# Numerical Linear Algebra

SIAM, Jun 1, 1997 - Mathematics - 361 pages
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.

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This is an excellent book on numerical linear algebra, a very good textbook for a senior undergraduate course. I like the writing style and have been enjoying the reading. Very often it explains "why", not just gives "what" and "how". Highly recommended.

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The strength of this book is in the conceptual discussions. This isn't the book to use to learn the mechanics of the methods described. It's one of my three favorite numerical linear algebra books.

### Contents

 Fundamentals 8 MatrixVector Multiplication 9 Orthogonal Vectors and Matrices 11 Norms 17 The Singular Value Decomposition 25 More on the SVD 32 QR Factorization and Least Squares 39 Projectors 41
 Cholesky Factorization 172 Eigenvalues 179 Eigenvalue Problems 181 Overview of Eigenvalue Algorithms 190 Reduction to Hessenberg or Tridiagonal Form 196 Rayleigh Quotient Inverse Iteration 202 QR Algorithm without Shifts 211 QR Algorithm with Shifts 219

 QR Factorization 48 Gram Schmidt Orthogonalization 56 MATLAB 63 Householder Triangularization 69 Least Squares Problems 77 Conditioning and Stability 87 Conditioning and Condition Numbers 89 Floating Point Arithmetic 97 Stability 102 More on Stability 108 Stability of Householder Triangularization 114 Stability of Back Substitution 121 Conditioning of Least Squares Problems 129 Stability of Least Squares Algorithms 137 Systems of Equations 145 Gaussian Elimination 147 Pivoting 155 Stability of Gaussian Elimination 163
 Other Eigenvalue Algorithms 225 Computing the SVD 234 Iterative Methods 241 Overview of Iterative Methods 243 The Arnoldi Iteration 250 How Arnoldi Locates Eigenvalues 257 GMRES 266 The Lanczos Iteration 276 From Lanczos to Gauss Quadrature 285 Conjugate Gradients 293 Biorthogonalization Methods 303 Preconditioning 313 Appendix The Definition of Numerical Analysis 321 Notes 329 Bibliography 343 Index 353 Copyright