Global Arnoldi Type Algorithms For Large Non-hermitian Eigenproblems With Special Applications To Multiple Eigenvalue Problems

This thesis is concerned with global Arnoldi type algorithms for large non-Hermitian eigenproblems. Global projection methods have been used for solving nu-merous large matrix equations, but it is not the case for the large eigenproblems. In thisthesis, based on the global Arnoldi process that generates an F-orthonormal basis of amatrix Krylov subspace, a global Arnoldi method is proposed for large eigenproblems.It computes certain F-Ritz pairs that are used to approximate some eigenpairs.Though the global Arnoldi method can be used to compute exterior eigenpairs,it does not work well for interior eigenvalue problems. Based on the global Arnoldiprocess, we propose a global harmonic Arnoldi method. We propose computing the F-Rayleigh quotients of the large non-Hermitian matrix with respect to harmonic F-Ritzvectors and take them as new approximate eigenvalues. They are better and more reli-able than the harmonic F-Ritz values. However, the approximate eigenvectors obtainedby the above methods may converge erratically and may even fail to converge. To cor-rect this deficiency, global refined Arnoldi type methods are proposed that replace theapproximate eigenvectors by certain refined eigenvector approximations, called refinedF-Ritz vectors and refined harmonic F-Ritz vectors, respectively.The global Arnoldi type methods inherit convergence properties of the standardArnoldi type methods. It is shown that the global Arnoldi type methods are able tosolve multiple eigenproblems both in theory and practice. To be practical, the implicitrestart technique advocated by Sorensen is applied to the global Arnoldi type methodsand the implicitly restarted global Arnoldi type algorithms are developed. The selectionof shifts is one of the keys for the success of the algorithms and overall performance.We propose corresponding shifts for use within each algorithm and show why theywork. Numerical experiments show e?ciency and reliability of these new algorithms.