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.
目录 1 Introduction l 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 C0r刚iOn Equation 6.5 Asymptotic Condition Numbers 6.6 Numerical Examples
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