General algebraic equation cannot be solved CALSSICALLY after degree 4 (Galois theory). Some case seems tracktable for the quintic using hypergeometric function and other technics. BUT I would know what is actually the best algorithm for solving numerically any algebraic equation (technic + calgorithmic complexity) ? for the two case Real and Real+Complex. Thx in advance for any answer.
HP
Dear Herve Poulard
To find roots of polynomials is an active area of mathematics, for their importance in many applications in all branches of sciences. Mathematicians struggle for the most profound understanding of many results and conjectures about polynomial roots. There is no best algorithm to find the roots of a given polynomial. Several methods and techniques are available; each has advantages and disadvantages. Some of them are faster than the others. One may provide more precision for the largest or the minimum root but not the other ones. Many algorithms locate and determine proper bounds for the solutions.
In general, there are thousands of excellent articles about the topic.
It is a good idea to start a project that collects all and offer a fair comparison between them.
A brief review is available on the following link
https://en.wikipedia.org/wiki/Geometrical_properties_of_polynomial_roots
this offer more helpful links
Best regards
Hi, there is an interesting way to compute simultaneously all the roots of an univariate polynomial: the Weierstrass method. You can have a look for example to the following paper:
SIMULTANEOUS COMPUTATION OF ALL THE ZERO-CLUSTERS OF A UNIVARIATE POLYNOMIAL , Jean-Claude YAKOUBSOHN
in the file attached (proceeding of a relatively old conference). Best
Dear Professor thank you for this paper.
It seems really interesting.
Cheers
HP
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