Iterative Methods for Large Linear Systems.

By: Kincaid, David RContributor(s): Hayes, Linda JMaterial type: TextTextSeries: eBooks on DemandPublisher: Kent : Elsevier Science & Technology, 2014Copyright date: ©1990Description: 1 online resource (350 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9781483260204Subject(s): Iterative methods (Mathematics) -- Congresses | Parallel processing (Electronic computers) -- Congresses | Vector processing (Computer science) -- CongressesGenre/Form: Electronic books.Additional physical formats: Print version:: Iterative Methods for Large Linear SystemsDDC classification: 519.542 LOC classification: QA432.I84 1990Online resources: Click here to view this ebook.
Contents:
Front Cover -- Iterative Methods for Large Linear Systems -- Copyright Page -- Table of Contents -- Preface -- Authors of Chapters -- Papers Presented at Conference -- Professor David M. Young, Jr. -- Photographs from Conference -- Chapter 1. Fourier Analysis of Two-Level Hierarchical Basis Preconditioners -- 1 Introduction -- 2 1D, 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 -- References -- Chapter 2. 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 -- 3 Recursive Definition of Preconditioner -- 4 The Relative Condition Number of M with Respect to A -- 5 Concluding Remarks -- References -- Chapter 3. ELLPACK and ITPACK as Research Tools for Solving Elliptic Problems -- 1 Background -- 2 ELLPACK and ITPACK -- 3 Some Basic Questions -- 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 -- References -- Chapter 4. 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 -- References -- Chapter 5. A Local Relaxation Scheme (Ad-Hoc SOR) Applied to Nine Pointand Block Difference Equations -- 1 History -- 2 The Method -- 3 Nine Point Application: Cross Derivatives -- 4 Block Iteration -- Acknowledgements -- References -- Chapter 6. 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 Expe -- Acknowledgements -- References -- Chapter 7. 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 -- Appendix -- Chapter 8. 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 w -- 5 Numerical Results -- Acknowledgements -- References -- Chapter 9. Hodie Approximation of Boundary Conditions -- 1 Introduction -- 2 Approximation 'A way from the Boundary' -- 3 Hodie as Interpolation -- 4 Boundary Conditions -- 5 Extension of Ui,j to Ω -- 6 Indexing of Unknowns -- 7 Eigenproblems -- Acknowledgements -- References -- Chapter 10. Iterative Methods for Nonsymmetric Linear Systems -- 1 Introduction -- 2 Projection Methods -- 3 Krylov Projection Methods -- 4 Semi-Krylov Projection Methods -- 5 Non-polynomial Projection Methods -- 6 Non-projection Polynomial Methods -- 7 Conclusion -- Acknowledgements -- References -- Chapter 11. Solution of Three-Dimensional Generalized Poisson Equations on Vector Computers -- 1 Introduction -- 2 Discretization -- 3 The SSOR Preconditioned Conjugate Gradient Method -- 4 Numerical Results -- 5 Summary and Conclusions -- Acknowledgements -- References -- Chapter 12. Multi-Level Asynchronous Iteration for PDEs -- 1 Introduction -- 2 Multiple Level Asynchronous PDE Algorithms -- 3 A Unified Model of Parallel Computation -- 4 Model of Multi-Level IterationOn a Hypercube Machine -- 5 Mapping Multi-Level Structures Onto a Hypercube.
6 Analysis of the Iteration and its Performance -- Acknowledgements -- References -- Chapter 13. An Adaptive Algorithm for Richardson's Method -- 1 Introduction -- 2 The Numerical Framework -- 3 The Power Method for Eigenvalues -- 4 Finding the Optimal Richardson Parameters -- 5 The Minimum Residual Method -- 6 Algorithm -- Summary -- Acknowledgements -- References -- Chapter 14. A Note on the SSOR and USSOR Iterative Methods Applied to p-Cyclic Matrices -- 1 Introduction -- 2 Statement of Main Result and Discussion -- 3 Proof of the Theorem -- Acknowledgements -- References -- Chapter 15. The ADI Minimax Problem for Complex Spectra -- 1 Introduction and Review of Results for Real Spectra -- 2 Early Analysis of Complex Spectra -- 3 The Family of Elliptic Function Domains -- 4 Spectral Boundary -- 5 Spectrum Partitioning -- 6 Subspace Refinement -- Acknowledgements -- References -- Chapter 16. Some Domain Decomposition Algorithms for Elliptic Problems -- 1 Introduction -- 2 Substructures, Subspaces and Projection -- 3 Schwarz Methods -- 4 Analysis of an Additive Schwarz Method -- 5 Iterative Substructuring Methods -- Acknowledgements -- References -- Chapter 17. The Search for Omega -- 1 Introduction -- 2 Iterative Algorithms and Iteration Parameters -- 3 A Priori Techniques -- 4 Adaptive Techniques -- 5 The Nonsymmetric Case -- Acknowlegements -- References -- Index.
Summary: Iterative Methods for Large Linear Systems contains a wide spectrum of research topics related to iterative methods, such as searching for optimum parameters, using hierarchical basis preconditioners, utilizing software as a research tool, and developing algorithms for vector and parallel computers. This book provides an overview of the use of iterative methods for solving sparse linear systems, identifying future research directions in the mainstream of modern scientific computing with an eye to contributions of the past, present, and future. Different iterative algorithms that include the successive overrelaxation (SOR) method, symmetric and unsymmetric SOR methods, local (ad-hoc) SOR scheme, and alternating direction implicit (ADI) method are also discussed. This text likewise covers the block iterative methods, asynchronous iterative procedures, multilevel methods, adaptive algorithms, and domain decomposition algorithms. This publication is a good source for mathematicians and computer scientists interested in iterative methods for large linear systems.
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Front Cover -- Iterative Methods for Large Linear Systems -- Copyright Page -- Table of Contents -- Preface -- Authors of Chapters -- Papers Presented at Conference -- Professor David M. Young, Jr. -- Photographs from Conference -- Chapter 1. Fourier Analysis of Two-Level Hierarchical Basis Preconditioners -- 1 Introduction -- 2 1D, 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 -- References -- Chapter 2. 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 -- 3 Recursive Definition of Preconditioner -- 4 The Relative Condition Number of M with Respect to A -- 5 Concluding Remarks -- References -- Chapter 3. ELLPACK and ITPACK as Research Tools for Solving Elliptic Problems -- 1 Background -- 2 ELLPACK and ITPACK -- 3 Some Basic Questions -- 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 -- References -- Chapter 4. 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 -- References -- Chapter 5. A Local Relaxation Scheme (Ad-Hoc SOR) Applied to Nine Pointand Block Difference Equations -- 1 History -- 2 The Method -- 3 Nine Point Application: Cross Derivatives -- 4 Block Iteration -- Acknowledgements -- References -- Chapter 6. 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 Expe -- Acknowledgements -- References -- Chapter 7. 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 -- Appendix -- Chapter 8. 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 w -- 5 Numerical Results -- Acknowledgements -- References -- Chapter 9. Hodie Approximation of Boundary Conditions -- 1 Introduction -- 2 Approximation 'A way from the Boundary' -- 3 Hodie as Interpolation -- 4 Boundary Conditions -- 5 Extension of Ui,j to Ω -- 6 Indexing of Unknowns -- 7 Eigenproblems -- Acknowledgements -- References -- Chapter 10. Iterative Methods for Nonsymmetric Linear Systems -- 1 Introduction -- 2 Projection Methods -- 3 Krylov Projection Methods -- 4 Semi-Krylov Projection Methods -- 5 Non-polynomial Projection Methods -- 6 Non-projection Polynomial Methods -- 7 Conclusion -- Acknowledgements -- References -- Chapter 11. Solution of Three-Dimensional Generalized Poisson Equations on Vector Computers -- 1 Introduction -- 2 Discretization -- 3 The SSOR Preconditioned Conjugate Gradient Method -- 4 Numerical Results -- 5 Summary and Conclusions -- Acknowledgements -- References -- Chapter 12. Multi-Level Asynchronous Iteration for PDEs -- 1 Introduction -- 2 Multiple Level Asynchronous PDE Algorithms -- 3 A Unified Model of Parallel Computation -- 4 Model of Multi-Level IterationOn a Hypercube Machine -- 5 Mapping Multi-Level Structures Onto a Hypercube.

6 Analysis of the Iteration and its Performance -- Acknowledgements -- References -- Chapter 13. An Adaptive Algorithm for Richardson's Method -- 1 Introduction -- 2 The Numerical Framework -- 3 The Power Method for Eigenvalues -- 4 Finding the Optimal Richardson Parameters -- 5 The Minimum Residual Method -- 6 Algorithm -- Summary -- Acknowledgements -- References -- Chapter 14. A Note on the SSOR and USSOR Iterative Methods Applied to p-Cyclic Matrices -- 1 Introduction -- 2 Statement of Main Result and Discussion -- 3 Proof of the Theorem -- Acknowledgements -- References -- Chapter 15. The ADI Minimax Problem for Complex Spectra -- 1 Introduction and Review of Results for Real Spectra -- 2 Early Analysis of Complex Spectra -- 3 The Family of Elliptic Function Domains -- 4 Spectral Boundary -- 5 Spectrum Partitioning -- 6 Subspace Refinement -- Acknowledgements -- References -- Chapter 16. Some Domain Decomposition Algorithms for Elliptic Problems -- 1 Introduction -- 2 Substructures, Subspaces and Projection -- 3 Schwarz Methods -- 4 Analysis of an Additive Schwarz Method -- 5 Iterative Substructuring Methods -- Acknowledgements -- References -- Chapter 17. The Search for Omega -- 1 Introduction -- 2 Iterative Algorithms and Iteration Parameters -- 3 A Priori Techniques -- 4 Adaptive Techniques -- 5 The Nonsymmetric Case -- Acknowlegements -- References -- Index.

Iterative Methods for Large Linear Systems contains a wide spectrum of research topics related to iterative methods, such as searching for optimum parameters, using hierarchical basis preconditioners, utilizing software as a research tool, and developing algorithms for vector and parallel computers. This book provides an overview of the use of iterative methods for solving sparse linear systems, identifying future research directions in the mainstream of modern scientific computing with an eye to contributions of the past, present, and future. Different iterative algorithms that include the successive overrelaxation (SOR) method, symmetric and unsymmetric SOR methods, local (ad-hoc) SOR scheme, and alternating direction implicit (ADI) method are also discussed. This text likewise covers the block iterative methods, asynchronous iterative procedures, multilevel methods, adaptive algorithms, and domain decomposition algorithms. This publication is a good source for mathematicians and computer scientists interested in iterative methods for large linear systems.

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