# Verified partial eigenvalue computations using contour integrals for Hermitian generalized eigenproblems

## TL;DR

This paper introduces a verified computational approach for partial eigenvalues of Hermitian generalized eigenproblems using contour integrals, providing rigorous error bounds and efficient verification techniques.

## Contribution

It develops a method that rigorously bounds errors in contour integral-based eigenvalue computations, improving accuracy and efficiency over standard methods.

## Key findings

- The proposed method provides rigorous error bounds for complex moments.
- Numerical experiments demonstrate superior performance compared to standard methods.
- The approach is potentially efficient for parallel computations.

## 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.