Fast Inexact Implicitly Restarted Arnoldi Method for Generalized Eigenvalue Problems with Spectral Transformation

We study an inexact implicitly restarted Arnoldi (IRA) method for computing a few eigenpairs of generalized non-Hermitian eigenvalue problems with spectral transformation, where in each Arnoldi step (outer iteration) the matrix-vector product involving the transformed operator is performed by iterative solution (inner iteration) of the corresponding linear system of equations. We provide new perspectives and analysis of two major strategies that help reduce the inner iteration cost: a special type of preconditioner with “tuning”, and gradually relaxed tolerances for the solution of the linear systems. We study a new tuning strategy constructed from vectors in both previous and the current IRA cycles, and we show how tuning is used in a new two-phase algorithm to greatly reduce inner iteration counts. We give an upper bound of the allowable tolerances of the linear systems and propose an alternative estimate of the tolerances. In addition, the inner iteration cost can be further reduced through the use of subspace recycling with iterative linear solvers. The effectiveness of these strategies is demonstrated by numerical experiments.