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Spectra and Pseudospectra : The Behavior of Nonnormal Matrices and Operators - Lloyd N. Trefethen

Spectra and Pseudospectra

The Behavior of Nonnormal Matrices and Operators

Hardcover Published: 7th August 2005
ISBN: 9780691119465
Number Of Pages: 624

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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.

Industry Reviews

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 B?ttcher, 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

Preface xiiiAcknowledgments xvI. Introduction 11.Eigenvalues 32.Pseudospectra of matrices 123.A matrix example 224.Pseudospectra of linear operators 275.An operator example 346.History of pseudospectra 41II. Toeplitz Matrices 477.Toeplitz matrices and boundary pseudomodes 498.Twisted Toeplitz matrices and wave packet pseudomodes 629.Variations on twisted Toeplitz matrices 74III. Differential Operators 8510.Differential operators and boundary pseudomodes 8711.Variable coeffcients and wave packet pseudomodes 9812.Advection-diffusion operators 11513.Lewy Hormander nonexistence of solutions 126IV. Transient Effects and Nonnormal Dynamics 13314.Overviewof transients and pseudospectra 13515.Exponentials of matrices and operators 14816.Powers of matrices and operators 15817.Numerical range, abscissa, and radius 16618.The Kreiss Matrix Theorem 17619.Growth bound theorem for semigroups 185V. Fluid Mechanics 19320.Stability of fluid flows 19521.A model of transition to turbulence 20722.Orr--Sommerfeld and Airy operators 21523.Further problems in fluid mechanics 224VI. Matrix Iterations 22924.Gauss--Seidel and SOR iterations 23125.Upwind effects and SOR convergence 23726.Krylov subspace iterations 24427.Hybrid iterations 25428.Arnoldi and related eigenvalue iterations 26329.The Chebyshev polynomials of a matrix 278VII. Numerical Solution of Differential Equations 28730.Spectral differentiation matrices 28931.Nonmodal instability of PDE discretizations 29532.Stability of the method of lines 30233.Stiffness of ODEs 31434.GKS-stability of boundary conditions 322VIII. Random Matrices 33135.Random dense matrices 33336.Hatano--Nelson matrices and localization 33937.Random Fibonacci matrices 35138.Random triangular matrices 359IX. Computation of Pseudospectra 36939.Computation of matrix pseudospectra 37140.Projection for large-scale matrices 38141.Other computational techniques 39142.Pseudospectral abscissae and radii 39743.Discretization of continuous operators 40544.A flowchart of pseudospectra algorithms 416X. Further Mathematical Issues 42145.Generalized eigenvalue problems 42346.Pseudospectra of rectangular matrices 43047.Do pseudospectra determine behavior? 43748.Scalar measures of nonnormality 44249.Distance to singularity and instability 44750.Structured pseudospectra 45851.Similarity transformations and canonical forms 46652.Eigenvalue perturbation theory 47353.Backward error analysis 48554.Group velocity and pseudospectra 492XI. Further Examples and Applications 49955.Companion matrices and zeros of polynomials 50156.Markov chains and the cutoff phenomenon 50857.Card shuffing 51958.Population ecology 52659.The Papkovich--Fadle operator 53460.Lasers 542References 555Index 597

ISBN: 9780691119465
ISBN-10: 0691119465
Series: International Studen
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 624
Published: 7th August 2005
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 4.24
Weight (kg): 1.01

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