Fast inexact implicitly restarted Arnoldi method for generalized eigenvalue problems with spectral transformation (Q2910964)

The authors study an inexact implicitly restarted Arnoldi (IRA) method for computing a few eigenpairs of generalized non-Hermitian eigenvalue problems with spectral transformation. In each Arnoldi step (outer iteration) the matrix-vector product involving the transformed operator is performed by iterative solution (inner iteration) of the corresponding system of linear equations. The authors provide new perspectives and analysis of two major strategies that help reduce the inner iteration cost, namely, a special type of preconditioner with ``tuning,'' and gradually relaxed tolerances for the solution of the linear systems. A new tuning strategy constructed from vectors in both previous and current IRA cycles is also discussed, and it is shown how tuning is used in a new two-phase algorithm that greatly reduces inner iteration counts. An upper bound on the allowable tolerances of the linear systems is given and an alternative estimate of the tolerances is presented as well. Finally, it is shown that the inner iteration cost can be further reduced through the use of subspace recycling with iterative linear solvers. Numerical experiments demonstrate the effectiveness of the proposed strategies.