Perturbation Theory for Matrix Equations

By

  • M. Konstantinov, University of Architecture, Sofia, Bulgaria
  • D. Wei Gu, University of Leicester, Department of Engineering, Leicester, UK
  • V. Mehrmann, Institut für Mathematik, Berlin, Germany
  • P. Petkov, Technical University of Sofia, Department of Systems and Control, Bulgaria

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

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Audience

Scientists, Specialists and Postgraduate Students in Applied Mathematics, Scientific Computing and Control Engineering, Developers of Mathematical Software

 

Book information

  • Published: May 2003
  • Imprint: JAI Press (Elsevier)
  • ISBN: 978-0-444-51315-1

Reviews

Since the matrix equations studied in this book appear in various applications and perturbation theory is essential for understanding the problems and estimating the accuracy of the computed results, the book will be an excellent reference for a wide audience. It will also be useful to researchers who would like to use the techniques presented in this book to carry out perturbation analysis for new types of matrix equations
Chun-Hua Gua, Mathematical Reviews



Table of Contents

1 Introduction.
2 Perturbation problems.
3 Problems with explicit solutions.
4 Problems with implicit solutions.
5 Lyapunov majorants.
6 Singular problems.
7 Perturbation bounds.
8 General Sylvester equations.
9 Specific Sylvester equations.
10 General Lyapunov equations.
11 Lyapunov equations in control theory.
12 General quadratic equations.
13 Continuous­time Riccati equations.
14 Coupled Riccati equations.
15 General fractional­afine equations.
16 Symmetric fractional­afine equations.
A Elements of algebra and analysis.
B Unitary and orthogonal decompositions.
C Kronecker product of matrices.
D Fixed point principles.
E Sylvester operators.
F Lyapunov operators.
G Lyapunov­like operators.
H Notation.
References.
Index.