Ebook: Recent Advances in Iterative Methods

This IMA Volume in Mathematics and its Applications RECENT ADVANCES IN ITERATIVE METHODS is based on the proceedings of a workshop that was an integral part of the 1991-92 IMA program on "Applied Linear Algebra. " Large systems of matrix equations arise frequently in applications and they have the prop­ erty that they are sparse and/or structured. The purpose of this workshop was to bring together researchers in numerical analysis and various ap­ plication areas to discuss where such problems arise and possible meth­ ods of solution. The last two days of the meeting were a celebration dedicated to Gene Golub on the occasion of his sixtieth birthday, with the program arranged by Jack Dongarra and Paul van Dooren. We are grateful to Richard Brualdi, George Cybenko, Alan George, Gene Golub, Mitchell Luskin, and Paul Van Dooren for planning and implementing the year-long program. We especially thank Gene Golub, Anne Greenbaum, and Mitchell Luskin for organizing this workshop and editing the proceed­ ings. The financial support of the National Science Foundation and the Min­ nesota Supercomputer Institute made the workshop possible. A vner Friedman Willard Miller, Jr. xi PREFACE The solution of very large linear algebra problems is an integral part of many scientific computations.

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The solution of very large sparse or structured linear algebra problems is an integral part of many scientific computations. Direct methods for solving such problems are often infeasible because of computation time and memory requirements, and so iterative techniques are used instead. In recent years much research has focussed on the efficient solution of large systems of linear equations, least squares problems, and eigenvalue problems using iterative methods. This volume on iterative methods for sparse and structured problems brings together researchers from all over the world to discuss topics of current research. Areas addressed included the development of efficient iterative techniques for solving nonsymmetric linear systems and eigenvalue problems, estimating the convergence rate of such algorithms, and constructing efficient preconditioners for special classes of matrices such as Toeplitz and Hankel matrices. Iteration strategies and preconditioners that could exploit parallelism were of special interest. This volume represents the latest results of mathematical and computational research into the development and analysis of robust iterative methods for numerical linear algebra problems. This volume will be useful for both mathematicians and for those involved in applications using iterative methods.

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The solution of very large sparse or structured linear algebra problems is an integral part of many scientific computations. Direct methods for solving such problems are often infeasible because of computation time and memory requirements, and so iterative techniques are used instead. In recent years much research has focussed on the efficient solution of large systems of linear equations, least squares problems, and eigenvalue problems using iterative methods. This volume on iterative methods for sparse and structured problems brings together researchers from all over the world to discuss topics of current research. Areas addressed included the development of efficient iterative techniques for solving nonsymmetric linear systems and eigenvalue problems, estimating the convergence rate of such algorithms, and constructing efficient preconditioners for special classes of matrices such as Toeplitz and Hankel matrices. Iteration strategies and preconditioners that could exploit parallelism were of special interest. This volume represents the latest results of mathematical and computational research into the development and analysis of robust iterative methods for numerical linear algebra problems. This volume will be useful for both mathematicians and for those involved in applications using iterative methods.
Content:
Front Matter....Pages i-xvi
Some Themes in Gene H. Golub’s Work on Iterative Methods....Pages 1-11
Computing the Sparse Singular Value Decomposition via SVDPACK....Pages 13-29
Gaussian Quadrature Applied to Adaptive Chebyshev Iteration....Pages 31-44
Ordering Effects on Relaxation Methods Applied to the Discrete Convection-Diffusion Equation....Pages 45-57
On the Error Computation for Polynomial Based Iteration Methods....Pages 59-67
Transpose-Free Quasi-Minimal Residual Methods for Non-Hermitian Linear Systems....Pages 69-94
Matrices that Generate the same Krylov Residual Spaces....Pages 95-118
Incomplete Block Factorizations as Preconditioners for Sparse SPD Matrices....Pages 119-133
How Fast Can Iterative Methods Be....Pages 135-147
Rational Krylov Algorithms for Nonsymmetric Eigenvalue Problems....Pages 149-164
Highly Parallel Preconditioners for General Sparse Matrices....Pages 165-199
A Two-Stage Iteration for Solving Nearly Completely Decomposable Markov Chains....Pages 201-216
Minimum Residual Modifications to BI-CG and to the Preconditioner....Pages 217-225

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The solution of very large sparse or structured linear algebra problems is an integral part of many scientific computations. Direct methods for solving such problems are often infeasible because of computation time and memory requirements, and so iterative techniques are used instead. In recent years much research has focussed on the efficient solution of large systems of linear equations, least squares problems, and eigenvalue problems using iterative methods. This volume on iterative methods for sparse and structured problems brings together researchers from all over the world to discuss topics of current research. Areas addressed included the development of efficient iterative techniques for solving nonsymmetric linear systems and eigenvalue problems, estimating the convergence rate of such algorithms, and constructing efficient preconditioners for special classes of matrices such as Toeplitz and Hankel matrices. Iteration strategies and preconditioners that could exploit parallelism were of special interest. This volume represents the latest results of mathematical and computational research into the development and analysis of robust iterative methods for numerical linear algebra problems. This volume will be useful for both mathematicians and for those involved in applications using iterative methods.
Content:
Front Matter....Pages i-xvi
Some Themes in Gene H. Golub’s Work on Iterative Methods....Pages 1-11
Computing the Sparse Singular Value Decomposition via SVDPACK....Pages 13-29
Gaussian Quadrature Applied to Adaptive Chebyshev Iteration....Pages 31-44
Ordering Effects on Relaxation Methods Applied to the Discrete Convection-Diffusion Equation....Pages 45-57
On the Error Computation for Polynomial Based Iteration Methods....Pages 59-67
Transpose-Free Quasi-Minimal Residual Methods for Non-Hermitian Linear Systems....Pages 69-94
Matrices that Generate the same Krylov Residual Spaces....Pages 95-118
Incomplete Block Factorizations as Preconditioners for Sparse SPD Matrices....Pages 119-133
How Fast Can Iterative Methods Be....Pages 135-147
Rational Krylov Algorithms for Nonsymmetric Eigenvalue Problems....Pages 149-164
Highly Parallel Preconditioners for General Sparse Matrices....Pages 165-199
A Two-Stage Iteration for Solving Nearly Completely Decomposable Markov Chains....Pages 201-216
Minimum Residual Modifications to BI-CG and to the Preconditioner....Pages 217-225
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