Perturbation Theory for Matrix Equations (Volume 9) (Studies in Computational Mathematics, Volume 9)

✍ Scribed by M. Konstantinov, D. Wei Gu, V. Mehrmann, P. Petkov

• Publisher: JAI Press
• Year: 2003
• Tongue: English
• Leaves: 443
• Edition: 1
• Category: Library

✦ Synopsis

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

✦ Table of Contents

Perturbation Theory for Matrix Equations
Copyright Page
Preface
Contents
Chapter 1. Introduction
Chapter 2. Perturbation problems
2.1 Introductory remarks
2.2 Problem statement
2.3 Numerical considerations
2.4 Component-wise and backward analysis
2.5 Error estimates
2.6 Scaling
2.7 Notes and references
Chapter 3. Problems with explicit solutions
3.1 Introductory remarks
3.2 Perturbation function
3.3 Regularity and linear bounds
3.4 Norilocal bounds
3.5 Case study
3.6 Notes and references
Chapter 4. Problems with implicit solutions
4.1 Introductory remarks
4.2 Posedness and regularity
4.3 Linear bounds
4.4 Equivalent operator equation
4.5 Linear equations
4.6 Case study
4.7 Notes and references
Chapter 5. Lyapunov majorants
5.1 Introductory remarks
5.2 General theory
5.3 Case study
5.4 Notes and references
Chapter 6. Singular problems
6.1 Introductory remarks
6.2 Distance to singularity
6.3 Classification
6.4 Regularization
6.5 Notes arid references
Chapter 7. Perturbation bounds
7.1 Introductory remarks
7.2 Definitions and properties
7.3 Conservativeness of “worst case” bounds
7.4 Notes and references
Chapter 8. General Sylvester equations
8.1 Introductory remarks
8.2 Motivating examples
8.3 General linear equations
8.4 Perturbation problem
8.5 Local perturbation analysis
8.6 Nonlocal perturbation analysis
8.7 Notes and references
Chapter 9. Specific Sylvester equations
9.1 Standard linear equation
9.2 General equations
9.3 Continuous-time equations
9.4 Discrete-time equations
9.5 Notes and references
Chapter 10. General Lyapunov equations
10.1 Introductory remarks
10.2 Application to descriptor systems
10.3 Additive matrix operators
10.4 Perturbation problem
10-5 Local perturbation analysis
10.6 Nonlocal perturbation analysis
10.7 Notes and references
Chapter 11. Lyapunov equations in control theory
11.1 Iritroductory remarks
11.2 General equation
11.3 Continuous-time equations
11.4 Continuous-time equations in descriptor form
11.5 Discrete-time equations
11:6 Discrete-time equations in descriptor form
11.7 Notes and references
Chapter 12. General quadratic equations
12.1 Introductory remarks
12.2 Problem statement
12.3 Motivating example
12.4 Local perturbation analysis
12.5 Nonlocal perturbation analysis
12.6 Notes and references
Chapter 13. Continuous-time Riccati equations
13.1 Introductory remarks
13.2 Motivating example
13.3 Standard equation
13.4 Descriptor equation
13.5 Notes and references
Chapter 14. Coupled Riccati equations
14.1 Problem statement
14.2 Local perturbation analysis
14.3 Nonlocal perturbation analysis
14.4 Notes arid references
Chapter 15. General fractional-affine equations
15.1 Introductory remarks
15.2 Problem statement
15.3 Local perturbation analysis
15.4 Non-local perturbation analysis
15.5 Notes and references
Chapter 16. Symmetric fractional-affine equations
16.1 Introductory remarks
16.2 Discretc-time Riccati equations
16.3 Symmetric fractional-linear equation
16.4 Notes and references
Appendix A. Elements of algebra and analysis
A.1 Introductory remarks
A.2 Sets and functions
A.3 Algebraic systems
A.4 Linear algebra
A.5 Normed spaces
A.6 Matrix functions
A.7 Transformation groups
A.8 Notes and references
Appendix B. Unitary and orthogonal decompositions
B.1 Introductory remarks
B.2 Elementary unitary matrices
B.3 QR decomposition
B.4 Schur decomposition
B.5 Polar decomposition
B.6 Singular value decomposition
B.7 Notes and references
Appendix C. Kronecker product of matrices
C.1 Introductory remarks
C.2 Definitions and properties
C.3 Notes and references
Appendix D. Fixed point principles
D.1 Introductory remarks
D.2 Banach principle
D.3 Generalized Banach principle
D.4 Schauder principle
D.5 Notes and references
Appendix E. Sylvester operators
E.1 Introductory
E.2 Basic concepts
E.3 Representations
E.4 Notes and references
Appendix F. Lyapunov operators
F.1 Introductory remarks
F.2 Real operators
F.3 Complex operators
F.4 Sensitivity and error analysis
F.5 Notes and references
Appendix G. Lyapunov-like operators
G.1 Introductory remarks
G.2 Skew-Lyapunov operators
G.3 Associated Lyapunov operators
G.4 Associated skew-Lyapunov operators
G.5 Notes and references
Appendix H. Notation
H.l Sets and spaces
H.2 Matrices
H.3 Matrix operators
H.4 Norms
H.5 Perturbation analysis
H.6 Other notation
Bibliography
Index