The refined harmonic Arnoldi method and an implicitly restarted refined algorithm for computing interior eigenpairs of large matrices

Abstract

The harmonic Arnoldi method can be used to compute some eigenpairs of a large matrix, and it is more suitable for finding interior eigenpairs. However, the harmonic Ritz vectors obtained by the method may converge erratically and may even fail to converge, so that resulting algorithms may not perform well. To improve convergence, a refined harmonic Arnoldi method is proposed that replaces the harmonic Ritz vectors by certain refined eigenvector approximations. The implicit restart technique advocated by Sorensen is applied to the refined harmonic Arnoldi method and an implicitly restarted refined harmonic Arnoldi algorithm (IRRHA) is presented. For the refined algorithm a new shift scheme, called refined shifts, is proposed that is shown to be better than the exact shifts scheme used in literature. The refined shifts can be computed reliably and cheaply. Extensive numerical experiments are made on real world problems, and comparisons are drawn on IRRHA and the implicitly restarted harmonic Arnoldi algorithm (IRHA) of Morgan as well as the implicitly restarted Arnoldi algorithm (IRA) of Sorensen and the implicitly restarted refined Arnoldi algorithm (IRRA) of Jia. They indicate that the refined algorithms outperform their standard counterparts considerably. Furthermore, for exterior eigenvalue problems IRRA is preferable and more powerful, and for interior eigenvalue problems IRRHA is more attractive and can be much more efficient.