Iterative Solution of Large Sparse Systems of Equations - Info and Reading Options

The Open Library:

This book presents the description of the state of modern iterative techniques together with systematic analysis. The first chapters discuss the classical methods. Comprehensive chapters are devoted to semi-iterative techniques (Chebyshev methods), transformations, incomplete decompositions, gradient and conjugate gradient methods, multi-grid methods and domain decomposition techniques (including e.g. the additive and multiplicative Schwartz method). In contrast to other books all techniques are described algebraically. For instance, for the domain decomposition method this is a new but helpful approach. Every technique described is illustrated by a Pascal program applicable to a class of model problem.

Open Data:

1. Introduction -- 1.1 Historical Remarks Concerning Iterative Methods -- 1.2 Model Problem (Poisson Equation) -- 1.3 Amount of Work for the Direct Solution of the System of Equations -- 1.4 Examples of Iterative Methods -- 2. Recapitulation of Linear Algebra -- 2.1 Notations for Vectors and Matrices -- 2.2 Systems of Linear Equations -- 2.3 Permutation Matrices -- 2.4 Eigenvalues and Eigenvectors -- 2.5 Block-Vectors and Block-Matrices -- 2.6 Norms -- 2.7 Scalar Product -- 2.8 Normal Forms -- 2.9 Correlation Between Norms and the Spectral Radius -- 2.10 Positive Definite Matrices -- 3. Iterative Methods -- 3.1 General Statements Concerning Convergence -- 3.2 Linear Iterative Methods -- 3.3 Effectiveness of Iterative Methods -- 3.4 Test of Iterative Methods -- 3.5 Comments Concerning the Pascal Procedures -- 4. Methods of Jacobi and Gauß-Seidel and SOR Iteration in the Positive Definite Case -- 4.1 Eigenvalue Analysis of the Model Problem -- 4.2 Construction of Iterative Methods -- 4.3 Damped Iterative Methods -- 4.4 Convergence Analysis -- 4.5 Block Versions -- 4.6 Computational Work of the Methods -- 4.7 Convergence Rates in the Case of the Model Problem -- 4.8 Symmetric Iterations -- 5. Analysis in the 2-Cyclic Case -- 5.1 2-Cyclic Matrices -- 5.2 Preparatory Lemmata -- 5.3 Analysis of the Richardson Iteration -- 5.4 Analysis of the Jacobi Method -- 5.5 Analysis of the Gauß-Seidel Iteration -- 5.6 Analysis of the SOR Method -- 5.7 Application to the Model Problem -- 5.8 Supplementary Remarks -- 6. Analysis for M-Matrices -- 6.1 Positive Matrices -- 6.2 Graph of a Matrix and Irreducible Matrices -- 6.3 Perron-Frobenius Theory of Positive Matrices -- 6.4 M-Matrices -- 6.5 Regular Splittings -- 6.6 Applications -- 7. Semi-Iterative Methods -- 7.1 First Formulation -- 7.2 Second Formulation of a Semi-Iterative Method -- 7.3 Optimal Polynomials -- 7.4 Application to Iterations Discussed Above -- 7.5 Method of Alternating Directions (ADI) -- 8. Transformations, Secondary Iterations, Incomplete Triangular Decompositions -- 8.1 Generation of Iterations by Transformations -- 8.2 Kaczmarz Iteration -- 8.3 Preconditioning -- 8.4 Secondary Iterations -- 8.5 Incomplete Triangular Decompositions -- 8.6 A Superfluous Term: Time-Stepping Methods -- 9. Conjugate Gradient Methods -- 9.1 Linear Systems of Equations as Minimisation Problem -- 9.2 Gradient Method -- 9.3 The Method of the Conjugate Directions -- 9.4 Conjugate Gradient Method (cg Method) -- 9.5 Generalisations -- 10. Multi-Grid Methods -- 10.1 Introduction -- 10.2 Two-Grid Method -- 10.3 Analysis for a One-Dimensional Example -- 10.4 Multi-Grid Iteration -- 10.5 Nested Iteration -- 10.6 Convergence Analysis -- 10.7 Symmetric Multi-Grid Methods -- 10.8 Combination of Multi-Grid Methods with Semi-Iterations -- 10.9 Further Comments -- 11. Domain Decomposition Methods -- 11.1 Introduction -- 11.2 Formulation of the Domain Decomposition Method -- 11.3 Properties of the Additive Schwarz Iteration -- 11.4 Analysis of the Multiplicative Schwarz Iteration -- 11.5 Examples -- 11.6 Multi-Grid Methods as Subspace Decomposition Method -- 11.7 Schur Complement Methods

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