The Projection Methods for Computing the Pseudospectra of Large Scale Matrices

Authors: Zhengsheng Wang, Xiangyong Ji, Yong Du

Abstract:

The projection methods, usually viewed as the methods for computing eigenvalues, can also be used to estimate pseudospectra. This paper proposes a kind of projection methods for computing the pseudospectra of large scale matrices, including orthogonalization projection method and oblique projection method respectively. This possibility may be of practical importance in applications involving large scale highly nonnormal matrices. Numerical algorithms are given and some numerical experiments illustrate the efficiency of the new algorithms.

Keywords: Pseudospectra, eigenvalue, projection method, Arnoldi, IOM(q)

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1061800

References:

[1] L.N.Trefethen. Spectra and pseudospectra: The behavior of nonnormal matrices and operator. 2005, Princeton University Press, Princeton.
[2] L.N.Trefethen. Pseudospectra of linear operators. SIAM Review, 1997, 39(3):383-406.
[3] L.N.Trefethen. Computation of pseudospectra. Acta Numerica ,1999, Cambridge:Cambridge University Press, 247-295.
[4] T.G.Wright,L.N.Trefethen. Large-scale computation of pseudospectra using ARPACK and eigs. SIAM J.Sci. Comput., 2001, 23(2):591-605.
[5] K.C.Toh,L.N.Trefethen. Calculation of pseudospectra by the Arnoldi iteration. SIAM J.Sci. Comput., 1996, 17(1):1-15.
[6] Y.M.Shen, J.X.Zhao, and H.J.Fan. Properties and computations of matrix pseudospectra. Applied Mathematics and Computation, 2005, 161:385- 393.
[7] T.Braconnier, N.J.Higham. Computing the field of values and pseudospectra using the Lanczos method with continuation. BIT, 1996, 36(3):422-440.
[8] M.Bruhl. A curve tracing algorthim for computing the pseudospectrum. BIT, 1996, 36(3):441-454.
[9] P.Lancaster, P.Psarrokos. On the pseudospectra of matrix polynomials. SIAM J. Matrix Anal. Appl., 2005, 27(1):115-129.
[10] F.Tisseur, N.J.Highm. Structured pseudospectra for polynomial eigenvalue problems, with applications. SIAM J. Matrix Anal. Appl., 2001, 23(1):187-208.
[11] G.H.Golub, C.F.Van Loan. Matrix computations, 2nd ed., Johns Hopkins University Press,Baltimore,MD,1989.
[12] Y.Saad. Numerical methods for large eigenvalue problems, Manchester University Press,Manchester,UK,1992.
[13] Z.J.Bai, J.Demmel, J.Dongarra, A.Ruhe, H. van der Vorst. Templates for the solution of algebraic eigenvalue problems: A practical guide. 2000, SIAM Philadelphia.
[14] H.A. van Vorst. Computational methods for large eigenvalue problems. 2002, North-Holland(Elsevier), Amsterdam.
[15] N. M. Nachtigal, S. C. Reddy, L.N.Trefethen. A hybrid GMRES algorithm for nonsymmetric liner systems. SIAM J.Matrix Anal. Appl., 1992,13:796-825.
[16] F. chattin-Chatelin, V. Toumazou, E. Traviesas. Accuracy assessment for eigencomputations:Variety of backward errors and pseudospectra. Linear Algebra Appl.,2000,309:73-83.
[17] S.H.Lui, Computation of pseudospectra by continuation. SIAM J. Sci. Comput., 1997, 18:565-573
[18] N.J.Higham, F.Tisseur, More on pseudospectra for polynomial eigenvalue problems and applications in control theory. Linear Algebra Appl., 2002,351-352:435-453.
[19] S.M.Rump. Eigenvalues,pseudospectrum and structured perturbations. Linear Algebra and its Applications,2006,413: 567-593.
[20] L.N.Trefethen,Marco Contedini,Mark Embree. Spectra ,Pseudospectra, and Localization for Random Bidiagonal Matrices. Communications on Pure and Applied Mathematics,2000,54: 595-623.
[21] Stef Graillat. A note on structured pseudospectra. Journal of Computational and Applied Mathematics,2006,191: 68-76.
[22] T.G.Wright,L.N.Trefethen. Pseudospectra of rectangular matrices. IMA Journal of Numerical Analysis,2002,22: 501-519.