Preconditioned Locally Harmonic Residual Method for Computing Interior Eigenpairs of Certain Classes of Hermitian Matrices

TL;DR

The paper introduces the PLHR method, a new approach for efficiently computing multiple interior eigenpairs of Hermitian matrices without spectral transformations, leveraging preconditioning and short-term recurrence.

Contribution

It presents the PLHR algorithm, a novel preconditioned iterative method that avoids traditional spectral transformations for interior eigenvalue problems.

Findings

PLHR is efficient for large-scale problems.

PLHR is robust for Laplacian and Hamiltonian operators.

Memory-efficient compared to existing methods.

Abstract

We propose a Preconditioned Locally Harmonic Residual (PLHR) method for computing several interior eigenpairs of a generalized Hermitian eigenvalue problem, without traditional spectral transformations, matrix factorizations, or inversions. PLHR is based on a short-term recurrence, easily extended to a block form, computing eigenpairs simultaneously. PLHR can take advantage of Hermitian positive definite preconditioning, e.g., based on an approximate inverse of an absolute value of a shifted matrix, introduced in [SISC, 35 (2013), pp. A696-A718]. Our numerical experiments demonstrate that PLHR is efficient and robust for certain classes of large-scale interior eigenvalue problems, involving Laplacian and Hamiltonian operators, especially if memory requirements are tight.