Good non-zeros of polynomials

Abstract

In numerical applications, one needs sometimes a point x* with F(x*) # O, if F # 0. The usual procedure which helps in exact arithmetic (at most deg(F) unsuccessful trials) cannot be applied, since small values F(x*) ,~ 0 are numerically useless. Hence the problem is how to find an x* with IF(x*)[ in a reasonable order of magnitude compared to a norm of F ~ 0. In this exposition, I describe two ways of finding good points x*. The appropriate norm in this context is certainly the maximum norm over an interval or a compact region in the complex plane and x* chosen from that region. However, the computation of this norm requires the solution of a maximum problem. Therefore it is more convenient to use a norm of the coefficient vector for fixing the norm of the polynomial and to use'known results on the comparison of different polynomial norms like in the book [2]. Unfortunately, in [2] the maximum norm for polynomials iis not considered. In the formulas (2) and (6) below, I compare a maximum norm with the euclidean norm. I hope, that the beauty of the following identity (1) and its elegant proof is also of some interest for the reader. In Schbnhage's article on quasi-gcd computations [4] x* is constructed by means of the identity

DOI: 10.1145/347127.347131

Cite this paper

@article{Mller1999GoodNO, title={Good non-zeros of polynomials}, author={H. Michael M{\"{o}ller}, journal={ACM SIGSAM Bulletin}, year={1999}, volume={33}, pages={10-11} }