Partially Orthogonal Generalized Arnoldi Method And Its Applications On Nonlinear Eigenvalue Problems

Polynomial eigenvalue problems, rational eigenvalue problems and general nonlinear eigenvalue problems arise in a variety of applications, such as stability analysis of control systems, dynamic analysis of structures, fluid-solid vibration, stability analysis of time-delay systems and so on. This thesis deals with numerical methods for solving polynomial eigenvalue problems, rational eigenvalue problems and general nonlinear eigenvalue problems. The following results are obtained.Based on the semi-orthogonal generalized Arnoldi method for solving quadratic eigenvalue problems, we propose a partially orthogonal generalized Arnoldi method for solving the polynomial eigenvalue problems. In order to enhance the method, we combine the partially orthogonal generalized Arnoldi method with the refine scheme, implicitly restarted strategy and deflation technique, and present an implicitly restarted refined partially orthogonal generalized Arnoldi method with deflation technique.Based on the theory of low-rank modification, we analyze the spectral properties and distribution of the rational eigenvalue problems, and present the k-value iteration method and k-value iteration method with range transformation for the rational eigenvalue problems combined with the partially orthogonal generalized Arnoldi method.We present a successive high-order approximation method for solving the general nonlinear eigenvalue problems, and analyze its local convergence. Combining with the partially orthogonal generalized Arnoldi method, we give the successive high-order approximated partially orthogonal generalized Arnoldi method for solving the nonlinear eigenvalue problems.Numerical results show that the obtained results are correct and the proposed methods are efficient.