Applied Iterative Methods
- 1st Edition - September 28, 1981
- Author: Louis A. Hageman
- Language: English
- eBook ISBN:9 7 8 - 1 - 4 8 3 2 - 9 4 3 7 - 7
Applied Iterative Methods discusses the practical utilization of iterative methods for solving large, sparse systems of linear algebraic equations. The book explains different… Read more
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Request a sales quoteApplied Iterative Methods discusses the practical utilization of iterative methods for solving large, sparse systems of linear algebraic equations. The book explains different general methods to present computational procedures to automatically determine favorable estimates of any iteration parameters, as well as when to stop the iterative process. The text also describes the utilization of iterative methods to solve multidimensional boundary-value problems (such as discretization stencil, mesh structure, or matrix partitioning) which affect the cost-effectiveness of iterative solution procedures. The book cites case studies involving iterative methods applications, including those concerning only three particular boundary-value problems. The text explains polynomial acceleration procedures (for example, Chebyshev acceleration and conjugate gradient acceleration) which can be applied to certain basic iterative methods or to the successive overtaxation (SOR) method. The book presents other case studies using the iterative methods to solve monoenergetic transport and nonlinear network flow multidimensional boundary-value problems. The text also describes the procedures for accelerating basic iterative methods which are not symmetrizable. The book will prove beneficial for mathematicians, students, and professor of calculus, statistics, and advanced mathematics.
Preface
Acknowledgments
Notation
Chapter 1 Background on Linear Algebra and Related Topics
1.1 Introduction
1.2 Vectors and Matrices
1.3 Eigenvalues and Eigenvectors
1.4 Vector and Matrix Norms
1.5 Partitioned Matrices
1.6 The Generalized Dirichlet Problem
1.7 The Model Problem
Chapter 2 Background on Basic Iterative Methods
2.1 Introduction
2.2 Convergence and Other Properties
2.3 Examples of Basic Iterative Methods
2.4 Comparison of Basic Methods
2.5 Other Methods
Chapter 3 Polynomial Acceleration
3.1 Introduction
3.2 Polynomial Acceleration of Basic Iterative Methods
3.3 Examples of Nonpolynomial Acceleration Methods
Chapter 4 Chebyshev Acceleration
4.1 Introduction
4.2 Optimal Chebyshev Acceleration
4.3 Chebyshev Acceleration with Estimated Eigenvalue Bounds
4.4 Sensitivity of the Rate of Convergence to the Estimated Eigenvalues
Chapter 5 An Adaptive Chebyshev Procedure Using Special Norms
5.1 Introduction
5.2 The Pseudoresidual Vector δ(n)
5.3 Basic Assumptions
5.4 Basic Adaptive Parameter and Stopping Relations
5.5 An Overall Computational Algorithm
5.6 Treatment of the W-Norm
5.7 Numerical Results
Chapter 6 Adaptive Chebyshev Acceleration
6.1 Introduction
6.2 Eigenvector Convergence Theorems
6.3 Adaptive Parameter and Stopping Procedures
6.4 An Overall Computational Algorithm Using the 2-Norm
6.5 The Estimation of the Smallest Eigenvalue μN
6.6 Numerical Results
6.7 Iterative Behavior When ME > μ1
6.8 Singular and Eigenvector Deficient Problems
Chapter 7 Conjugate Gradient Acceleration
7.1 Introduction
7.2 The Conjugate Gradient Method
7.3 The Three-Term Form of the Conjugate Gradient Method
7.4 Conjugate Gradient Acceleration
7.5 Stopping Procedures
7.6 Computational Procedures
7.7 Numerical Results
Chapter 8 Special Methods for Red/Black Partitionings
8.1 Introduction
8.2 The RS-SI and RS-CG Methods
8.3 The CCSI and CCG Procedures
8.4 Numerical Results
8.5 Arithmetic and Storage Requirements
8.6 Combined (Hybrid) Chebyshev and Conjugate Gradient Iterations
8.7 Proofs
Chapter 9 Adaptive Procedures for the Successive Overrelaxation Method
9.1 Introduction
9.2 Consistently Ordered Matrices and Related Matrices
9.3 The SOR Method
9.4 Eigenvector Convergence of the SOR Difference Vector
9.5 SOR Adaptive Parameter and Stopping Procedures
9.6 An Overall Computational Algorithm
9.7 The SOR Method for Problems with Red/Black Partitionings
9.8 Numerical Results
9.9 On the Relative Merits of Certain Partitionings and Certain Iterative Procedures
9.10 Proofs of Theorems and Discussion of the Strategy Condition (9-5.21)
Chapter 10 The Use of Iterative Methods in the Solution of Partial Differential Equations
10.1 Introduction
10.2 The Time-Independent Two-Dimensional Problem
10.3 The Time-Independent Three-Dimensional Problem
10.4 The Time-Dependent Problem
Chapter 11 Case Studies
11.1 Introduction
11.2 The Two-Group Neutron Diffusion Problem
11.3 The Neutron Transport Equation in x-y Geometry
11.4 A Nonlinear Network Problem
Chapter 12 The Nonsymmetrizable Case
12.1 Introduction
12.2 Chebyshev Acceleration
12.3 Generalized Conjugate Gradient Acceleration Procedures
12.4 Lanczos Acceleration
12.5 Acceleration Procedures for the GCW Method
12.6 An Example
Appendix A Chebyshev Acceleration Subroutine
Appendix B CCSI Subroutine
Appendix C SOR Subroutine
Bibliography
Index
- No. of pages: 386
- Language: English
- Edition: 1
- Published: September 28, 1981
- Imprint: Academic Press
- eBook ISBN: 9781483294377
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