Calculation of Pseudospectra by the Arnoldi Iteration

The Arnoldi iteration, usually viewed as a method for calculating eigenvalues, can also be used to estimate pseudospectra. This possibility may be of practical importance, because in applications involving highly nonnormal matrices or operators, such as hydrodynamic stability, pseudospectra may be physically more significant than spectra.

[1]  R. Freund.  Quasi-kernel polynomials and their use in non-Hermitian matrix iterations , 1992 .

[2]  E. Davidson.  The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices , 1975 .

[3]  Danny C. Sorensen,et al.  Implicit Application of Polynomial Filters in a k-Step Arnoldi Method , 1992, SIAM J. Matrix Anal. Appl..

[4]  Karl Meerbergen,et al.  Matrix transformations for computing rightmost eigenvalues of large sparse nonsymmetric matrices , 1994 .

[5]  Y. Saad,et al.  Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems , 1984 .

[6]  H. A. V. D. Vorsty.  University Utrecht a Generalized Jacobi-davidson Iteration Method for Linear Eigenvalue Problems a Generalized Jacobi-davidson Iteration Method for Linear Eigenvalue Problems , 1994 .

[7]  Diem Ho.  Tchebychev acceleration technique for large scale nonsymmetric matrices , 1989 .

[8]  Y. Saad.  Analysis of some Krylov subspace approximations to the matrix exponential operator , 1992 .

[9]  Anne E. Trefethen,et al.  Hydrodynamic Stability Without Eigenvalues , 1993, Science.

[10]  W. Arnoldi.  The principle of minimized iterations in the solution of the matrix eigenvalue problem , 1951 .

[11]  Lloyd N. Trefethen,et al.  A Hybrid GMRES Algorithm for Nonsymmetric Linear Systems , 1992, SIAM J. Matrix Anal. Appl..

[12]  O. Nevanlinna.  Convergence of Iterations for Linear Equations , 1993 .

[13]  L. Böberg,et al.  Onset of Turbulence in a Pipe , 1988 .

[14]  W. S. Edwards,et al.  Krylov methods for the incompressible Navier-Stokes equations , 1994 .

[15]  Kathryn M. Butler,et al.  Three‐dimensional optimal perturbations in viscous shear flow , 1992 .

[16]  Bernard Philippe,et al.  The Davidson Method , 1994, SIAM J. Sci. Comput..

[17]  Thomas A. Manteuffel,et al.  On hybrid iterative methods for nonsymmetric systems of linear equations , 1996 .

[18]  Axel Ruhe.  Rational Krylov algorithms for nonsymmetric eigenvalue problems. II. matrix pairs , 1994 .

[19]  C. Lubich,et al.  On Krylov Subspace Approximations to the Matrix Exponential Operator , 1997 .

[20]  Dan S. Henningson,et al.  Pseudospectra of the Orr-Sommerfeld Operator , 1993, SIAM J. Appl. Math..

[21]  Lloyd N. Trefethen,et al.  Pseudospectra of the Convection-Diffusion Operator , 1993, SIAM J. Appl. Math..

[22]  S. C. Reddy,et al.  Energy growth in viscous channel flows , 1993, Journal of Fluid Mechanics.

[23]  A. Spence,et al.  Shift-invert and Cayley transforms for detection of rightmost eigenvalues of nonsymmetric matrices , 1994 .

[24]  Meza.  A modification to the GMRES method for ill-conditioned linear systems , 1995 .

[25]  Jennifer A. Scott,et al.  An Arnoldi code for computing selected eigenvalues of sparse, real, unsymmetric matrices , 1995, TOMS.