Computing pseudospectra using block implicitly restarted Arnoldi iteration

The pseudospectra is a useful tool to study the behavior of systems associated with non-normal matrices. In the past decade, different projection Krylov methods have been used to calculate the pseudospectra of large matrices, rather than earlier approaches which require the application of SVD decomposition at each point of a grid. Inverse Lanczos is a better choice, but still requires previous Schur factorization, which is prohibited in the large scale setting. In this work, we investigate the practical applicability and the performance of a block implicitly restarted Arnoldi method to approximate the pseudospectrum of large matrices, as was suggested by Wright and Trefethen in “Pseudospectra of rectangular matrices” (Wright and Trefethen, 2002 [16]). We present a case study of this idea, using a block version of the Implicitly Restarted Arnoldi Method (IRAM) (Sorensen, 1992 [5]). Numerical results, on several test matrices from the literature, are encouraging and show a reduction in time of this block method compared with its counterpart single version IRAM.