Spectra and pseudospectra : the behavior of nonnormal matrices and operators


Spectra and pseudospectra : the behavior of nonnormal matrices and operators

Lloyd N. Trefethen and Mark Embree

Princeton University Press, c2005

大学図書館所蔵 件 / 23



Includes bibliographical references (p. [555]-595) and index



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.


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

「Nielsen BookData」 より