Summary: 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 thesis we consider the most general nonlinear eigenvalue problem without assumptions on the structure 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 the thesis connects Newton-type methods for nonlinear eigenvalue problems and nonlinear Rayleigh functionals.
The one-sided and the two-sided/generalized Rayleigh functional are introduced with complex range, in contrast to the one-sided functional of the literature. Such functionals are the generalizations of Rayleigh quotients for matrices. Local uniqueness and bounds for the distance to the exact eigenvalue in terms of the angles of eigenvectors and approximations to eigenvectors are derived. We obtain a first order perturbation bound which is used to show stationarity, and which implies the Lipschitz continuity of the functionals.
With the so gained knowledge on Rayleigh functionals, we design new basic methods for the computation of eigenvalues and -vectors: The two-sided Rayleigh functional iteration is the analogon to the two-sided Rayleigh quotient iteration for nonnormal matrices, and is shown to converge locally cubically as well. These methods are important for subspace extension methods like Jacobi-Davidson and nonlinear Arnoldi, that need to solve small dimensional projected nonlinear problems. We compare the methods regarding computational costs and initial assumptions, and show numerical results.
A technique to accelerate convergence for methods computing left and right eigenvector is introduced and the convergence improvement is shown in terms of the R-order. The new methods are called half-step methods. The half-step two-sided Rayleigh functional iteration is shown to converge with R-order 4.
Using the previous results, we discuss various nonlinear Jacobi-Davidson methods. The proposed methods are compared theoretically regarding asymptotic condition numbers, and in practice with examples.
The special case of nonlinear complex symmetric eigenvalue problems is examined. We show the appropriate definition of a complex symmetric Rayleigh functional, which is used to derive a complex symmetric Rayleigh functional iteration which converges locally cubically, and the complex symmetric residual inverse iteration method. A complex symmetric Jacobi-Davidson algorithm is formulated and tested numerically.