Numerical matrix analysis : linear systems and least squares

This self-contained textbook presents matrix analysis in the context of numerical computation with numerical conditioning of problems and numerical stability of algorithms at the forefront. Using a unique combination of numerical insight and mathematical rigor, it advances readers understanding of two phenomena: sensitivity of linear systems and least squares problems, and numerical stability of algorithms. This book differs in several ways from other numerical linear algebra texts. It offers a systematic development of numerical conditioning; a simplified concept of numerical stability in exact arithmetic; simple derivations; a high-level view of algorithms; and results for complex matrices. The material is presented at a basic level, emphasizing ideas and intuition, and each chapter offers simple exercises for use in the classroom and more challenging exercises for student practice. Audience: This book is intended for first-year graduate students in engineering, operations research, computational science, and all areas of mathematics. It also is appropriate for self-study. Contents: Preface; Introduction; Chapter 1: Matrices; Chapter 2: Sensitivity, Errors, and Norms; Chapter 3: Linear Systems; Chapter 4: Singular Value Decomposition; Chapter 5: Least Square Problems; Chapter 6: Subspaces; Index.

The purpose of this book is to promote understanding of two phenomena: sensitivity of linear systems and least squares problems, and numerical stability of algorithms. Sensitivity and stability are analyzed as mathematical properties, without reference to finite precision arithmetic. The material is presented at a basic level, emphasizing ideas and intuition, but in a mathematically rigorous fashion. The derivations are simple and elegant, and the results are easy to understand and interpret. The book is self-contained. It was written for students in all areas of mathematics, engineering, and the computational sciences, but can easily be used for self-study. This text differs from other numerical linear algebra texts by offering the following: a systematic development of numerical conditioning; a simplified concept of numerical stability in exact arithmetic; simple derivations; a high-level view of algorithms; and results for complex matrices.

Contents......Page 7 Preface......Page 9 Introduction......Page 13 1. Matrices......Page 15 2. Sensitivity, Errors, and Norms......Page 37 3. Linear Systems......Page 57 4. Singular Value Decomposition......Page 91 5. Least Squares Problems......Page 105 6. Subspaces......Page 119 Index......Page 135