Accelerated Approximation of the Complex Roots and Factors of a Univariate Polynomial

TL;DR

This paper introduces an accelerated, nearly optimal algorithm for approximating complex roots of univariate polynomials, especially effective when roots are initially isolated, and applicable to black-box polynomials, improving stability and efficiency.

Contribution

It extends existing root-finding algorithms to work efficiently with less initial precision and applies to polynomials given as black boxes, enhancing practical stability and applicability.

Findings

Algorithm achieves nearly optimal complexity bounds.

Effective for polynomials with roots in a fixed region.

Applicable to black-box polynomial evaluations.

Abstract

The algorithms of Pan (1995) and(2002) approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and Boolean time but require precision of computing that exceeds the degree of the polynomial. This causes numerical stability problems when the degree is large. We observe, however, that such a difficulty disappears at the initial stage of the algorithms, and in our present paper we extend this stage to root-finding within a nearly optimal arithmetic and Boolean complexity bounds provided that some mild initial isolation of the roots of the input polynomial has been ensured. Furthermore our algorithm is nearly optimal for the approximation of the roots isolated in a fixed disc, square or another region on the complex plane rather than all complex roots of a polynomial. Moreover the algorithm can be applied to a polynomial given by a black box for its evaluation (even if its coefficients are not known); it promises to be of practical value for polynomial root-finding and factorization, the latter task being of interest on its own right. We also provide a new support for a winding number algorithm, which enables extension of our progress to obtaining mild initial approximations to the roots. We conclude with summarizing our algorithms and their extension to the approximation of isolated multiple roots and root clusters.