Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators

by Lloyd N. Trefethen, Mark Embree

Overview

Pure and applied mathematicians, physicists, scientists, and engineers use matrices and operators and their eigenvalues in quantum mechanics, fluid mechanics, structural analysis, acoustics, ecology, numerical analysis, and many other areas. However, in some applications the usual analysis based on eigenvalues fails. For example, eigenvalues are often ineffective for analyzing dynamical systems such as fluid flow, Markov chains, ecological models, and matrix iterations. That's where this book comes in.


This is the authoritative work on nonnormal matrices and operators, written by the authorities who made them famous. Each of the sixty sections is written as a self-contained essay. Each document is a lavishly illustrated introductory survey of its topic, complete with beautiful numerical experiments and all the right references. The breadth of included topics and the numerous applications that provide links between fields will make this an essential reference in mathematics and related sciences.

Product Details

ISBN-13:	9780691119465
Publisher:	Princeton University Press
Publication date:	08/07/2005
Pages:	624
Product dimensions:	6.00(w) x 9.25(h) x (d)

About the Author

Lloyd N. Trefethen is Professor of Numerical Analysis and Head of the Numerical Analysis Group at the University of Oxford. Mark Embree is Assistant Professor of Computational and Applied Mathematics at Rice University.

Table of Contents

Preface xiii
Acknowledgments xv

I. Introduction 1

1.Eigenvalues 3
2.Pseudospectra of matrices 12
3.A matrix example 22
4.Pseudospectra of linear operators 27
5.An operator example 34
6.History of pseudospectra 41
II. Toeplitz Matrices 47
7.Toeplitz matrices and boundary pseudomodes 49
8.Twisted Toeplitz matrices and wave packet pseudomodes 62
9.Variations on twisted Toeplitz matrices 74

III. Differential Operators 85

10.Differential operators and boundary pseudomodes 87
11.Variable coeffcients and wave packet pseudomodes 98
12.Advection-diffusion operators 115
13.Lewy Hörmander nonexistence of solutions 126

IV. Transient Effects and Nonnormal Dynamics 133

14.Overviewof transients and pseudospectra 135
15.Exponentials of matrices and operators 148
16.Powers of matrices and operators 158
17.Numerical range, abscissa, and radius 166
18.The Kreiss Matrix Theorem 176
19.Growth bound theorem for semigroups 185

V. Fluid Mechanics 193

20.Stability of fluid flows 195
21.A model of transition to turbulence 207
22.Orr—Sommerfeld and Airy operators 215
23.Further problems in fluid mechanics 224

VI. Matrix Iterations 229

24.Gauss—Seidel and SOR iterations 231
25.Upwind effects and SOR convergence 237
26.Krylov subspace iterations 244
27.Hybrid iterations 254
28.Arnoldi and related eigenvalue iterations 263
29.The Chebyshev polynomials of a matrix 278

VII. Numerical Solution of Differential Equations 287

30.Spectral differentiation matrices 289
31.Nonmodal instability of PDE discretizations 295
32.Stability of the method of lines 302
33.Stiffness of ODEs 314
34.GKS-stability of boundary conditions 322

VIII. Random Matrices 331

35.Random dense matrices 333
36.Hatano—Nelson matrices and localization 339
37.Random Fibonacci matrices 351
38.Random triangular matrices 359

IX. Computation of Pseudospectra 369

39.Computation of matrix pseudospectra 371
40.Projection for large-scale matrices 381
41.Other computational techniques 391
42.Pseudospectral abscissae and radii 397
43.Discretization of continuous operators 405
44.A flowchart of pseudospectra algorithms 416

X. Further Mathematical Issues 421

45.Generalized eigenvalue problems 423
46.Pseudospectra of rectangular matrices 430
47.Do pseudospectra determine behavior? 437
48.Scalar measures of nonnormality 442
49.Distance to singularity and instability 447
50.Structured pseudospectra 458
51.Similarity transformations and canonical forms 466
52.Eigenvalue perturbation theory 473
53.Backward error analysis 485
54.Group velocity and pseudospectra 492

XI. Further Examples and Applications 499

55.Companion matrices and zeros of polynomials 501
56.Markov chains and the cutoff phenomenon 508
57.Card shuffing 519
58.Population ecology 526
59.The Papkovich—Fadle operator 534
60.Lasers 542

References 555
Index 597

Editorial Reviews

Honorable Mention for the 2005 Award for Best Professional/Scholarly Book in Mathematics and Statistics, Association of American Publishers

From the Publisher

"This is the authoritative work on nonnormal matrices and operators, written by the authorities who made them famous. Their decision to prepare this volume is indeed a momentous event."

Current Engineering Practice

The book has been written at a level to be accessible to a wide audience of students of the applied sciences. The subject matter has been carefully referenced, Many illustrations are provided showing an amazing diversity of spectra end pseudospectra. A detailed chapter is provided for those who wish to generate software to approximate the spectrum and pseudospectrum in a particular application.

Zentralblatt Math - J. B. Butler

We suggest . . . strongly that the book be opened and read. . . . One will profit by getting considerable insight into a rich variety of phenomena and being acquainted with a large number of beautiful mathematical thoughts.

Monatshefte für Mathematik - H. Muthsam Wien

The book contains good introductions to a wide variety of application areas and research topics and is a very appropriate text for a graduate-level seminar. For those interested in pursuing these topics further, the bibliography is an absolute treasure!. . . It is an invaluable resource for anyone working in the area of nonnormal matrices and linear operators or for anyone involved in an application where nonnormality is important.

SIAM Review - Anne Greenbaum

The book has been written at a level to be accessible to a wide audience of students of the applied sciences. The subject matter has been carefully referenced, Many illustrations are provided showing an amazing diversity of spectra end pseudospectra. A detailed chapter is provided for those who wish to generate software to approximate the spectrum and pseudospectrum in a particular application.

Zentralblatt Math - J.B. Butler

The overall organization of the book is a methodological masterpiece. The splitting of the huge amount of material into sixty short self-contained essays is extremely reader-friendly. The writing is extremely lucid and intriguing. . . . The many figures illustrate the mathematics in an unusually fascinating way.

Linear Algebra and its Applications - Albrecht Bottcher

The book has been written at a level to be accessible to a wide audience of students of the applied sciences. The subject matter has been carefully referenced, Many illustrations are provided showing an amazing diversity of spectra end pseudospectra. A detailed chapter is provided for those who wish to generate software to approximate the spectrum and pseudospectrum in a particular application.
— J. B. Butler

Zentralblatt MATH Database

We suggest . . . strongly that the book be opened and read. . . . One will profit by getting considerable insight into a rich variety of phenomena and being acquainted with a large number of beautiful mathematical thoughts.
— H. Muthsam Wien

Monatshefte fur Mathematik fur Mathematik

The book contains good introductions to a wide variety of application areas and research topics and is a very appropriate text for a graduate-level seminar. For those interested in pursuing these topics further, the bibliography is an absolute treasure!. . . It is an invaluable resource for anyone working in the area of nonnormal matrices and linear operators or for anyone involved in an application where nonnormality is important.
— Anne Greenbaum

SIAM Review

The overall organization of the book is a methodological masterpiece. The splitting of the huge amount of material into sixty short self-contained essays is extremely reader-friendly. The writing is extremely lucid and intriguing. . . . The many figures illustrate the mathematics in an unusually fascinating way.
— Albrecht Böttcher

Linear Algebra and its Applications

The book has been written at a level to be accessible to a wide audience of students of the applied sciences. The subject matter has been carefully referenced, Many illustrations are provided showing an amazing diversity of spectra end pseudospectra. A detailed chapter is provided for those who wish to generate software to approximate the spectrum and pseudospectrum in a particular application.
— J. B. Butler

Zentralblatt Math