New Structured Matrix Methods for Real and Complex Polynomial Root-finding

TL;DR

This paper introduces new structured matrix methods that improve the numerical solution of univariate polynomial root-finding, especially for approximating real roots, combining existing techniques with novel approaches.

Contribution

It presents novel structured matrix algorithms that enhance the efficiency of polynomial root-finding, integrating known methods with new techniques for better numerical approximation.

Findings

Algorithms demonstrate improved efficiency in root-finding tasks.

Effective approximation of real roots shown through analysis and experiments.

Structured matrix methods outperform some existing approaches.

Abstract

We combine the known methods for univariate polynomial root-finding and for computations in the Frobenius matrix algebra with our novel techniques to advance numerical solution of a univariate polynomial equation, and in particular numerical approximation of the real roots of a polynomial. Our analysis and experiments show efficiency of the resulting algorithms.