Novel Approach to Real Polynomial Root-finding and Matrix Eigen-solving

TL;DR

This paper introduces a novel method that significantly speeds up real polynomial root-finding and real eigenvalue approximation by leveraging matrix-polynomial correlations and matrix sign iteration techniques.

Contribution

It extends existing algorithms by exploiting the structure of companion matrices and matrix sign iteration to improve efficiency in real root and eigenvalue computations.

Findings

Accelerates real polynomial root-finding algorithms

Enhances real eigenvalue approximation for nonsymmetric matrices

Utilizes matrix-polynomial correlation and structure exploitation

Abstract

Univariate polynomial root-finding is both classical and 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 typically nonreal roots are much more numerous than the real ones. We dramatically accelerate the known algorithms in this case by exploiting the correlation between the computations with matrices and polynomials, extending the techniques of the matrix sign iteration, and exploiting the structure of the companion matrix of the input polynomial. We extend some of the proposed techniques to the approximation of the real eigenvalues of a real nonsymmetric matrix.