Titre : | Spectra and pseudospectra : the behavior of nonnormal matrices and operators | Type de document : | document imprimé | Auteurs : | Lloyd N. Trefethen, Auteur ; M. Embree, Auteur | Editeur : | Princeton, NJ : Princeton University Press | Année de publication : | 2005 | Importance : | XVII-606 p. : ill. ; 24 cm | ISBN/ISSN/EAN : | 0-691-11946-5 | Note générale : | Hardback
| Langues : | Français (fre) | Index. décimale : | 02.60 Numerical differentiation and analysis | Résumé : | Summary: 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. 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 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
| Note de contenu : | Includes bibliographical references (p. [555]-595) and index.
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Spectra and pseudospectra : the behavior of nonnormal matrices and operators [document imprimé] / Lloyd N. Trefethen, Auteur ; M. Embree, Auteur . - Princeton, NJ : Princeton University Press, 2005 . - XVII-606 p. : ill. ; 24 cm. ISBN : 0-691-11946-5 Hardback
Langues : Français ( fre) Index. décimale : | 02.60 Numerical differentiation and analysis | Résumé : | Summary: 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. 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 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
| Note de contenu : | Includes bibliographical references (p. [555]-595) and index.
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