| > I'm aware of the Jenkins Traub algorithm - I figured that was the last
> word on polynomial roots but this uses real number representation (real
> in the context of FORTRAN).
> What is the algorithm that MAPLE uses to find the exact roots of
> polynomials?
There is a theoretical limit to the *exact* solution of nth-degree
polynomials; regrettably, that limit is n=4. Beyond that, the solutions
cannot be expressed as radicals, i.e. there *are no* exact solutions, in
general. What MAPLE will do is break down the problem and report the
results in the form
RootOf(expression)
which begs the question a lot. Jenkins-Traub is a good computational
rootfinder but makes no pretensions about finding exact roots.
The algorithms MAPLE uses for n = 2,3,4 are those found by Cardan et al a
couple of centuries ago, and are given in most high school or college
algebra texts.
|
| ref .0
in Maple also, for an ill-condition problem such as
4 -8
(x-2) - 10 = 0
to find roots you should use fsolve, as it tries to find all roots for
an ill-conditioned problem, but it still could miss some. this relates
to the nature of the problem not the method.
FUNCTION : fsolve - solve using floating-point arithmetic
For a general equation, fsolve attempts to compute a single real root.
However for polynomials it will compute all real (non-complex) roots,
although exceptionally ill-conditioned polynomials may cause fsolve
to miss some roots.
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