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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 | |