Numerical Linear Algebra


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Table of Contents

Preface; Part I. Fundamental: 1. Matrix-vector multiplication; 2. Orthogonal vectors and matrices; 3. Norms; 4. The singular value decomposition; 5. More on the SVD; Part II. QR Factorization and Least Squares: 6. Projectors; 7. QR factorization; 8. Gram-Schmidt orthogonalization; 9. MATLAB; 10. Householder triangularization; 11. Least squares problems; Part III. Conditioning and Stability: 12. Conditioning and condition numbers; 13. Floating point arithmetic; 14. Stability; 15. More on stability; 16. Stability of householder triangularization; 17. Stability of back substitution; 18. Conditioning of least squares problems; 19. Stability of least squares algorithms; Part IV. Systems of Equations: 20. Gaussian elimination; 21. Pivoting; 22. Stability of Gaussian elimination; 23. Cholesky factorization; Part V. Eigenvalues: 24. Eigenvalue problems; 25. Overview of Eigenvalue algorithms; 26. Reduction to Hessenberg or tridiagonal form; 27. Rayleigh quotient, inverse iteration; 28. QR algorithm without shifts; 29. QR algorithm with shifts; 30. Other Eigenvalue algorithms; 31. Computing the SVD; Part VI. Iterative Methods: 32. Overview of iterative methods; 33. The Arnoldi iteration; 34. How Arnoldi locates Eigenvalues; 35. GMRES; 36. The Lanczos iteration; 37. From Lanczos to Gauss quadrature; 38. Conjugate gradients; 39. Biorthogonalization methods; 40. Preconditioning; Appendix; Notes; Bibliography; Index.

Promotional Information

Numerical Linear Algebra is a concise, insightful, and elegant introduction to the field of numerical linear algebra.

About the Author

Lloyd N. Trefethen is a Professor of Computer Science at Cornell University. Starting October 1, 1997, he will be the Professor of Numerical Analysis at Oxford University in England. He has won teaching awards at both MIT and Cornell. In addition to editorial positions on such journals as SIAM Journal on Numerical Analysis, Journal of Computational and Applied Mathematics, Numerische Mathematik, and SIAM Review, he has been an invited lecturer at two dozen international conferences. While at Cornell, David Bau was a student of Trefethen. He is currently a Software Engineer at Google Inc., where he helped develop Google Talk, Google's IM and VOIP service.


' The authors are to be congratulated on producing a fresh and lively introduction to a fundamental area of numerical analysis.' G. W. Stewart, Mathematics of Computation '...Each lecture in the textbook is pleasantly written in a conversational style and concludes with a set of related exercises. This low-cost textbook emphasizes many important and relevant topics in numerical linear algebra and seems ideal for a graduate course as long as it is accompanied by a textbook with more mathematical details.' Ricardo D. Fierro, SIAM Review 'Trefethen and Bau clear the dark clouds from numerical problems associated with factoring matrices, solving linear equations, and finding eigenvalues.' P. Cull, CHOICE 'Just exactly what I might have expected - an absorbing look at the familiar topics through the eyes of a master expositor. I have been reading it and learning a lot.' Paul Saylor, University of Illinois, Urbana-Champaign 'A beautifully written textbook offering a distinctive and original treatment. It will be of use to all who teach or study the subject.' Nicholas J. Higham, Professor of Applied Mathematics, University of Manchester "This is a beautifully written book which carefully brings to the reader the important issues connected with the computational issues in matrix computations. The authors show a broad knowledge of this vital area and make wonderful connections to a variety of problems of current interest. The book is like a delicate souffle --- tasteful and very light." -Gene Golub, Stanford University. "I have used Numerical Linear Algebra in my introductory graduate course and I have found it to be almost the perfect text to introduce mathematics graduate students to the subject. I like the choice of topics and the format: a sequence of lectures. Each chapter (or lecture) carefully builds upon the material presented in previous chapters, providing new concepts in a very clear manner. Exercises at the end of each chapter reinforce the concepts, and in some cases introduce new ones. ...The emphasis is on the mathematics behind the algorithms, in the understanding of why the algorithms work. ...The text is sprinkled with examples and explanations, which keep the student focused." -Daniel B. Szyld, Department of Mathematics, Temple University. "...This is an ideal book for a graduate course in numerical linear algebra (either in mathematics or in computer science departments); it presents the topics in such a way that background material comes along with the course. ...I will use it again next time I teach this course!" -Suely Oliveira, Texas A&M University.

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