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List of Algorithms | |
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Preface | |
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Introduction | |
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Brief Overview of the State of the Art | |
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Notation | |
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Review of Relevant Linear Algebra | |
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Krylov Subspace Approximations | |
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Some Iteration Methods | |
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Simple Iteration | |
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Orthomin(1) and Steepest Descent | |
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Orthomin(2) and CG | |
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Orthodir, MINRES, and GMRES | |
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Derivation of MINRES and CG from the Lanczos Algorithm | |
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Error Bounds for CG, MINRES, and GMRES | |
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Hermitian Problems-CG and MINRES | |
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Non-Hermitian Problems-GMRES | |
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Effects of Finite Precision Arithmetic | |
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Some Numerical Examples | |
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The Lanczos Algorithm | |
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A Hypothetical MINRES/CG Implementation | |
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A Matrix Completion Problem | |
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Orthogonal Polynomials | |
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BiCG and Related Methods | |
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The Two-Sided Lanczos Algorithm | |
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The Biconjugate Gradient Algorithm | |
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The Quasi-Minimal Residual Algorithm | |
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Relation Between BiCG and QMR | |
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The Conjugate Gradient Squared Algorithm | |
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The BiCGSTAB Algorithm | |
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Which Method Should I Use? | |
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Is There A Short Recurrence for a Near-Optimal Approximation? | |
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The Faber and Manteuffel Result | |
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Implications | |
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Miscellaneous Issues | |
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Symmetrizing the Problem | |
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Error Estimation and Stopping Criteria | |
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Attainable Accuracy | |
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Multiple Right-Hand Sides and Block Methods | |
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Computer Implementation | |
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Preconditioners | |
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Overview and Preconditioned Algorithms | |
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Two Example Problems | |
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The Diffusion Equation | |
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The Transport Equation | |
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Comparison of Preconditioners | |
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Jacobi, Gauss--Seidel, SOR | |
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The Perron--Frobenius Theorem | |
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Comparison of Regular Splittings | |
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Regular Splittings Used with the CG Algorithm | |
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Optimal Diagonal and Block Diagonal Preconditioners | |
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Incomplete Decompositions | |
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Incomplete Cholesky Decomposition | |
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Modified Incomplete Cholesky Decomposition | |
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Multigrid and Domain Decomposition Methods | |
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Multigrid Methods | |
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Basic Ideas of Domain Decomposition Methods | |