Preconditioned Locally Harmonic Residual Method for Computing Interior Eigenpairs of Certain Classes of Hermitian Matrices
This work addresses the need for memory-efficient computation of interior eigenpairs in large-scale Hermitian eigenvalue problems, offering an alternative to traditional methods that require spectral transformations.
The paper proposes the PLHR method for computing interior eigenpairs of generalized Hermitian eigenvalue problems without spectral transformations or matrix inversions, demonstrating efficiency and robustness for large-scale problems involving Laplacian and Hamiltonian operators, especially under tight memory constraints.
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.