Numerical Algebra

Basic Numerical Mathematics, Volume II: Numerical Algebra focuses on numerical algebra, with emphasis on the ideas of "controlled computational experiments" and "bad examples". The existence of an orthogonal matrix which diagonalizes a real symmetric matrix is highlighted, and partitioned or block matrices are discussed, along with induced norms and inversion problems. Comprised of 12 chapters, this volume begins with an overview of the manipulation of vectors and matrices, followed by an analysis of induced norms. The reader is then introduced to the direct solution of the inversion problem, first in the context of theoretical arithmetic (that is, when round-off is disregarded) and second in the context of practical computation. Various methods of handling the characteristic value problems are also considered, together with several iterative methods for the solution of a system of linear equations. Two applications are described: the solution of a two-point boundary value problem and the solution of least squares curve fitting. The book concludes with an account of the singular value decomposition and pseudo-inverses. This monograph will be of interest to mathematicians and students of mathematics.

Notations and Abbreviations

Preface

Chapter 1. Manipulation of Vectors and Matrices

Chapter 2. Norms of Vectors and Matrices

Chapter 3. Induced Norms

Chapter 4. The Inversion Problem I: Theoretical Arithmetic

Chapter 5. The Inversion Problem II: Practical Computation

Chapter 6. The Characteristic Value Problem — Generalities

Chapter 7. The Power Method, Deflation, Inverse Iteration

Chapter 8. Characteristic Values

Chapter 9. Iterative Methods for the Solution of Systems Ax=b

Chapter 10. Application: Solution of a Boundary Value Problem

Chapter 11. Application: Least Squares Curve Fitting

Chapter 12. Singular Value Decomposition and Pseudo-Inverses

Solutions to Selected Problems

Bibliographical Remarks

Index