Palindromic quadratization and structure-preserving algorithm for palindromic matrix polynomials of even degree

Abstract

In this paper, we propose a palindromic quadratization approach, transforming a palindromic matrix polynomial of even degree to a palindromic quadratic pencil. Based on the $${(\mathcal{S}+ \mathcal{S}^{-1})}$$ -transform and Patel’s algorithm, the structure-preserving algorithm can then be applied to solve the corresponding palindromic quadratic eigenvalue problem. Numerical experiments show that the relative residuals for eigenpairs of palindromic polynomial eigenvalue problems computed by palindromic quadratized eigenvalue problems are better than those via palindromic linearized eigenvalue problems or $${{\texttt {polyeig}}}$$ in MATLAB.