Bibliothèque de L'IMIST/CNRST

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

Includes bibliographical references and index.

Description based on print version record.

Cover -- 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 perturbati.