> -----Original Message-----
>> From: maxima-bounces at math.utexas.edu
>> [mailto:maxima-bounces at math.utexas.edu] On Behalf Of sen1 at math.msu.edu
>> Sent: Tuesday, August 22, 2006 6:59 PM
>> To: maxima at math.utexas.edu
>> Subject: cubics and quartics
>>
>> Hello,
>> I was curious if the Cardan formulas are used in the computation of
>> roots of cubic and quartic polynomials.
>
> Yes. Look in the source for psolve.lisp, solvecubic and solvequartic, called
> by solve1a in file solve.lisp
>
>> I could not find them.
>>
>> Implementation of these seems straightforward for an
>> experienced lisp programmer (which I, unfortunately, am
>> not).
>
> The math is tricky, so getting the algorithm right is trick. The lisp is
> easy.
>
>> Are there problems in implementing the Cardan formulas
>> (potential disadvantages, etc)?
>
> Yes, you want to try to keep the real roots real without any complex numbers
> if possible. Since all CAS screw up the simplification of radicals and
> nested radicals when they pretend they have a single value in all cases,
> these formulas can be screwed up too.
>
>>
>> In general, one would like to find symbolic solutions of
>> rational polynomials when possible (i.e., computing Galois
>> groups when feasable). Of course, here, I am thinking of
>> solutions by extracting roots (radicals).
>
> There is usually little merit to finding exact expressions for roots when
> those expressions are huge messes. What are you going to do with them?
> Evaluate them numerically? Why not do that in the first place? Paste them
> on your wall?
>>
>> Anyone know if this is done in other computer algebra programs?
>
> All of them, probably.
>
>>
>> It would be nice to allow users to choose to implement
>> symbolic or numerical methods for eigenvalues. Is this
>> possible now, say by setting some variable true or false?
>
> Exact eigenvalues tend to be impossible to find except for very simple
> examples. It might even be plausible to have only numeric routines, and just
> point out that the symbolic ones are the roots of the characteristic
> polynomial, a one-line maxima program. This transformation is a bad idea
> for numerics, however, since it is probably harder to find the roots than
> the eigenvalues.
>
>
>> RJF
>
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>
--
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| Sheldon E. Newhouse | e-mail: sen1 at math.msu.edu |
| Mathematics Department | |
| Michigan State University | telephone: 517-355-9684 |
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