非线性特征值问题——牛顿型方法与非线性瑞利函数

本书考虑了最一般的非线性特征值问题，而没有对结构或谱进行假设。该书首先提供了有关简单特征值和奇异值分解的逆算子表示条件的基本事实；其次研究了非线性复对称特征值问题的特例，给出了复对称瑞利函数的适当定义，该定义用于导出局部三次收敛的复对称瑞利函数的迭代，以及复对称剩余逆迭代方法，制定了复对称雅可比—戴维森算法，并进行了数值测试；最后还介绍了一种用于计算左右特征向量的方法的加速收敛技术，并以R阶代表了收敛改进。

1 Introduction
1.1 Motivation Ⅰ
1.2 The NonlineaLr Eigenvalue Pmblem
1.3 Variational Principle
1.4 Outline
1.5 Motivation Ⅱ
2 Preliminaries and Basic Results
2.1 Basic Properties
2.2 Assumptions
2.2.1 Real-valued Problems
2.2.2 Complex-valued Problems
2.3 Eigenvalue Condition Numbers
2.4 Representation of the Inverse Operator
2.5 Angles and Distances
3 Nonlinear Rayleigh Functionals
3.1 Introduction and Historical Review
3.2 Existence and Stationarity of the Generalized Rayleigh Functional
3.2.1 Real-valued Problems
3.2.2 Complex-valued Problems
3.3 The Standard Nonlinear Rayleigh Functional
3.3.1 Structured Problems
3.3.2 General Problems
3.3.3 Perturbation Expansion
3.4 Generalized Quotient vs.Functional
3.4.1 Two-sided Quotient and Functional
3.4.2 One-sided Quotient and Functional
3.5 Conclusion
4 Newton-type Methods
4.1 Methods for Approximating One Eigenpair
4.2 Methods for Approximating One Eigentriple
4.2.1 Two-sided Rayleigh Functional Iteration
4.2.2 Two-sided Residual Inverse Iteration
4.2.3 Alternating Rayleigh Functional Ireration
4.2.4 Generalized Rayleigh Functional Iteration
4.3 Theoretical Comparison of the Methods
4.4 Computation of the Rayleigh Functional
4.5 Numerical Experiments
4.6 Conclusion
5 Half-step Methods
5.1 Half-step Rwleigh Functional Iteration
5.2 Half-step Generalized Rayleigh Functional Iteration
5.3 Half-step Residual hwerse Iteration
5.4 Numerical Experiments
6 Jacobi-Davidson-type Methods
6.1 Introduction
6.2 Nonlinear Jacobi-Davidson
6.3 Generalized Jacobi-Davidson-type Methods
6.4 Solving the Preconditioned Correction Equation
6.5 Asymptotic Condition Numbers
6.6 Numerical Examples
6.6.1 A Linear Problem
6.6.2 Two Exceptional Cases
6.6.3 More Examples
6.7 Conclusion
7 Nonlinear Complex Symmetric Jacobi-Davidson
7.1 Introduction
7.2 The Rayleigh Functional
7.3 Complex Symmetric Rayleigh Functional Iteration
7.4 Complex Symmetric Residual Inverse Iteration
7.5 Nonlinear Complex Symmetric Jacobi-Davidson
7.6 Numerical Examples
7.7 Conclusion
Summary and Outlook
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