Computing sharp and scalable bounds on errors in approximate zeros of univariate polynomials
P. H. D. Ramakrishna, Sudebkumar Prasant Pal, Samir Bhalla, Hironmay Basu, Sudhir Kumar Singh
Abstract
There are several numerical methods for computing approximate zeros of a given univariate polynomial. In this paper, we develop a simple and novel method for determining sharp upper bounds on errors in approximate zeros of a given polynomial using Rouche's theorem from complex analysis. We compute the error bounds using non-linear optimization. Our bounds are scalable in the sense that we compute sharper error bounds for better approximations of zeros. We use high precision computations using the LEDA/real floating-point filter for computing our bounds robustly.