×

Computing pseudospectra using block implicitly restarted Arnoldi iteration. (English) Zbl 1286.65045

Summary: 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 T. G. Wright and L. N. Trefethen [IMA J. Numer. Anal. 22, No. 4, 501–519 (2002; Zbl 1038.65033)]. We present a case study of this idea, using a block version of the Implicitly Restarted Arnoldi Method (IRAM) [D. C. Sorensen, SIAM J. Matrix Anal. Appl. 13, No. 1, 357–385 (1992; Zbl 0763.65025)]. 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.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Grone, R.; Johnson, C. R.; Sá, E. M.; Wolkowicz, H., Normal matrices, Linear Algebra and its Applications, 87, 213-225 (1987) · Zbl 0613.15021
[2] Elsner, L.; Ikramov, K. D., Normal matrices: an update, Linear Algebra and its Applications, 285, 291-303 (1998) · Zbl 0931.15019
[3] Trefethen, L. N.; Embree, M., Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators (2005), Princeton University Press: Princeton University Press New Jersey, USA · Zbl 1085.15009
[5] Sorensen, D., Implicit application of polynomial filters in a \(k\)-step Arnoldi method, SIAM Journal on Matrix Analysis and Applications, 13, 1, 357-385 (1992) · Zbl 0763.65025
[6] Godunov, S. K.; Ryabenkii, V. S., Theory of Difference Schemes: An Introduction (1964), North-Holland: North-Holland Amsterdam · Zbl 0116.33102
[8] Landau, H., Computation of pseudospectra, Journal d’Analyse Mathematique, 98, 98, 243-251 (1975)
[9] Kostin, V.; Razzakov, I., On convergence of the power orthogonal method of spectrum cumputing, Inst. Math. Sib. Branch Acad. Sci. (1985), Report · Zbl 0597.65025
[10] Trefethen, L. N., Pseudospectra of matrices, (Griffiths, D. F.; Watson, G. A., Numerical Analysis, 1991: Proceedings of the 14th Dundee Conference, vol. 260 (1991), Longman Scientific and Technical: Longman Scientific and Technical Harlow, Essex), 234-266 · Zbl 0798.15005
[13] Lui, S., Computation of pseudospectra by continuation, SIAM Journal on Scientific Computing, 18, 2, 565-573 (1997) · Zbl 0874.65027
[15] Toh, K.; Trefethen, L. N., Calculation of pseudospectra by the Arnoldi iteration, SIAM Journal on Scientific Computing, 17, 1, 1-15 (1996) · Zbl 0842.65022
[16] Wright, T. G.; Trefethen, L. N., Pseudospectra of rectangular matrices, IMA Journal of Numerical Analysis, 22, 4, 501-519 (2002) · Zbl 1038.65033
[18] Lehoucq, R.; Sorensen, D.; Yang, C., ARPACK USERS GUIDE: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods (1998), SIAM: SIAM Philadelphia · Zbl 0901.65021
[19] Wright, T. G.; Trefethen, L. N., Large-scale computation of pseudospectra using ARPACK and, SIAM Journal on Scientific Computing, 23, 2, 591-605 (2002) · Zbl 0992.65030
[20] Simoncini, V.; Gallopoulos, E., Transfer functions and resolvent norm approximation of large matrices, Electronic Transactions on Numerical Analysis, 7, 190-201 (1998) · Zbl 0915.65029
[21] Baglama, J., Augmented block householder Arnoldi method, Linear Algebra and its Applications, 429, 10, 2315-2334 (2008) · Zbl 1153.65034
[24] Hassard, B.; Kazarinoff, N.; Wan, Y., Theory and Applications of Hopf Bifurcation (1981), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0474.34002
[27] Yin, Q.; Lu, L., An implicitly restarted block Arnoldi method in a vector-wise fashion, Numerical Mathematics, 15, 3, 268-277 (2006) · Zbl 1132.65031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.