Real Polynomial Root-finding by Means of Matrix and Polynomial Iterations

TL;DR

This paper introduces novel matrix and polynomial iteration techniques for efficiently finding real roots of high-degree polynomials, significantly accelerating existing algorithms by leveraging matrix-polynomial correlations and complex geometry.

Contribution

It presents new algorithms that dramatically speed up real root-finding for high-degree polynomials using matrix-based methods and complex plane analysis.

Findings

Number of iterations grows very slowly with polynomial degree

Algorithms perform well on benchmark and random matrices

Significant acceleration over traditional methods

Abstract

Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial has no nonreal roots, but for high degree polynomials, nonreal roots are typically much more numerous than the real ones. The challenge is known for a long time, and the subject has been intensively studied. Nevertheless, we produce some novel ideas and techniques and obtain dramatic acceleration of the known algorithms. In order to achieve our progress we exploit the correlation between the computations with matrices and polynomials, randomized matrix computations, and complex plane geometry, extend the techniques of the matrix sign iterations, and use the structure of the companion matrix of the input polynomial. The results of our extensive tests with benchmark polynomials and random matrices are quite encouraging. In particular in our tests the number of iterations required for convergence of our algorithms grew very slowly (if at all) as we increased the degree of the univariate input polynomials and the dimension of the input matrices from 64 to 1024.