Perturbation Theory for Matrix Equations (eBook)

D. Wei Gu, M. Konstantinov, V. Mehrmann, P. Petkov (Autoren)

eBook Download: PDF

2003 | 1. Auflage
442 Seiten
Elsevier Science (Verlag)
978-0-08-053867-9 (ISBN)

Infos

The book is devoted to the perturbation analysis of matrix equations. The importance of perturbation analysis is that it gives a way to estimate the influence of measurement and/or parametric errors in mathematical models together with the rounding errors done in the computational process. The perturbation bounds may further be incorporated in accuracy estimates for the solution computed in finite arithmetic. This is necessary for the development of reliable computational methods, algorithms and software from the viewpoint of modern numerical analysis.

In this book a general perturbation theory for matrix algebraic equations is presented. Local and non-local perturbation bounds are derived for general types of matrix equations as well as for the most important equations arising in linear algebra and control theory. A large number of examples, tables and figures is included in order to illustrate the perturbation techniques and bounds.

Key features:

&bull, The first book in this field
&bull, Can be used by a variety of specialists
&bull, Material is self-contained
&bull, Results can be used in the development of reliable computational algorithms
&bull, A large number of examples and graphical illustrations are given
&bull, Written by prominent specialists in the field

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The book is devoted to the perturbation analysis of matrix equations. The importance of perturbation analysis is that it gives a way to estimate the influence of measurement and/or parametric errors in mathematical models together with the rounding errors done in the computational process. The perturbation bounds may further be incorporated in accuracy estimates for the solution computed in finite arithmetic. This is necessary for the development of reliable computational methods, algorithms and software from the viewpoint of modern numerical analysis.In this book a general perturbation theory for matrix algebraic equations is presented. Local and non-local perturbation bounds are derived for general types of matrix equations as well as for the most important equations arising in linear algebra and control theory. A large number of examples, tables and figures is included in order to illustrate the perturbation techniques and bounds.Key features:* The first book in this field * Can be used by a variety of specialists * Material is self-contained * Results can be used in the development of reliable computational algorithms * A large number of examples and graphical illustrations are given * Written by prominent specialists in the field

Inhalt

Front Cover 1
Perturbation Theory for Matrix Equations 4
Copyright Page 5
Preface 6
Contents 8
Chapter 1. Introduction 14
Chapter 2. Perturbation problems 22
2.1 Introductory remarks 22
2.2 Problem statement 23
2.3 Numerical considerations 31
2.4 Component-wise and backward analysis 33
2.5 Error estimates 37
2.6 Scaling 40
2.7 Notes and references 41
Chapter 3. Problems with explicit solutions 42
3.1 Introductory remarks 42
3.2 Perturbation function 42
3.3 Regularity and linear bounds 48
3.4 Norilocal bounds 60
3.5 Case study 61
3.6 Notes and references 63
Chapter 4 .Problems with implicit solutions 64
4.1 Introductory remarks 64
4.2 Posedness and regularity 64
4.3 Linear bounds 75
4.4 Equivalent operator equation 77
4.5 Linear equations 80
4.6 Case study 84
4.7 Notes and references 88
Chapter 5. Lyapunov majorants 90
5.1 Introductory remarks 90
5.2 General theory 90
5.3 Case study 112
5.4 Notes and references 113
Chapter 6. Singular problems 116
6.1 Introductory remarks 116
6.2 Distance to singularity 117
6.3 Classification 118
6.4 Regularization 121
6.5 Notes arid references 124
Chapter 7. Perturbation bounds 126
7.1 Introductory remarks 126
7.2 Definitions and properties 126
7.3 Conservativeness of “worst case” bounds 131
7.4 Notes and references 133
Chapter 8. General Sylvester equations 134
8.1 Introductory remarks 134
8.2 Motivating examples 136
8.3 General linear equations 140
8.4 Perturbation problem 142
8.5 Local perturbation analysis 149
8.6 Nonlocal perturbation analysis 158
8.7 Notes and references 166
Chapter 9. Specific Sylvester equations 168
9.1 Standard linear equation 168
9.2 General equations 181
9.3 Continuous-time equations 181
9.4 Discrete-time equations 184
9.5 Notes and references 185
Chapter 10. General Lyapunov equations 188
10.1 Introductory remarks 188
10.2 Application to descriptor systems 188
10.3 Additive matrix operators 192
10.4 Perturbation problem 200
10-5 Local perturbation analysis 202
10.6 Nonlocal perturbation analysis 208
10.7 Notes and references 213
Chapter 11. Lyapunov equations in control theory 214
11.1 Iritroductory remarks 214
11.2 General equation 214
11.3 Continuous-time equations 216
11.4 Continuous-time equations in descriptor form 223
11.5 Discrete-time equations 229
11:6 Discrete-time equations in descriptor form 230
11.7 Notes and references 234
Chapter 12. General quadratic equations 236
12.1 Introductory remarks 236
12.2 Problem statement 236
12.3 Motivating example 240
12.4 Local perturbation analysis 242
12.5 Nonlocal perturbation analysis 246
12.6 Notes and references 251
Chapter 13. Continuous-time Riccati equations 252
13.1 Introductory remarks 252
13.2 Motivating example 252
13.3 Standard equation 254
13.4 Descriptor equation 268
13.5 Notes and references 278
Chapter 14. Coupled Riccati equations 280
14.1 Problem statement 280
14.2 Local perturbation analysis 285
14.3 Nonlocal perturbation analysis 291
14.4 Notes arid references 298
Chapter 15. General fractional-affine equations 300
15.1 Introductory remarks 300
15.2 Problem statement 300
15.3 Local perturbation analysis 304
15.4 Non-local perturbation analysis 308
15.5 Notes and references 315
Chapter 16. Symmetric fractional-affine equations 316
16.1 Introductory remarks 316
16.2 Discretc-time Riccati equations 316
16.3 Symmetric fractional-linear equation 329
16.4 Notes and references 339
Appendix A. Elements of algebra and analysis 340
A.l Introductory remarks 340
A.2 Sets and functions 340
A.3 Algebraic systems 342
A.4 Linear algebra 344
A.5 Normed spaces 348
A.6 Matrix functions 350
A.7 Transformation groups 355
A.8 Notes and references 356
Appendix B. Unitary and orthogonal decompositions 358
B.l Introductory remarks 358
B.2 Elementary unitary matrices 359
B.3 QR decomposition 361
B.4 Schur decomposition 363
B.5 Polar decomposition 365
B.6 Singular value decomposition 367
B.7 Notes and references 368
Appendix C. Kronecker product of matrices 370
C.l Introductory remarks 370
C.2 Definitions and properties 370
C.3 Notes and references 374
Appendix D. Fixed point principles 376
D.l Introductory remarks 376
D.2 Banach principle 376
D.3 Generalized Banach principle 378
D.4 Schauder principle 381
D.5 Notes and references 382
Appendix E. Sylvester operators 384
E . l Introductory 384
E.2 Basic concepts 384
E.3 Representations 386
E.4 Notes and references 390
Appendix F. Lyapunov operators 392
F.1 Introductory remarks 392
F.2 Real operators 393
F.3 Complex operators 402
F.4 Sensitivity and error analysis 407
F.5 Notes and references 408
Appendix G. Lyapunov-like operators 410
G.l Introductory remarks 410
G.2 Skew-Lyapunov operators 410
G.3 Associated Lyapunov operators 411
G.4 Associated skew-Lyapunov operators 412
G.5 Notes and references 413
Appendix H. Notation 414
H.l Sets and spaces 414
H.2 Matrices 415
H.3 Matrix operators 416
H.4 Norms 417
H.5 Perturbation analysis 418
H.6 Other notation 419
Bibliography 420
Index 438

Produkt-Details

• Erscheint lt. Verlag: 20.5.2003
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik ► Theorie / Studium
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Finanz- / Wirtschaftsmathematik
• Themenwelt: Technik
• ISBN-10: 0-08-053867-3 / 0080538673
• ISBN-13: 978-0-08-053867-9 / 9780080538679