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 Gauร-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 Gauร-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

1. Mathematics
2. Numerical analysis
3. Mathematics
4. Numerical Analysis