"Numerical Linear Algebra" by Lloyd N. Trefethen and David Bau, III
"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.
Numerical Linear Algebra is a concise, insightful, and elegant introduction
to the field of numerical linear algebra. Designed for use as a stand-alone
textbook in a one-semester, graduate-level course in the topic, it has
already been class-tested by MIT and Cornell graduate students from all
fields of mathematics, engineering, and the physical sciences. The authors'
clear, inviting style and evident love of the field, along with their
eloquent presentation of the most fundamental ideas in numerical linear
algebra, make it popular with teachers and students alike.
Numerical Linear Algebra aims to expand the reader's view of the field and to
present the core, standard material in a novel way. It is a perfect companion
volume to the encyclopedic treatment of the topic that already exists in Golub
and Van Loan's now-classic Matrix Computations. All of the most important
topics in the field, including iterative methods for systems of equations and
eigenvalue problems and the underlying principles of conditioning and
stability, are covered. Trefethen and Bau offer a fresh perspective on these
and other topics, such as an emphasis on connections with polynomial
approximation in the complex plane.
Numerical Linear Algebra is presented in the form of 40 lectures, each of
which focuses on one or two central ideas. Throughout, the authors emphasize
the unity between topics, never allowing the reader to get lost in details and
technicalities. The book breaks with tradition by beginning not with Gaussian
elimination, but with the QR factorization--a more important and fresher idea
for students, and the thread that connects most of the algorithms of numerical
linear algebra, including methods for least squares, eigenvalue, and singular
value problems, as well as iterative methods for all of these and for systems
of equations.
Students will benefit from the many exercises that follow each lecture.
Well-chosen references and extensive notes enrich the presentation and
provide historical context.
OUTLINE OF TEXT:
Contents; 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.
Audience: Written on the graduate or advanced undergraduate level, this book
can be used widely for teaching. Professors looking for an elegant
presentation of the topic will find it an excellent teaching tool for a
one-semester graduate or advanced undergraduate course. A major contribution
to the applied mathematics literature, most researchers in the field will
consider it a necessary addition to their personal collections.
About the Authors: Lloyd N. Trefethen is a Professor of Computer Science at
Cornell University. 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 Developer at Microsoft Corporation,
where he works in the Internet Division.
Available May 1997 / xii + 361 pages / Softcover / ISBN 0-89871-361-7 / List
price $34.50 / SIAM Member Price $27.60 / Order Code OT50
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