Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators

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Princeton University Press, 2005/08/07 - 606 ページ
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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.

 

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目次

1 Eigenvalues
1
2 Pseudospectra of matrices
10
3 A matrix example
20
4 Pseudospectra of linear operators
25
5 An operator example
32
6 History of pseudospectra
39
Toeplitz Matrices
45
7 Toeplitz matrices and boundary pseudomodes
47
32 Stability of the method of lines
300
33 Stiffness of ODEs
312
34 GKSstability of boundary conditions
320
Random Matrices
329
35 Random dense matrices
331
36 HatanoNelson matrices and localization
337
37 Random Fibonacci matrices
349
38 Random triangular matrices
357

8 Twisted Toeplitz matrices and wave packet pseudomodes
60
9 Variations on twisted Toeplitz matrices
72
Differential Operators
83
10 Differential operators and boundary pseudomodes
85
11 Variable coefficients and wave packet pseudomodes
96
12 Advectiondiffusion operators
113
13 LewyHörmander nonexistence of solutions
124
Transient Effects and Nonnormal Dynamics
131
14 Overview of transients and pseudospectra
133
15 Exponentials of matrices and operators
146
16 Powers of matrices and operators
156
17 Numerical range abscissa and radius
164
18 The Kreiss Matrix Theorem
174
19 Growth bound theorem for semigroups
183
Fluid Mechanics
191
20 Stability of fluid flows
193
21 A model of transition to turbulence
205
22 OrrSommerfeld and Airy operators
213
23 Further problems in fluid mechanics
222
Matrix Iterations
227
24 GaussSeidel and SOR iterations
229
25 Upwind effects and SOR convergence
235
26 Krylov subspace iterations
242
27 Hybrid iterations
252
28 Arnoldi and related eigenvalue iterations
261
29 The Chebyshev polynomials of a matrix
276
Numerical Solution of Differential Equations
285
30 Spectral differentiation matrices
287
31 Nonmodal instability of PDE discretizations
293
Computation of Pseudospectra
367
39 Computation of matrix pseudospectra
369
40 Projection for largescale matrices
379
41 Other computational techniques
389
42 Pseudospectral abscissae and radii
395
43 Discretization of continuous operators
403
44 A flow chart of pseudospectra algorithms
414
Further Mathematical Issues
419
45 Generalized eigenvalue problems
421
46 Pseudospectra of rectangular matrices
428
47 Do pseudospectra determine behavior?
435
48 Scalar measures of nonnormality
440
49 Distance to singularity and instability
445
50 Structured pseudospectra
456
51 Similarity transformations and canonical forms
464
52 Eigenvalue perturbation theory
471
53 Backward error analysis
483
54 Group velocity and pseudospectra
490
Further Examples and Applications
497
55 Companion matrices and zeros of polynomials
499
56 Markov chains and the cutoff phenomenon
506
57 Card shuffling
517
58 Population ecology
524
59 The PapkovichFadle operator
532
60 Lasers
540
References
553
Index
595
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著者について (2005)

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.

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