On the Convergence of Ritz Pairs and Refined Ritz Vectors for Quadratic Eigenvalue Problems
Provides theoretical convergence guarantees for a numerical method used in solving quadratic eigenvalue problems, addressing a known limitation of Ritz vectors.
The paper analyzes convergence of Rayleigh-Ritz method for quadratic eigenvalue problems, proving Ritz values converge unconditionally but Ritz vectors may not; it proposes refined Ritz vectors that converge unconditionally.
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.