Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators

Εξώφυλλο
Princeton University Press, 7 Αυγ 2005 - 606 σελίδες

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.

 

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Περιεχόμενα

Eigenvalues
3
Pseudospectra of matrices
12
A matrix example
22
An operator example
34
History of pseudospectra
41
Toeplitz matrices and boundary pseudomodes
49
Twisted Toeplitz matrices and wave packet pseudomodes
62
Variations on twisted Toeplitz matrices
74
GKSstability of boundary conditions
322
Random Matrices
331
Random dense matrices
333
HatanoNelson matrices and localization
339
Random Fibonacci matrices
351
Random triangular matrices
359
Computation of Pseudospectra
369
Computation of matrix pseudospectra
371

Differential operators and boundary pseudomodes
87
Variable coefficients and wave packet pseudomodes
98
Advectiondiffusion operators
115
LewyHörmander nonexistence of solutions
126
Overview of transients and pseudospectra
135
Exponentials of matrices and operators
148
Powers of matrices and operators
158
Numerical range abscissa and radius
166
The Kreiss Matrix Theorem
176
Growth bound theorem for semigroups
185
Stability of fluid flows
195
A model of transition to turbulence
207
OrrSommerfeld and Airy operators
215
Further problems in fluid mechanics
224
Matrix Iterations
229
GaussSeidel and SOR iterations
231
Upwind effects and SOR convergence
237
Krylov subspace iterations
244
Hybrid iterations
254
Arnoldi and related eigenvalue iterations
263
The Chebyshev polynomials of a matrix
278
Numerical Solution of Differential Equations
287
Spectral differentiation matrices
289
Nonmodal instability of PDE discretizations
295
Stability of the method of lines
302
Stiffness of ODEs
314
Projection for largescale matrices
381
Other computational techniques
391
Pseudospectral abscissae and radii
397
Discretization of continuous operators
405
A flow chart of pseudospectra algorithms
416
Further Mathematical Issues
421
Generalized eigenvalue problems
423
Pseudospectra of rectangular matrices
430
Do pseudospectra determine behavior?
437
Scalar measures of nonnormality
442
Distance to singularity and instability
447
Structured pseudospectra
458
Similarity transformations and canonical forms
466
Eigenvalue perturbation theory
473
Backward error analysis
485
Group velocity and pseudospectra
492
Companion matrices and zeros of polynomials
501
Markov chains and the cutoff phenomenon
508
Card shuffling
519
Population ecology
526
The PapkovichFadle operator
534
Lasers
542
References
555
Index
597
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Σχετικά με τον συγγραφέα (2005)

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.

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