●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
編輯手記
