# Applied Iterative Methods

## 1st Edition

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eBook ISBN: 9781483294377
Published Date: 28th September 1981
Page Count: 386
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## Description

Applied 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

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

7.1 Introduction

7.3 The Three-Term Form of the Conjugate Gradient Method

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