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Table of Contents for

Solving Linear Systems

An Analysis of Matrix Prefactorization Iterative Methods

by Zbigniew Ignacy 

554 pages, 8 x 10 inches, hardcover, smythe-sewn binding
More than 230 figures and 140 tables, $89
ISBN 9780971576667

Table of contents in pdf

Foreword (by Richard Varga)

Chapter 0 Introduction to iterative methods and preconditioning  1

Chapter 1 Numerical linear algebra: background   5

1.1  Review of matrix theory
Basic matrix operations
Matrix partitionings
Basic concepts of matrix analysis
Vector norms and matrix norms
Computational work
1.2  Eigenvalues and eigenvectors
Relating norms and eigenvalues
Convergence of vector and matrix sequences
Perron-Frobenius theory of nonnegative matrices
Diagonally dominant matrices
Power method
1.3  General linear systems
Direct methods
Iterative refinement

Chapter 2 The theory of matrix splitting   41

2.1  General properties of matrix splittings
2.2  Regular splittings
2.3  Nonnegative and weak nonnegative splittings
2.4  Weak and weaker splittings
2.5  Summary

Chapter 3 Discretization of partial differential equations  70

3.1  Finite-difference approximations
3.2  One-dimensional problems
Forward elimination - backward substitution
Backward elimination - forward substitution
3.3  Band matrices
Pentadiagonal matrices
General band systems
3.4  Two-dimensional problems
Rectangular geometry
Triangular geometry
Hexagonal geometry
Reduced systems
Line orderings
Computational molecules
Irregular mesh structures
3.5  Test problems

Chapter 4 Standard iterative methods   129

4.1  General theory of iterative methods
Stopping criteria
Starting vectors
4.2   Point iterative methods
Basic algorithms
Consistent orderings
The successive overrelaxation method (SOR)
Determining the optimum relaxation parameter
Experimental examination of SOR convergence
Computational aspects
4.3  Line iterative methods
1-line algorithms
2-line algorithms
3-line algorithms
4.4  Results of numerical experiments

Chapter 5 Explicit prefactorization methods (AGA)   199

5.1  Matrix notation
Basic algorithms
AGA algorithms with point modification
AGA algorithms with line modification
Techniques for accelerating convergence
5.2  Implementing prefactorization algorithms in mesh structures
Rectangular geometry
Triangular geometry
Hexagonal geometry
5.3  Results of numerical experiments

Chapter 6 Semi-explicit prefactorization with implicit backward sweep  248

6.1  Matrix notation
Modified line methods
6.2  Implementation in mesh structures
Rectangular geometry
Triangular geometry
6.3  Numerical experiments
Self-adjoint problems
Non-self-adjoint problems
Concluding remarks

Chapter 7 Semi-explicit prefactorization with implicit forward sweep (OLA)  436

7.1  Implementing OLA methods in mesh structures
Rectangular geometry
Triangular geometry
Successive overrelaxation
7.2  Matrix notation
7.3  Numerical experiments
Self-adjoint problems
Non-self-adjoint problems
OLA algorithms: concluding remarks
7.4   Final discussion

Chapter 8  Advances in solving linear control systems   308

8.1  Sylvester equations
The SOR-like method
Numerical experiments
Sylvester equations: conclusion
8.2  Continuous-time algebraic Riccati equations
The SOR-like method
Numerical experiments
Concluding remarks


Appendix A  Numerical experiments for chapter 4  325

Appendix B  Numerical experiments for chapter 5  375

Appendix C  Numerical experiments for chapter 6  473

Appendix D  Numerical experiments for chapter 7  496

Bibliography  529

Index  535

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