Another applied problem arose today which requires me to compute all the roots of a polynomial
with real coefficients. The degree needs to be left arbitrary.
I only see vague references on the internet to the very abstract, symbolic "formula"
(if one wishes to call it that) for all the roots of a polynomial in terms of siegel
(siegal? seagel?) elliptic modular functions. No one ever seems to try to use it.
The only person I know who ever wrote out the formula was Hiroshi Umemura, professor
from Nagoya University in Japan. I actually contacted him once, back in 1994. But
we could not say much to each other over the internet. I saw his article in which
he expressed this formula in the back of a book called "Tata Lectures" Volume 2.
I'd like to at least see how far I can get with it before giving up.
I like the features of both Mathematica and Maple that, rather than "give up", they at least
express some formula in some sort of symbolic fashion, rather than "FAIL". I still find
it helpful, say, when I am computing partial fractions.
I am currently using Maple for college courses in microbiology and biotechnology.
I wrote my own little heuristic algorithm of how to choose functions (I chose
certain very specific rational functions) to model probability distribution functions.