矩陣迭代分析

《國外數學名著系列（影印版）13：矩陣迭代分析（第二版）》的作者現任英國肯特大學教授，多種國際雜誌主編或編委。

《國外數學名著系列（影印版）13：矩陣迭代分析（第二版）》第一版1962年由Prentice Hall出版，是矩陣迭代分析方面的經典教材。此次修訂，有些章節吸收了新的研究成果，如弱正則分裂方面的結果；有些章節則增添了新的內容，引述了最近的定理，更新了參考文獻，讀者從中可以瞭解一些新的發展方向。此次修訂，新的章節的內容基本上都是自含的，並添加了習題。原版主要基於線性代數方法，而修訂版強調藉助其他領域的工具，如逼近論和共型映射理論，得到更加新穎的結果。《國外數學名著系列（影印版）13：矩陣迭代分析（第二版）》尤其適合從事數值分析的科研人員和研究生閲讀。 ^[2] 

1． Matrix Properties and Concepts

1．1 Introduction

1．2 A Simple Example

1．3 Norms and Spectral Radii

1．4 Bounds for the Spectral Radius of a Matrix and Directed Graphs

1．5 Diagonally Dominant Matrices

1．6 Ovals of Cassini

2． Nonnegative Matrices

2．1 Spectral Radii of Nonnegative Matrices

2．2 Cyclic and Primitive Matrices

2．3 Reducible Matrices

2．4 Nonnegative Matrices and Directed Graphs

3． Basic Iterative Methods and Comparison Theorems

3．1 The Point Jacobi， Gauss-Seidel， and Successive Overrelaxation Iterative Methods

3．2 Average Rates of Convergence

3．3 The Stein-Rosenberg Theorem

3．4 The Ostrowski-Reich Theorem

3．5 Stieltjes Matrices， M-Matrices and H-Matrices

3．6 Regular and Weak Regular Splittings of Matrices

4． Successive Overrelaxation Iterative Methods

4．1 p-Cyclic Matrices

4．2 The Successive Overrelaxation Iterative Method for p-Cyclic Matrices

4．3 Theoretical Determination of an Optimum Relaxation Factor

4．4 Extensions of the 2-Cyclic Theory of Matrices

4．5 Asymptotic Rates of Convergence

4．6 CO(q，r) and GCO(q，r)： Generalized Consistent Orderings

5． Semi-Iterative Methods

5．1 Semi-Iterative Methods and Chebyshev Polynomials

5．2 Relationship of Semi-Iterative Methods to Successive Overrelaxation Iterative Methods

5．3 Comparison of Average Rates of Convergence： the Weakly Cyclic Case

5．4 Cyclic Reduction and Related Iterative Methods

5．5 Semi-Iterative Methods Applied to the Successive Overrelaxation Method

6． Derivation and Solution of Elliptic Difference Equations

6．1 A Simple Two-Point boundary-Value Problem

6．2 General Second-Order Ordinary Differential Equations

6．3 Derivation of Finite Difference Approximations in Higher Dimensions

6．4 Factorization Techniques and block Iterative Methods

6．5 Asymptotic Convergence Rates for the Model Problem

7． Alternating-Direction Implicit Iterative Methods

7．1 The Peaceman-Rachford Iterative Method

7．2 The Commutative Case

7．3 The Noncommutative Case

7．4 Variants of the Peaceman-Rachford Iterative Method

8． Matrix Methods for Parabolic Partial Differential Equations

8．1 Semi-Discrete Approximation

8．2 Essentially Positive Matrices

8．3 Matrix Approximations for exp (-tS)

8．4 Relationship with Iterative Metholds for Solving Elliptic Difference Equations

8．5 Chebyshev Rational Approximations for exp (-tS)

9． Estimation of Acceleration Parameters

9．1 Application of the Theory of Nonnegative Matrices

9．2 Application of Isoperimetric Inequalities

A． Appendix

B． Appendix

References

Index ^[2]