Spectra and Pseudospectra

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.

• 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 Hormander 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

Honorable Mention for the 2005 Award for Best Professional/Scholarly Book in Mathematics and Statistics, Association of American Publishers "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 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 Bottcher, Linear Algebra and its Applications "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 "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 "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