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Numerical Linear Algebra [Paperback]

Lloyd N. Trefethen , David Bau III
4.2 out of 5 stars  See all reviews (19 customer reviews)

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Book Description

June 1, 1997 0898713617 978-0898713619
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.

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.


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Numerical Linear Algebra + Matrix Computations (Johns Hopkins Studies in the Mathematical Sciences) + Applied Numerical Linear Algebra
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Editorial Reviews

Review

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.

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 at Urbana-Champaign

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 soufflé --- tasteful and very light. --Gene Golub, Stanford University.

Book Description

This is a concise, insightful introduction to the field of numerical linear algebra. 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.

Product Details

  • Paperback: 373 pages
  • Publisher: SIAM: Society for Industrial and Applied Mathematics (June 1, 1997)
  • Language: English
  • ISBN-10: 0898713617
  • ISBN-13: 978-0898713619
  • Product Dimensions: 10.1 x 7 x 0.8 inches
  • Shipping Weight: 1.4 pounds (View shipping rates and policies)
  • Average Customer Review: 4.2 out of 5 stars  See all reviews (19 customer reviews)
  • Amazon Best Sellers Rank: #191,591 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
49 of 54 people found the following review helpful
5.0 out of 5 stars Excellent: the best book I have seen on the subject. February 24, 1999
By Tia
Format:Paperback
This book should be required reading for anyone interested in computational numerics, especially those who are starting in the field. The authors concentrate on the few fundamental topics that underlie and unite the subject. The presentation, while rigorous, is simple, clear and friendly. The authors follow a logical thread and eliminate unnecessary and disorienting aspects that plague other books on the subject. It is easy to pick up the book, read several chapters at a stretch without looking up, and come away with new insights. Unquestionably the most valuable book I have read to date on the subject.
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23 of 24 people found the following review helpful
5.0 out of 5 stars Excellent...with a few caveats May 13, 2005
Format:Paperback
This book on Linear Algebra is excellent. In particular chapters seven through thirty (as far as I have read) are great for self-directed study. However, I found chapters one through six ( through Projectors) a bit terse. Therefore I would highly recommend this book for self-study ONLY IF you already have a good idea of the concept of basic linear algebra including matrix norms, the singular value decomposition, and projectors, and also the correct way to perform a proof...and by a "good idea" I mean you already know how to use these ideas in a practical way. Otherwise, you should only use this book if you have a truly good instructor to guide you through the early material. I started out taking a class using this book four years ago from a poor instructor, and I and the entire class, as far as I could tell from casual conversation, were completely lost. I dropped the class and retook it just recently with an excellent instructor. Her help and insight made a world of difference. It will also help to have a copy of "Matrix Computations" by Golub and Van Loan for reference, especially when you get to the later chapters and eigenproblems.
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11 of 11 people found the following review helpful
5.0 out of 5 stars A must for computational mathematicians June 24, 2002
Format:Hardcover
The chapters on numerical stability of algorithms and conditioning of numerical problems are excellent. While the focus is of course linear algebra, these principles can be readily extended to all computational mathematics. If you regularly use computational methods and have not yet studied elementary error analysis, this book may revolutionize how you perceive numerical problems.
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6 of 6 people found the following review helpful
5.0 out of 5 stars great math text July 24, 2006
Format:Paperback
I used this book at NYU in a graduate class on numerical linear algebra and it was great. The book is incredibly clear, starts from the basics and just goes from there. You won't be lost or feel like it has too little (and I usually have one of those two feelings about a math textbook).

The book is focused around matrix decompositions and does quite a bit of theoretical matrix algebra before it gets into accurate computation of decompositions, what this means and how various algorithms achieve it.

The theorems are clear and the proofs concise and easy to read.
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12 of 15 people found the following review helpful
Format:Paperback|Verified Purchase
This is a fairly good, concise analysis of numerical linear algebra. It covers topics in a logical manner and overall has fairly good explanations. However, I feel that it has a few notable shortcomings.

Although explanations are fairly good, I found that they were too short. A minimum amount of detail is given, and there are very few examples. Also, there are very few end-of-chapter problems, and the ones given are almost all quite difficult and there are no answers available.

For these reasons I would only recommend this book for people with a strong background in linear algebra. I was an engineering undergraduate with a limited knowledge of linear algebra when I read it, but an upper-year math major or a math graduate would probably find this book to be ok, as long as their course instructor gives many supplementary resources, such as practice problems.
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6 of 7 people found the following review helpful
4.0 out of 5 stars Excellent for its purpose August 30, 2010
By Xiao Hu
Format:Paperback|Verified Purchase
I am reviewing the book by Trefethen not by Demmel. There has been some confusion about these two books.

Many people commented that this book is logic and easy to read. Compared with some other books, this is true. However, the perquisite for this book is a sound understanding of linear algebra. Without that, you will need to be a math genius to find this book easy to read. So, this is an excellent book for your (at least my) first exposure to numerical linear algebra. It picks up about 40 important topics and cover them in such details that are not too overwhelming.

With the above positive side, I wish the author could expand the book and cover more topics or some topics in greater details. This book is good for an one semester course (Note that it has only just more than 300 pages for the main content). Many books are easily enough for two courses if not more. So, as a text book, this is excellent. But as a reference, you will need another book to go with it. Unfortunately, other books are not so easy to read.

By the way, for CG method the book is not excellent. You will need the article by Jonathan Richard Shewchuk. You can find it online. This is the best for CG.
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3 of 3 people found the following review helpful
5.0 out of 5 stars A math textbook that's a joy to read June 4, 2013
By K. Long
Format:Paperback|Verified Purchase
Face it, most math textbooks are awful: boring to read, not much insight, little more than a compendium of definitions, theorems, proofs, and examples. Trefethen and Bau is an exception to that rule. Indeed, the field of numerical linear algebra is unusual in having available several top-notch textbooks: Golub and Van Loan, Stewart's two volumes, Saad's books on iterative methods, Demmel's introduction, Watkins' undergraduate level treatment, and T&B. All of these are excellent (and any student in numerical analysis should delve into all of them) but to my tastes T&B and Stewart are the standouts for insight and simply being fun to read.

If you're a student using T&B in a course, to use it effectively you need to understand that T&B is a book to be read carefully for understanding; it's not a typical textbook suited only for "mining" for examples and solutions to homework problems. My students have sometimes complained -- accurately -- that T&B is short on details and worked examples, and many of the proofs are just sketches. But that's a feature, not a bug: you can learn much by filling in the missing steps. This is book for reading, so take the time to read it, to think about what you've read, and to fill in the gaps; it's worth it. If you need some worked examples, Watkins has them in great detail and would be a good supplement to T&B (though see the caveat below).

The only minor gripe I have about T&B is that the order of topics (QR before LU before Cholesky) is unusual, which makes it a little awkward to coordinate with other books such as Watkins which do Cholesky before LU before QR.
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Most Recent Customer Reviews
2.0 out of 5 stars Not enough for a solid preparation
The book has practically no examples and exercises solved. The lectures are very schematic and take a lot of notions for granted. Read more
Published 5 months ago by Francy_B
1.0 out of 5 stars All theory, few examples, and all Matlab.
I'm using this textbook for a computational math class I'm taking at Illinois Institute of Technology. Read more
Published 8 months ago by Douglas Lewit
5.0 out of 5 stars Definitely the best!!!
The best numerical linear algebra text book ever! The book is organized into various chapters each for one lecture. Read more
Published 11 months ago by Amirhossein Arzani
5.0 out of 5 stars Read it like Rudin's Analysis
This book will teach you nothing if you expect it to tell you everything. Just like Rudin's Analysis textbook, this book needs to be read slowly and thoughtfully. Read more
Published 19 months ago by Tyson A. Dilorenzo
4.0 out of 5 stars This is a good book
The book is still in a very good condition. It was shipped very fast, and I got the book three days after class started. However, the book is really hard to understand. Read more
Published on September 7, 2011 by Jade
5.0 out of 5 stars Arrive at perfect condition!
The book arrive in time and is in perfect condition(just like its online description "Like New"). I've read chapters of this book and find that it is very suitable for reader who... Read more
Published on September 4, 2011 by cwei
5.0 out of 5 stars It couldn't be better
Beautiful! Very simply, if you want to have an insight on linear algebraic procedures, and why this and that happens so and so, this is the book. Read more
Published on January 9, 2009 by Ali Civril
5.0 out of 5 stars Great Book for Self-Learning
I am a second year PhD student in Operations Research and for long I had been looking for a book in linear algebra to help me learn it myself (as I see that I need it no matter... Read more
Published on February 13, 2008 by gumma60
4.0 out of 5 stars Must be strong in Linear Algebra to use this!
The reader msut have a strong grasp on linear algebra before using this book. Many algorithms are written in pseudo-code which is nice, but sometimes important details lack. Read more
Published on January 5, 2007 by Eddie Van Halen
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