On the Convergence of Ritz Pairs and Refined Ritz Vectors for Quadratic Eigenvalue Problems

TL;DR

This paper analyzes the convergence properties of Ritz pairs and introduces refined Ritz vectors for quadratic eigenvalue problems, ensuring reliable convergence of eigenvector approximations.

Contribution

It provides a theoretical analysis of Ritz pair convergence and proposes refined Ritz vectors that guarantee convergence for quadratic eigenvalue problems.

Findings

Unconditional convergence of Ritz values to the eigenvalue.

Conditional convergence of Ritz vectors, which may fail.

Refined Ritz vectors always converge unconditionally.

Abstract

For a given subspace, the Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the angle between the subspace and the desired eigenvector converges to zero. We prove that there is a Ritz value that converges to the desired eigenvalue unconditionally but the Ritz vector converges conditionally and may fail to converge. To remedy the drawback of possible non-convergence of the Ritz vector, we propose a refined Ritz vector that is mathematically different from the Ritz vector and is proved to converge unconditionally. We construct examples to illustrate our theory.