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Iterative Methods for Sparse Linear Systems Second Edition

Iterative Methodsfor SparseLinear SystemsSecond Edition +07 Yousef SaadCopyrightc 2003 by the Society for Industrial and Applied MathematicsContentsPrefacexiiiPreface to Second Edition ..xiiiPreface to first Edition ..xix1 Background in Linear .. matrices and Eigenvalues .. of matrices .. Inner Products and Norms .. Norms .. , Range, and Kernel .. Vectors and Subspaces .. Forms of matrices .. to the Diagonal Form .. Jordan Canonical Form .. Schur Canonical Form .. to Powers of matrices .

13.2 Matrices and spectra of model problems . . . . . . . . . . . . 424 ... iterative methods for linear systems have made good progress in scientific an d engi-neering disciplines. This is due in great part to the increased complexity and size of xiii. methods). ...

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Transcription of Iterative Methods for Sparse Linear Systems Second Edition

1 Iterative Methodsfor SparseLinear SystemsSecond Edition +07 Yousef SaadCopyrightc 2003 by the Society for Industrial and Applied MathematicsContentsPrefacexiiiPreface to Second Edition ..xiiiPreface to first Edition ..xix1 Background in Linear .. matrices and Eigenvalues .. of matrices .. Inner Products and Norms .. Norms .. , Range, and Kernel .. Vectors and Subspaces .. Forms of matrices .. to the Diagonal Form .. Jordan Canonical Form .. Schur Canonical Form .. to Powers of matrices .

2 And Hermitian matrices .. matrices .. matrices .. matrices , M- matrices .. matrices .. Operators .. and Null Space of a Projector .. Representations .. and Oblique Projectors .. of Orthogonal Projectors .. Concepts in Linear Systems .. of a Solution .. Analysis ..412 Discretization of Differential Equations .. Operators .. Convection Diffusion Equation .. Difference Methods .. Approximations .. Schemes for the Laplacean Operator . Differences for 1-D Problems.

3 Schemes .. Differences for 2-D Problems .. Poisson Solvers .. Finite Element Method .. Generation and Refinement .. Volume Method ..693 Sparse .. Representations .. and Adjacency Graphs .. of PDE matrices .. and Reorderings .. Concepts .. with the Adjacency Graph .. Reorderings .. Schemes .. Sparse Matrix Operations .. Direct Solution Methods .. degree ordering .. Dissection ordering .. Problems ..984 Basic Iterative , Gauss-Seidel, and SOR .. Relaxation Schemes.

4 matrices and Preconditioning .. Convergence Result .. Splittings .. Dominant matrices .. Positive Definite matrices .. A and Consistent Orderings .. Direction Methods .. 1275 Projection Definitions and Algorithms .. Projection Methods .. Representation .. Theory .. Optimality Results .. in Terms of Projectors .. Error Bound .. Projection Processes .. Descent .. Residual (MR) Iteration .. Norm Steepest Descent.

5 And Multiplicative Processes .. 1476 Krylov Subspace Methods Part .. Subspaces .. s Method .. Basic Algorithm .. Implementations .. s Method for Linear Systems (FOM) .. 1: Restarted FOM .. 2: IOM and DIOM .. Basic GMRES Algorithm .. Householder Version .. Implementation Issues .. of GMRES .. 1: Restarting .. 2: Truncated GMRES Versions .. between FOM and GMRES .. smoothing .. for complex Systems .

6 Symmetric Lanczos Algorithm .. Algorithm .. with Orthogonal Polynomials .. Conjugate Gradient Algorithm .. and Theory .. Formulations .. Estimates from the CG Coefficients .. Conjugate Residual Method .. , ORTHOMIN, and ORTHODIR .. Faber-Manteuffel Theorem .. Analysis .. Chebyshev Polynomials .. Chebyshev Polynomials .. of the CG Algorithm .. of GMRES .. Krylov Methods .. 2187 Krylov Subspace Methods Part Biorthogonalization.

7 Algorithm .. Implementations .. Lanczos Algorithm for Linear Systems .. BCG and QMR Algorithms .. Biconjugate Gradient Algorithm .. Residual Algorithm .. Variants .. Gradient Squared .. QMR (TFQMR) .. 2478 Methods Related to the Normal Normal Equations .. Projection Methods .. on the Normal Equations .. s Method .. Gradient and Normal Equations .. Problems .. 2689 Preconditioned .. Conjugate Gradient.

8 Symmetry .. Implementations .. GMRES .. GMRES .. GMRES .. Preconditioning .. of Right and Left Preconditioning .. Variants .. GMRES .. CG for the Normal Equations .. Concus, Golub, and Widlund Algorithm .. 29210 Preconditioning .. , SOR, and SSOR Preconditioners .. Factorization Preconditioners .. LU Factorizations .. Fill-in ILU (ILU(0)) .. of Fill and ILU(p) .. with Regular Structure .. ILU (MILU).

9 Strategies and ILUT .. ILUT Approach .. Details .. ILUTP Approach .. ILUS Approach .. Crout ILU Approach .. Inverse Preconditioners .. the Inverse of a Sparse Matrix .. Iteration .. Algorithms .. Considerations .. of Self Preconditioned MR .. Inverses via bordering .. inverses via orthogonalization: AINV .. a Preconditioner .. for ILU .. permutations .. reorderings .. Preconditioners .. matrices .

10 matrices .. for the Normal Equations .. , SOR, and Variants .. (0) for the Normal Equations .. Gram-Schmidt and ILQ .. 36011 Parallel .. of Parallelism .. Functional Units .. Processors .. and Distributed Computing .. of Parallel Architectures .. Memory Computers .. Memory Architectures .. of Operations .. Products .. CSR and CSC Formats .. in the Diagonal Format .. Ellpack-Itpack Format .. Jagged Diagonal Format.

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