Functions of matrices: theory and computation

Nicholas J. Higham0898716462, 9780898716467

The only book devoted exclusively to matrix functions, this research monograph gives a thorough treatment of the theory of matrix functions and numerical methods for computing them. The author s elegant presentation focuses on the equivalent definitions of f(A) via the Jordan canonical form, polynomial interpolation, and the Cauchy integral formula, and features an emphasis on results of practical interest and an extensive collection of problems and solutions. Functions of Matrices: Theory and Computation is more than just a monograph on matrix functions; its wide-ranging content including an overview of applications, historical references, and miscellaneous results, tricks, and techniques with an f(A) connection makes it useful as a general reference in numerical linear algebra. Other key features of the book include development of the theory of conditioning and properties of the Fréchet derivative; an emphasis on the Schur decomposition, the block Parlett recurrence, and judicious use of Padé approximants; the inclusion of new, unpublished research results and improved algorithms; a chapter devoted to the f(A)b problem; and a MATLAB® toolbox providing implementations of the key algorithms.

Audience: This book is for specialists in numerical analysis and applied linear algebra as well as anyone wishing to learn about the theory of matrix functions and state of the art methods for computing them. It can be used for a graduate-level course on functions of matrices and is a suitable reference for an advanced course on applied or numerical linear algebra. It is also particularly well suited for self-study.

Contents: List of Figures; List of Tables; Preface; Chapter 1: Theory of Matrix Functions; Chapter 2: Applications; Chapter 3: Conditioning; Chapter 4: Techniques for General Functions; Chapter 5: Matrix Sign Function; Chapter 6: Matrix Square Root; Chapter 7: Matrix pth Root; Chapter 8: The Polar Decomposition; Chapter 9: Schur-Parlett Algorithm; Chapter 10: Matrix Exponential; Chapter 11: Matrix Logarithm; Chapter 12: Matrix Cosine and Sine; Chapter 13: Function of Matrix Times Vector: f(A)b; Chapter 14: Miscellany; Appendix A: Notation; Appendix B: Background: Definitions and Useful Facts; Appendix C: Operation Counts; Appendix D: Matrix Function Toolbox; Appendix E: Solutions to Problems; Bibliography; Index.

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Table of contents :
Functions of Matrices: Theory and Computation……Page 1
Contents……Page 8
List of Figures……Page 14
List of Tables……Page 16
Preface……Page 18
1 Theory of Matrix Functions……Page 22
2 Applications……Page 56
3 Conditioning……Page 76
4 Techniques for General Functions……Page 92
5 Matrix Sign Function……Page 128
6 Matrix Square Root……Page 154
7 Matrix pth Root……Page 194
8 The Polar Decomposition……Page 214
9 Schur-Parlett Algorithm……Page 242
10 Matrix Exponential……Page 254
11 Matrix Logarithm……Page 290
12 Matrix Cosine and Sine……Page 308
13 Function of Matrix Times Vector: f (A)b……Page 322
14 Miscellany……Page 334
Appendix A Notation……Page 340
Appendix B Background: Definitions and Useful Facts……Page 342
Appendix C Operation Counts……Page 356
Appendix D Matrix Function Toolbox……Page 360
Appendix E Solutions to Problems……Page 364
Bibliography……Page 400
Index……Page 436