Nonlinear eigenvalue problems arise in many fields of natural and engineering sciences. Theoretical and practical results are scattered in the literature and in most cases they have been developed for a certain type of problem. In this book we consider the most general nonlinear eigenvalue problem without assumptions on the struct.ure or spectrum. We start by providing basic facts on the conditioning of a simple eigenvalue and an inverse operator representation in terms of the singular value decomposition. The main part of this work connects Newton-type methods for nonlinear eigenvalue problems and nonlinear Rayleigh functionals.

1 Introduction
1．1 Motivation I
1．2 The Nonlinear Eigenvalue Problem
1．3 Variational Principles
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 lteration
4．2．2 Two-sided Residual Inverse Iteration
4．2．3 Alternating Rayleigh Functional Iteration
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
6 Jacobi-Davidson-type Methods
7 Nonlinear Complex Symmetric Jacobi-Davidson
8 Summary and Outlook

编辑手记