A Class Of Iterative Projection Methods For Nonlinear Large Sparse Eigenvalue Problems

A class of numerical methods for nonlinear large sparse eigenvalue problems are studied in this thesis.Rational Krylov method and nonlinear Arnoldi method arc effective methods to find all eigenvalues of a nonlinear eigenvalue problem in some region.Both algorithms are belong to iterative projection inethods and do indeed resemble each other.The difference between them lies only in the way of implementation.Nonlinear Arnoldi method has the advantage over nonlinear Krylov method that it can take a flexible and more effieient way to solve the projected eigenvalue problem.But nonlinear rational Krylov method does not require the explicit form of the projected eigenvalue problem and may be more suitable for some particular applications.Nonliner Krylov method use a linearization to approximate the nonlinear problem and solve the projected eigenvalue problem approximately.By making use of the idea of refined projection and adopting refined vector instead of Ritz vector,we present the refined nonlinear rational Krylov method.In view of the considerable expense of LU factorization for large system,we nse a inner iteration to solve the precondition system and propose the inexact nonlinear Arnotdi method. Practical algorithms are given for these methods and numerical examples with them indicate the improved two methods presented in this thesis are superior in speed of convergence.