Verified eigenvalue and eigenvector computations using complex moments and the Rayleigh-Ritz procedure for generalized Hermitian eigenvalue problems.

TL;DR: In this article, the authors proposed a verified computation method for eigenvalues in a region and the corresponding eigenvectors of generalized Hermitian eigenvalue problems, which uses complex moments to extract the eigencomponents of interest from a random matrix and uses the Rayleigh-Ritz procedure to project a given eigen value problem into a reduced Eigenvalue problem.

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--Ritz procedure to project a given eigenvalue problem into a reduced eigenvalue problem. The complex moment is given by contour integral and approximated by 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 requires half the number of quadrature points for the previous approach to reduce the truncation error to the same order. Moreover, the Rayleigh--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.