This is a concise, insightful introduction to the field of numerical linear algebra. The clarity and eloquence of the presentation make it popular with teachers and students alike. The text aims to expand the reader's view of the field and to present standard material in a novel way. All of the most important topics in the field are covered with a fresh perspective, including iterative methods for systems of equations and eigenvalue problems and the underlying principles of conditioning and stability. Presentation is in the form of 40 lectures, which each focus on one or two central ideas. The unity between topics is emphasized throughout, with no risk of getting lost in details and technicalities. The book breaks with tradition by beginning with the QR factorization - an important and fresh idea for students, and the thread that connects most of the algorithms of numerical linear algebra.
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.
Preface Acknowledgments Part I: Fundamentals. Lecture 1: Matrix-Vector Multiplication Lecture 2: Orthogonal Vectors and Matrices Lecture 3: Norms Lecture 4: The Singular Value Decomposition Lecture 5: More on the SVD Part II: QR Factorization and Least Squares. Lecture 6: Projectors Lecture 7: QR Factorization Lecture 8: Gram-Schmidt Orthogonalization Lecture 9: MATLAB Lecture 10: Householder Triangularization Lecture 11: Least Squares Problems Part III: Conditioning and Stability. Lecture 12: Conditioning and Condition Numbers Lecture 13: Floating Point Arithmetic Lecture 14: Stability Lecture 15: More on Stability Lecture 16: Stability of Householder Triangularization Lecture 17: Stability of Back Substitution Lecture 18: Conditioning of Least Squares Problems Lecture 19: Stability of Least Squares Algorithms Part IV: Systems of Equations. Lecture 20: Gaussian Elimination Lecture 21: Pivoting Lecture 22: Stability of Gaussian Elimination Lecture 23: Cholesky Factorization Part V: Eigenvalues. Lecture 24: Eigenvalue Problems Lecture 25: Overview of Eigenvalue Algorithms Lecture 26: Reduction to Hessenberg or Tridiagonal Form Lecture 27: Rayleigh Quotient, Inverse Iteration Lecture 28: QR Algorithm without Shifts Lecture 29: QR Algorithm with Shifts Lecture 30: Other Eigenvalue Algorithms Lecture 31: Computing the SVD Part VI: Iterative Methods. Lecture 32: Overview of Iterative Methods Lecture 33: The Arnoldi Iteration Lecture 34: How Arnoldi Locates Eigenvalues Lecture 35: GMRES Lecture 36: The Lanczos Iteration Lecture 37: From Lanczos to Gauss Quadrature Lecture 38: Conjugate Gradients Lecture 39: Biorthogonalization Methods Lecture 40: Preconditioning Appendix: The Definition of Numerical Analysis Notes Bibliography Index.
' 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.
