A quadrature-based eigensolver with a Krylov subspace method for shifted linear systems for Hermitian eigenproblems in lattice QCD

Abstract

We consider a quadrature-based eigensolver to find eigenpairs of Hermitian matrices arising in lattice quantum chromodynamics. To reduce the computational cost for finding eigenpairs of such Hermitian matrices, we propose a new technique for solving shifted linear systems with complex shifts by means of the shifted CG method. Furthermore, by using integration paths along horizontal lines corresponding to the real axis of the complex plane, the number of iterations for the shifted CG method is also reduced. Some numerical experiments illustrate the accuracy and efficiency of the proposed method by comparison with a conventional method.

A projection method for nonlinear eigenvalue problems using contour integrals

“…In numerical calculations, we evaluated the contour integrals using a numerical integral, refer to [9]. The approximations for S κ are represented as following,…”

“…5. ConstructŜm =Ŝ(:, 1 :m), wherem is the numerical rank whose tolerance is δ ofŜ According to [9], we can set z p and w p for any numerical integrals. The dimensionality of the generated subspace depends on the numerical rank ofŜ.…”

Designing rational filter functions for solving eigenvalue problems by contour integration

“…Later on Tetsuya Sakurai joined us in our study and co-authored some papers [13,17,12]. To solve eigenvalue problems, Sakurai and his co-authors applied the idea of the generalized eigenvalue problem involving the Hankel and shifted Hankel matrix using moments based on the resolvent function [18,10,15,9,19,25,1,2]. Eric Polizzi and co-authors also used contour integrals based on the resolvent function resulting in the FEAST algorithm [16,22,7].…”

Nonlinear eigenvalue problems and contour integrals

“…We reduced the problem to a generalized eigenvalue problem involving a Hankel matrix as well as a shifted Hankel matrix consisting of the moments of the analytic function T . For generalized linear eigenvalue problems corresponding to the pencil A − zB Tetsuya Sakurai and his co-authors [4][5][6][7] (see also [8] for the specific case where A, B ∈ R m×m are symmetric and B is positive definite) use 1 2π i  Γ (z − γ ) pûH (zB − A) −1v dz, p = 0, 1, 2, . .…”