Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators

Front Cover
Princeton University Press, 2005 - Mathematics - 606 pages
0 Reviews

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.

 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

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
Toeplitz Matrices
47
7 Toeplitz matrices and boundary pseudomodes
49
32 Stability of the method of lines
302
33 Stiffness of ODEs
314
34 GKSstability of boundary conditions
322
Random Matrices
331
35 Random dense matrices
333
36 HatanoNelson matrices and localization
339
37 Random Fibonacci matrices
351
38 Random triangular matrices
359

8 Twisted Toeplitz matrices and wave packet pseudomodes
62
9 Variations on twisted Toeplitz matrices
74
Differential Operators
85
10 Differential operators and boundary pseudomodes
87
11 Variable coefficients and wave packet pseudomodes
98
12 Advectiondiffusion operators
115
13 LewyHörmander nonexistence of solutions
126
Transient Effects and Nonnormal Dynamics
133
14 Overview of 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
Fluid Mechanics
193
20 Stability of fluid flows
195
21 A model of transition to turbulence
207
22 OrrSommerfeld and Airy operators
215
23 Further problems in fluid mechanics
224
Matrix Iterations
229
24 GaussSeidel 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
Numerical Solution of Differential Equations
287
30 Spectral differentiation matrices
289
31 Nonmodal instability of PDE discretizations
295
Computation of Pseudospectra
369
39 Computation of matrix pseudospectra
371
40 Projection for largescale matrices
381
41 Other computational techniques
391
42 Pseudospectral abscissae and radii
397
43 Discretization of continuous operators
405
44 A flow chart of pseudospectra algorithms
416
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
Further Examples and Applications
499
55 Companion matrices and zeros of polynomials
501
56 Markov chains and the cutoff phenomenon
508
57 Card shuffling
519
58 Population ecology
526
59 The PapkovichFadle operator
534
60 Lasers
542
References
555
Index
597
Copyright

Common terms and phrases

References to this book

All Book Search results »

About the author (2005)

Nick Trefethen is Professor of Numerical Analysis at the University of Oxford and a Fellow of the Royal Society. During 2011 12 he served as President of SIAM.

Mark Embree is Assistant Professor of Computational and Applied Mathematics at Rice University.

Bibliographic information