An iterative block Arnoldi algorithm with modified approximate eigenvectors for large unsymmetric eigenvalue problems

Abstract

A modified m-step Arnoldi algorithm proposed by Jia and Elsner for computing a few selected eigenpairs of large unsymmetric matrices is generalized to its block version. In the new method, we use new modified approximate eigenvectors obtained from a linear combination of Ritz vectors and the wasted ( m+1)th block basis vector, whose residual norms are satisfied with some ( p+1)-dimensional minimization problems. The resulting block Arnoldi algorithm is not only better than the standard m-step one both in theory and in practice but also cheaper than the standard ( m+1)-step one. The relationships among the residual norms of Ritz pairs and those of new approximate eigenpairs are analyzed. Theoretical results show that the new method can overcome the drawback of nonconvergence, which exists in the conventional one in some extent. We carry out some numerical experiments, which exhibit that the new algorithm is more efficient and works better than its counterparts.