Verified partial eigenvalue computations using contour integrals for Hermitian generalized eigenproblems

Abstract

We propose a verified computation method for partial eigenvalues of a Hermitian generalized eigenproblem. The block Sakurai-Sugiura Hankel method, a contour integral-type eigensolver, can reduce a given eigenproblem into a generalized eigenproblem of block Hankel matrices whose entries consist of complex moments. In this study, we evaluate all errors in computing the complex moments. We derive a truncation error bound of the quadrature. Then, we take numerical errors of the quadrature into account and rigorously enclose the entries of the block Hankel matrices. Each quadrature point gives rise to a linear system, and its structure enables us to develop an efficient technique to verify the approximate solution. Numerical experiments show that the proposed method outperforms a standard method and infer that the proposed method is potentially efficient in parallel.

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

“…Remark 2.2. The evaluations (2.18), (2.19) can also be used for the Hankel matrix approach [6] for the evaluation of eigenvectors.…”

“…Behnke [1] uses a variational principle, and Yamamoto [20] uses Sylvester's law of inertia. See [14,7,6] for further studies and references therein. Verified eigenvalue computations arise in applications, e.g., from the numerical verification of a priori error estimations for finite element solutions [21,19].…”

“…Our previous study proposes a verification method using complex moments [6]. This method is based on an eigensolver [17], which reduces a given generalized Hermitian matrix eigenvalue problem into another generalized eigenvalue problem with block Hankel matrices, and evaluates all the errors in the reduction for verification.…”

“…The Rayleigh-Ritz procedure projects a given eigenvalue problem into an (approximated) eigenspace of interest. We develop a Rayleigh-Ritz procedure version of the verified computation method for generalized Hermitian eigenvalue problems [6]. We first review the Rayleigh-Ritz procedure approach of a projection method using complex moment [18,3].…”

Fast verification for the Perron pair of an irreducible nonnegative matrix

“…For all eigenpairs, see [16,17,18]. For a few specified eigenvalues, see, e.g., [19,20,21,22,23,24,25]. For a few specified eigenvectors, see [26].…”