Product Description: 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.
Good reference I have been using this textbook for my Numerical Linear Algebra class, and it have been a great help, actually is a good complement to Numerical Linear Algebra by Lloyd N. Trefethen and David Bau III. I totally recommend it.
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 what research I want to do. It's just a good tool to know). One of my friends recommended this book to me, I got it and I am very happy with it. The book is great in different ways: -it is in the form of short lectures and for me who wants to learn linear algebra step by step, this is a perfect approach. You will have a 5-6 page lecture so whenever you start, you are set to finish that lecture. -It gives you intuition and understanding about what is really happenning geometrically which is amazing. To me, it is very important to have the "feeling" of what is happening because it is only then that you can think about bringing your real problem in this framework. -The examples in lectures clarify the subject while exercises give you a chance to learn even more. If you are new to linear algebra or know it but want to refresh your mind on intuitions and systematic thinking, I highly recommend this book.
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. I used this book as a required text in a graduate level course.
Fantastic book, with great insight I can't speak to the entire book, as I've only made significant use of the section of matrix solvers. Having said that, his explanation of Krylov methods was the most clear and well organized I've ever seen. His book is the first I've seen that so nicely ties together all such methods. It's true that his book is probably not going to be enough if you are planning to focus on this as your research topic. But for those of us who simply need to apply the field to their research, it is the best book I've found, and he goes out of his way to be helpful to the practitioner, a rare thing in a math book. (For example, he has a wonderful flowchart in Chapter 6 providing a rough guideline for selecting a linear system solver based on the properties of one's problem.)
great math text 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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