Verified eigenvalue and eigenvector computations using complex moments and the Rayleigh$\unicode{x2013}$Ritz procedure for generalized Hermitian eigenvalue problems

TL;DR

This paper introduces a verified eigenvalue and eigenvector computation method for generalized Hermitian eigenproblems using complex moments and the Rayleigh–Ritz procedure, improving efficiency and robustness over previous approaches.

Contribution

The paper presents a novel verified computation method combining complex moments and the Rayleigh–Ritz procedure, reducing quadrature points and enabling eigenvector verification.

Findings

Faster than previous methods while maintaining verification accuracy.

Effective for nearly singular matrix pencils and multiple eigenvalues.

Requires fewer quadrature points for error control.

Abstract

We propose a verified computation method for eigenvalues in a region and the corresponding eigenvectors of generalized Hermitian eigenvalue problems. The proposed method uses complex moments to extract the eigencomponents of interest from a random matrix and uses the Rayleigh$\unicode{x2013}$Ritz procedure to project a given eigenvalue problem into a reduced eigenvalue problem. The complex moment is given by contour integral and approximated using numerical quadrature. We split the error in the complex moment into the truncation error of the quadrature and rounding errors and evaluate each. This idea for error evaluation inherits our previous Hankel matrix approach, whereas the proposed method enables verification of eigenvectors and requires half the number of quadrature points for the previous approach to reduce the truncation error to the same order. Moreover, the Rayleigh$\unicode{x2013}$Ritz procedure approach forms a transformation matrix that enables verification of the eigenvectors. Numerical experiments show that the proposed method is faster than previous methods while maintaining verification performance and works even for nearly singular matrix pencils and in the presence of multiple and nearly multiple eigenvalues.