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Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators [Hardcover]

Lloyd N. Trefethen , Mark Embree

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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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