PRECONDITIONED LOCALLY HARMONIC RESIDUAL METHOD FOR COMPUTING INTERIOR EIGENPAIRS OF CERTAIN CLASSES OF HERMITIAN MATRICES

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 [E. Vecharynski and A. V. Knyazev, SIAM J. Sci. Comput., 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.