Univariate Polynomials: Nearly Optimal Algorithms for Numerical Factorization and Root-finding

Abstract

To approximate all roots (zeros) of a univariate polynomial, we develop two effective algorithms and combine them in a single recursive process. One algorithm computes a basic well isolated zero-free annulus on the complex plane, whereas another algorithm numerically splits the input polynomial of the nth degree into two factors balanced in the degrees and with the zero sets separated by the basic annulus. Recursive combination of the two algorithms leads to computation of the complete numerical factorization of a polynomial into the product of linear factors and further to the approximation of the roots. The new root-finder incorporates the earlier techniques of Schönhage, Neff/Reif, and Kirrinnis and our old and new techniques and yields nearly optimal (up to polylogarithmic factors) arithmetic and Boolean cost estimates for the computational complexity of both complete factorization and root-finding. The improvement over our previous record Boolean complexity estimates is by roughly the factor of n for complete factorization and also for the approximation of well-conditioned (well isolated) roots, whereas the same algorithm is also optimal (under both arithmetic and Boolean models of computing) for the worst case input polynomial, whose roots can be ill-conditioned, forming clusters. (The worst case complexity bounds for root-finding are supported by our previous algorithms as well.) All algorithms allow processor efficient acceleration to achieve solution in polylogarithmic parallel time.

Experimental evaluation and cross-benchmarking of univariate real solvers

“…The best worst case complexity bound for all these algorithms, after eliminating the (poly)logarithmic factors, is e OB(d 4 τ 2 ), where d is the degree of the polynomial and τ the maximum coefficient bitsize. From a theoretical point of view, the goal is to propose algorithms with complexity bounds that are close to, or match, the bound of the nearly optimal numerical algorithm of Pan [28]. The worst case bound of the latter is e OB(d 3 τ ), which can be further improved to e OB(d 2 τ ), using sophisticated splitting techniques.…”

On Application of the Ray-Shooting Method for LQR via Static-Output-Feedback

“…• computing the matrix P (K (t)) as in (5) p, by the algorithm of V. Y. Pan (see [26]). We end up with O p 3 for these operations.…”

A Descartes Algorithm for Polynomials with Bit-Stream Coefficients

“…Weyl [11] exhibited the first complete algorithm and Pan [12] surveys the development till about 1995. The currently best algorithm is due to Pan [13]. It applies to polynomials with bit-stream coefficients.…”