Iterative Methods for Large Linear Systems

¿1 Fourier Analysis of Two-Level Hierarchical Basis Preconditioners 1 Introduction 2 ID, Linear S 3 2D, Bilinear S, Bilinear A 4 2D, Bilinear, 5-Point A 5 3D, Trilinear S, 7-Point A 6 Concluding Remarks Acknowledgements References2 An Algebraic Framework for Hierarchical Basis Functions Multilevel Methods or the Search for 'Optimal' Preconditioners 1 Introduction 2 The Algebraic Framework for Two-Level Hierarchical Basis Function Methods Basic Assumptions 3 Recursive Definition of Preconditioner Forward Substitution Backward Substitution Computational Complexity Domain Decomposition 4 The Relative Condition Number of M(l) with Respect to A(l) Fixed-Point Analysis 5 Concluding Remarks References3 ELLPACK and ITPACK as Research Tools for Solving Elliptic Problems 1 Background 2 ELLPACK and ITPACK 3 Some Basic Question 4 Direct vs. Iterative Methods 5 Different Elliptic Problems 6 Symmetry? 7 Extended Network Analogy 8 Orders of Accuracy 9 Choice of Mesh 10 Computational Complexity 11 3D Problems Acknowledgement References4 Preconditioned Iterative Methods for Indefinite Symmetric Toeplitz Systems 1 Introduction 2 Toeplitz and Circulant Matrices 3 Solution Methods 4 Test Matrix Preconditioners 5 Test Matrices 6 Computed Spectra Acknowlegements References5 A Local Relaxation Scheme (Ad-Hoc SOR) Applied to Nine Point and Block Difference Equations 1 History 2 The Method 3 Nine Point Application: Cross Derivatives 4 Block Iteration Acknowledgements References6 Block Iterative Methods for Cyclically Reduced Non-Self-Adjoint Elliptic Problems 1 Introduction 2 The Reduced System for the Convection-Diffusion Equation 3 Bounds for Solving the Convection-Diffusion Equation 4 Numerical Experiments Acknowledgements References7 Toward an Effective Two-Parameter SOR Method 1 Background 2 Singular Value Decomposition and Orthogonal Similarities 3 Two-Parameter SOR 4 A Numerical Example Acknowledgements References Appendix8 Relaxation Parameters for the IQE Iterative Procedure for Solving Semi-Implicit Navier-Stokes Difference Equations 1 Introduction 2 The Continuous and Discrete Problems 3 The IQE Iterative Method 4 The Calculation of ¿ 5 Numerical Results Acknowledgements References9 Hodie Approximation of Boundary Conditions 1 Introduction 2 Approximation 'Away from the Boundary' 3 Hodie as Interpolation 4 Boundary Conditions 5 Extension of Ui,j to O 6 Indexing of Unknowns 7 Eigenproblems Acknowledgements References10 Iterative Methods for Nonsymmetric Linear Systems 1 Introduction 2 Projection Methods Balanced Projection Methods 3 Krylov Projection Methods 3.1 Computational Schemes for Krylov Projection Methods 3.2 Examples of Krylov Projection Method 4 Semi-Krylov Projection Methods 4.1 Balanced SKPM's: Truncated/Restarted Methods 4.