New Schemes for Solving the Principal Eigenvalue Problems of Perron-like Matrices via Polynomial Approximations of Matrix Exponentials

TL;DR

This paper introduces new polynomial approximation schemes for efficiently computing the principal eigenvalues and eigenspaces of Perron-like matrices, which include nonnegative and symmetric matrices, demonstrating practical effectiveness through numerical examples.

Contribution

The paper develops novel polynomial approximation methods for calculating principal eigenvalues and eigenspaces of Perron-like matrices, expanding computational tools in this area.

Findings

Schemes effectively compute principal eigenvalues.

Numerical examples confirm practical efficiency.

Applicable to nonnegative and symmetric matrices.

Abstract

A real square matrix is Perron-like if it has a real eigenvalue $s$, called the principal eigenvalue of the matrix, and $\mbox{Re}\,\mu<s$ for any other eigenvalue $\mu$. Nonnegative matrices and symmetric ones are typical examples of this class of matrices. The main purpose of this paper is to develop a set of new schemes to compute the principal eigenvalues of Perron-like matrices and the associated generalized eigenspaces by using polynomial approximations of matrix exponentials. Numerical examples show that these schemes are effective in practice.