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
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