## Computation of Pseudospectra (1999)

Venue: | ACTA NUMERICA |

Citations: | 83 - 2 self |

### BibTeX

@ARTICLE{Trefethen99computationof,

author = {Lloyd N. Trefethen},

title = {Computation of Pseudospectra},

journal = {ACTA NUMERICA},

year = {1999},

volume = {8},

pages = {247--295}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

844 |
Semigroups of Linear Operators and Applications to Partial Differential Equations
- Pazy
- 1983
(Show Context)
Citation Context ...ial or a rational function. If A is an unbounded operator, as with our example (4:1), the notion of exp(tA) can be made rigorous by various methods considered in the theory of semigroups (Kato, 1976; =-=Pazy, 1983-=-). If A is far from normal, pseudospectra are likely to do better than eigenvalues alone in the second of these two roles. It is known that pseudospectra cannot in general give exact information about... |

430 |
Numerical Methods for Large Eigenvalue Problems, Revised Version
- Saad
- 2011
(Show Context)
Citation Context ...ions, Krylov subspace iterations are generally used instead to determine not all the eigenvalues but those in the portion of the complex plane considered important (Lehoucq, Sorensen, and Yang, 1998; =-=Saad, 1992-=-). For our example problem, we can get away with dimensions small enough for direct methods to be appropriate, and Figure 1 shows eigenvalues calculated by standard methods for the spectral approximat... |

327 |
Iterative Methods for Solving Linear Systems
- Greenbaum
- 1997
(Show Context)
Citation Context ...ices of dimensions in the thousands, this conclusion seems inescapable. For information on Krylov subspace iterations see (Barrett, et al., 1994), (Dongarra, Duff, Sorensen and van der Vorst, 1998), (=-=Greenbaum, 1997-=-), (Lehoucq, Sorensen, and Yang, 1998), (Saad, 1992), and (Trefethen and Bau, 1997). The very many ideas of this kind that might be considered fall roughly into two classes. One can attempt to approxi... |

221 |
ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods
- Lehoucq, Sorensen, et al.
- 1998
(Show Context)
Citation Context ...ions. For matrices of larger dimensions, Krylov subspace iterations are generally used instead to determine not all the eigenvalues but those in the portion of the complex plane considered important (=-=Lehoucq, Sorensen, and Yang, 1998-=-; Saad, 1992). For our example problem, we can get away with dimensions small enough for direct methods to be appropriate, and Figure 1 shows eigenvalues calculated by standard methods for the spectra... |

172 | Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods - Barrett, Berry, et al. - 1994 |

135 |
The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices, J.Comput.Phys.17
- Davidson
- 1975
(Show Context)
Citation Context ...rring very fast. However, here is a quick outline. One of the first papers to discuss methods of this kind was by Carpraux, Erhel, and Sadkane (1994), who used a Davidson iteration with continuation (=-=Davidson, 1975-=-). This method was subsequently parallelized by Heuveline, Philippe, and Sadkane (1997) and applied by them to a matrix of dimension N = 8192. Other contributions in this area are due to Lui (1997), B... |

133 |
Hydrodynamic stability without eigenvalues
- Trefethen, Trefethen, et al.
- 1993
(Show Context)
Citation Context ...on decouples so readily that taking advantage of multiple processors requires little effort. The first parallel computations of pseudospectra appear to have been those by A. E. Trefethen reported in (=-=Trefethen, Trefethen, Reddy, and Driscoll, 1993-=-), and subsequent contributions in this area have been due to Braconnier (1996), A. E. Trefethen, et al. (1996), Frayss'e, Giraud, and Toumazou (1996), Heuveline, Philippe, and Sadkane (1997), Trefeth... |

114 | Pseudospectra of linear operators
- Trefethen
- 1997
(Show Context)
Citation Context ...ioners for fluid mechanics (Darmofal and Schmid, 1996) ffl absorbing boundary conditions (Driscoll and Trefethen, 1996) ffl waveform relaxation (Lumsdaine and Wu, 1997) ffl Papkovitch--Fadle problem (=-=Trefethen, 1997-=-) ffl Abel integral operators (Plato, 1997) ffl Ginzburg-Landau equations (Cossu and Chomaz, 1997) ffl non-Hermitian quantum mechanics (Davies, 1998) ffl differential operators (Davies, 1998b) ffl Mar... |

113 | Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators
- Trefethen, Embree
- 2005
(Show Context)
Citation Context ...son and Trefethen, 1998) ffl Chebyshev polynomials of matrices (Toh and Trefethen, 1998) ffl flow in a pipe (Trefethen, Trefethen and Schmid, 1999) ffl ionospheric instabilities (Flaherty, Seyler and =-=Trefethen, 1999-=-) ffl lasers and optical resonators (see Section 21) 4 Lloyd N. Trefethen As is common in the history of scientific computing, this progress has been made possible by developments in both hardware and... |

102 |
Spectral approximation of linear operators
- Chatelin
(Show Context)
Citation Context ...o vary all possible aspects of the discretization systematically until one is confident that the results are correct. In some cases convergence theorems will be available to give further reassurance (=-=Chatelin, 1983-=-). Figure 1 is reminiscent of Figure 3 and other figures in the paper by Computation of Pseudospectra 11 -100-50 0 50 0 50 100 150 -100-50 0 50 0 50 100 150 N = 140 N = 160 -100-50 0 50 0 50 100 150 -... |

85 |
Perturbation theory for linear operators, 2nd éd
- Kato
- 1976
(Show Context)
Citation Context ...is a polynomial or a rational function. If A is an unbounded operator, as with our example (4:1), the notion of exp(tA) can be made rigorous by various methods considered in the theory of semigroups (=-=Kato, 1976-=-; Pazy, 1983). If A is far from normal, pseudospectra are likely to do better than eigenvalues alone in the second of these two roles. It is known that pseudospectra cannot in general give exact infor... |

63 | Three-dimensional optimal perturbations in viscous shear flow - Butler, Farrell - 1992 |

58 |
Eigenvalues and pseudo-eigenvalues of Toeplitz matrices
- Reichel, Trefethen
- 1992
(Show Context)
Citation Context ... of them, ordered by year of publication. ffl spectral methods for differential equations (Reddy and Trefethen, 1990) ffl approximate Fourier analysis (Donato, 1991) ffl matrix iterations (Nachtigal, =-=Reichel and Trefethen, 1992-=-) ffl Toeplitz matrices and operators (Reichel and Trefethen, 1992) ffl control theory (Hinrichsen and Pritchard, 1992) ffl random matrices (Trefethen, 1992) ffl Orr--Sommerfeld operator (Reddy, Schmi... |

55 |
The cutoff phenomenon in finite Markov chains
- Diaconis
- 1996
(Show Context)
Citation Context ...rically is the study of Abel integral operators by Plato (1997). My own conversion to the importance of this subject came with a study of the `cutoff phenomenon' that occurs in certain Markov chains (=-=Diaconis, 1996-=-). Diaconis and others have shown that for various random processes such as random walk on a hypercube (Diaconis, Graham, and Morrison, 1990) and riffle shuffling of a deck of cards (Bayer and Diaconi... |

50 | The Test Matrix Toolbox for Matlab, version 3.0, Numerical Analysis - Higham - 1995 |

49 |
Approximation theory and numerical linear algebra, in Algorithms for Approximation
- Trefethen
- 1990
(Show Context)
Citation Context ... 1990s, and pseudospectra have now been computed for dozens of applications. Here is a list of some of them, ordered by year of publication. ffl spectral methods for differential equations (Reddy and =-=Trefethen, 1990-=-) ffl approximate Fourier analysis (Donato, 1991) ffl matrix iterations (Nachtigal, Reichel and Trefethen, 1992) ffl Toeplitz matrices and operators (Reichel and Trefethen, 1992) ffl control theory (H... |

46 | Rational Krylov: A practical algorithm for large sparse nonsymmetric matrix pencils - Ruhe - 1998 |

45 |
Trailing the dovetail shuffle to its lair
- Bayer, Diaconis
- 1992
(Show Context)
Citation Context ...s (Diaconis, 1996). Diaconis and others have shown that for various random processes such as random walk on a hypercube (Diaconis, Graham, and Morrison, 1990) and riffle shuffling of a deck of cards (=-=Bayer and Diaconis, 1992-=-), convergence to a uniform probability distribution, when measured in a certain way, occurs not gradually but in a sudden fashion after a certain number of steps. Since the processes in question invo... |

41 | A hybrid GMRES algorithm for nonsymmetric linear systems
- Nachtigal, Reichel, et al.
- 1992
(Show Context)
Citation Context ...cussed in Section 11. Instead of computing just eigenvalues (Ritz values) in these subspaces, why not compute pseudospectra? Preliminary ideas in this direction can be found in (Freund, 1992) and in (=-=Nachtigal, Reichel, and Trefethen, 1992-=-), and the method has been explored further in (Toh and Trefethen, 1996) and in (Simoncini and Gallopoulos, 1998). The simplest procedure, as described by Toh and Trefethen, goes as follows. Starting ... |

41 | Energy growth in viscous channel flow - Reddy, Henningson - 1993 |

39 |
Asymptotic analysis of a random walk on a hypercube with many dimensions, Random Structures Algorithms 1
- Diaconis, Graham, et al.
- 1990
(Show Context)
Citation Context ...ame with a study of the `cutoff phenomenon' that occurs in certain Markov chains (Diaconis, 1996). Diaconis and others have shown that for various random processes such as random walk on a hypercube (=-=Diaconis, Graham, and Morrison, 1990-=-) and riffle shuffling of a deck of cards (Bayer and Diaconis, 1992), convergence to a uniform probability distribution, when measured in a certain way, occurs not gradually but in a sudden fashion af... |

38 | D S: Pseudospectra of the OrrSommerfeld operator - Reddy, Schmid, et al. - 1993 |

37 |
Quasi-kernel polynomials and their use in non-hermitian matrix iterations
- Freund
- 1992
(Show Context)
Citation Context ...dospectra 31 those discussed in Section 11. Instead of computing just eigenvalues (Ritz values) in these subspaces, why not compute pseudospectra? Preliminary ideas in this direction can be found in (=-=Freund, 1992-=-) and in (Nachtigal, Reichel, and Trefethen, 1992), and the method has been explored further in (Toh and Trefethen, 1996) and in (Simoncini and Gallopoulos, 1998). The simplest procedure, as described... |

36 | Pseudozeros of polynomials and pseudospectra of companion matrices
- Toh, Trefethen
- 1994
(Show Context)
Citation Context ...ifferential equations (Higham and Trefethen, 1993) ffl convection-diffusion operators (Reddy and Trefethen, 1994) ffl Hille--Phillips and Zabczyk operators (Baggett, 1994) ffl polynomial zerofinding (=-=Toh and Trefethen, 1994-=-) ffl magnetohydrodynamics (Borba, et al., 1994) ffl aerodynamic flutter (Braconnier, Chatelin, and Dunyach, 1995) ffl flow down inclined plane (Olsson and Henningson, 1995) ffl rounding error analysi... |

35 | L.: Calculation of pseudospectra by the Arnoldi iteration
- Toh, Trefethen
- 1996
(Show Context)
Citation Context ...terest can be captured by the lower-dimensional projection. This technique was perhaps first employed by Reddy, Schmid and Henningson (1993) and is described in Appendix B of that paper and in x6 of (=-=Toh and Trefethen, 1996-=-). It is elementary, but crucial in practice, and too often overlooked. We can often get an improvement in this way by a factor of 10 or more. Following the authors just cited, we first describe a pro... |

34 | Bounds for iterates, inverses, spectral variation and field of values of non–normal matrices - Henrici - 1962 |

32 | A block algorithm for matrix 1-norm estimation, with an application to 1-norm pseudospectra - Higham, Tisseur |

28 | Computing the field of values and pseudospectra using the Lanczos method with continuation - Braconnier, Higham - 1996 |

28 |
A Practical Guide to Pseudospectral Methods. Cambridge monographs on applied and computational mathematics
- Fornberg
- 1996
(Show Context)
Citation Context ...spectral methods rather than finite differences or finite elements, since spectral methods have arbitrarily high order of accuracy for smooth problems (Canuto, et al., 1988; Fornberg and Sloan, 1994; =-=Fornberg, 1996-=-). For our tutorial example, (4:1) has been discretized by a Chebyshev collocation spectral method on a finite interval [\GammaL; L] with boundary conditions u(\SigmaL) = 0. (One could work on [0; L] ... |

27 | Matrix Eigensystem Routines-Eispack Guide - Smith, Boyle, et al. - 1976 |

26 | On the separation of two matrices - Varah - 1979 |

24 | the harmonic oscillator and complex resonances
- Davies, Pseudospectra
- 1999
(Show Context)
Citation Context ...ine and Wu, 1997) ffl Papkovitch--Fadle problem (Trefethen, 1997) ffl Abel integral operators (Plato, 1997) ffl Ginzburg-Landau equations (Cossu and Chomaz, 1997) ffl non-Hermitian quantum mechanics (=-=Davies, 1998-=-) ffl differential operators (Davies, 1998b) ffl Markov chain `cutoff phenomenon' (J'onsson and Trefethen, 1998) ffl Chebyshev polynomials of matrices (Toh and Trefethen, 1998) ffl flow in a pipe (Tre... |

24 | Trefethen L N: Pseudospectra of the convection-diffusion operator
- Reddy
- 1994
(Show Context)
Citation Context ...ling line vortex flow (Schmid, et al., 1993) ffl Wiener--Hopf operators (Reddy, 1993) ffl stiffness of ordinary differential equations (Higham and Trefethen, 1993) ffl convection-diffusion operators (=-=Reddy and Trefethen, 1994-=-) ffl Hille--Phillips and Zabczyk operators (Baggett, 1994) ffl polynomial zerofinding (Toh and Trefethen, 1994) ffl magnetohydrodynamics (Borba, et al., 1994) ffl aerodynamic flutter (Braconnier, Cha... |

23 |
Spectral portraits of matrices and criteria of spectrum dichotomy
- Godunov
- 1992
(Show Context)
Citation Context ...l-border region) ffl ! 1, respectively. Pseudospectra seem to have been invented independently (with different names) at least five times: by Landau (1975, 1976, 1977), Varah (1979), Godunov, et al. (=-=Godunov, 1992-=- and 1997; Godunov, Kiriljuk, and Kostin, Computation of Pseudospectra 3 1990; Godunov, Antonov, Kiriljuk, and Kostin, 1993; Kostin, 1991), myself (1990, 1992), and Hinrichsen, Pritchard and Kelb (199... |

23 |
Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren
- Kreiss
- 1962
(Show Context)
Citation Context ..., then it can be shown by a Laplace transform that k exp(tA)k must be as large as j=ffl for some t ? 0. Given A, let the Kreiss constant K for A be defined as the supremum of this ratio over all ffl (=-=Kreiss, 1962-=-). Equivalently, K is the smallest constant such that k(z \Gamma A) \Gamma1 ksK Re z 38 Lloyd N. Trefethen for all z with Re z ? 0. Then the inequality just mentioned takes the form sup t0 k exp(tA)ks... |

22 | Linear stability analysis in the numerical solution of initial value problems, Acta Numer - Dorsselaer, Kraaijevanger, et al. - 1993 |

22 | Spectral value sets: a graphical tool for robustness analysis - Hinrichsen, Kelb - 1993 |

21 | Computation of pseudospectra by continuation
- LUI
- 1997
(Show Context)
Citation Context ...bsets of C . 12. Triangularization + inverse iteration or Lanczos A third, major new idea for speeding up the computation of pseudospectra was introduced by S.-H. Lui in an article published in 1997 (=-=Lui, 1997-=-). Lui's method is described in his own paper and elsewhere as a method of `continuation ', but his key contribution is really the technique of triangularization followed by inverse iteration or inver... |

20 | The importance of eigenvectors for local preconditioners of the Euler equations
- Darmofal, Schmid
(Show Context)
Citation Context ...n and Henningson, 1995) ffl rounding error analysis (Chaitin-Chatelin and Frayss'e, 1996) ffl reaction-convection-diffusion equations (Higham and Owren, 1996) ffl preconditioners for fluid mechanics (=-=Darmofal and Schmid, 1996-=-) ffl absorbing boundary conditions (Driscoll and Trefethen, 1996) ffl waveform relaxation (Lumsdaine and Wu, 1997) ffl Papkovitch--Fadle problem (Trefethen, 1997) ffl Abel integral operators (Plato, ... |

19 |
A review of pseudo-spectral methods for solving partial differential equations
- Fornberg, Sloan
- 1994
(Show Context)
Citation Context ... one should discretize by spectral methods rather than finite differences or finite elements, since spectral methods have arbitrarily high order of accuracy for smooth problems (Canuto, et al., 1988; =-=Fornberg and Sloan, 1994-=-; Fornberg, 1996). For our tutorial example, (4:1) has been discretized by a Chebyshev collocation spectral method on a finite interval [\GammaL; L] with boundary conditions u(\SigmaL) = 0. (One could... |

18 | Pseudospectra and singular values of large convolution operators - Böttcher - 1994 |

18 | A counterexample for two conjectures about stability - Demmel - 1987 |

17 | der Vorst - Sleijpen, Fokkema, et al. - 1996 |

17 | Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues
- Reddy, Trefethen
- 1990
(Show Context)
Citation Context ...ely in the 1990s, and pseudospectra have now been computed for dozens of applications. Here is a list of some of them, ordered by year of publication. ffl spectral methods for differential equations (=-=Reddy and Trefethen, 1990-=-) ffl approximate Fourier analysis (Donato, 1991) ffl matrix iterations (Nachtigal, Reichel and Trefethen, 1992) ffl Toeplitz matrices and operators (Reichel and Trefethen, 1992) ffl control theory (H... |

15 | Spectral portrait for nonHermitian large sparse matrices - Carpraux, Erhel, et al. - 1994 |

14 | A curve tracing algorithm for computing the pseudospectrum - Brühl - 1996 |

14 | MultiMatlab: MATLAB on multiple processors Cornell Theory Center
- Trefethen
- 1996
(Show Context)
Citation Context ...rayss'e, 1996) ffl reaction-convection-diffusion equations (Higham and Owren, 1996) ffl preconditioners for fluid mechanics (Darmofal and Schmid, 1996) ffl absorbing boundary conditions (Driscoll and =-=Trefethen, 1996-=-) ffl waveform relaxation (Lumsdaine and Wu, 1997) ffl Papkovitch--Fadle problem (Trefethen, 1997) ffl Abel integral operators (Plato, 1997) ffl Ginzburg-Landau equations (Cossu and Chomaz, 1997) ffl ... |

13 |
On Szegő’s eigenvalue distribution theory and non-Hermitian kernels
- Landau
- 1975
(Show Context)
Citation Context ...o far as I know, the first person to define the notion of pseudospectra was Henry Landau at Bell Laboratories in the 1970s, who was motivated in part by applications in lasers and optical resonators (=-=Landau 1975-=-, 1976, 1977). One of the operators that Landau considered in detail was the complex symmetric (but non-Hermitian) compact integral operator Au(x) = q iF= Z 1 \Gamma1 e \GammaiF (x\Gammay) 2 u(y)dy; (... |

13 | The Chebyshev polynomials of a matrix - Toh, Trefethen - 1998 |

12 | A note on the normwise perturbation theory for the regular generalized eigenproblem Ax = λBx, Numerical Linear Algebra With Applications 5 - Frayssé, Toumazou - 1998 |