Computing the field of values and pseudospectra using the Lanczos method with continuation

Abstract

.The field of values and pseudospectra are useful tools for understanding the behaviour of various matrix processes . To compute these subsets of the complex plane it is necessary to estimate one or two eigenvalues of a large number of parametrized Hermitian matrices ; these computations are prohibitively expensive for large, possibly sparse, matrices, if done by use of the QR algorithm . We describe an approach based on the Lanczos method with selective reorthogonalization and Chebyshev acceleration that, when combined with continuation and a shift and invert technique, enables efficient and reliable computation of the field of values and pseudospectra for large matrices . The idea of using the Lanczos method with continuation to compute pseudospectra is not new, but in experiments reported here our algorithm is faster and more accurate than existing algorithms of this type .

Analysis of algebraic systems arising from fourth‐order compact discretizations of convection‐diffusion equations

“…where σ min (zI − A) is the smallest singular value of zI − A. Algorithms for computation of pseudospectra are described in [14][15][16][17]. The following qualitative measures of the nonnormality of a matrix A are given by Braconnier [18]: A matrix is…”

Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization

“…Recent needs in applications such as the ones mentioned earlier, however, have motivated research oriented towards the development of algorithms for the computation of the smallest singular triplets, a problem that is acknowledged to challenge the capabilities of current state-of-the-art software, e.g., see [1,8,14,18,20,21,34].…”

Computing Probabilistic Bounds for Extreme Eigenvalues of Symmetric Matrices with the Lanczos Method

“…For example, the stability of processes involving such matrices is often governed by the location of their eigenvalues. The extreme eigenvalues can also be used to determine condition numbers, the field of values, and ε-pseudospectra of arbitrary matrices (see, e.g., [1,12]). For smallsized matrices the eigenvalues can be computed by the QR-method (see, e.g., [2]), but this is not feasible for large matrices.…”