Date of Award

2021

Degree Type

Dissertation

Degree Name

Doctor of Philosophy in Applied Mathematics

Department

Mathematics and Applied Mathematical Sciences

First Advisor

James Baglama

Abstract

This thesis presents new hybrid restarted Lanczos methods for computing eigenpairs and singular triplets of large matrices. Our methods combine thick-restarting with Ritz or harmonic Ritz vectors with iteratively refined Ritz vectors to compute a few of the extreme eigenpairs of symmetric matrices or singular triplets of rectangular matrices. The refined process improves the (harmonic) Ritz values/vectors yielding better approximations, i.e., this process results in a “smaller” residual norm compared to just using Ritz/harmonic vectors. The iterative refined process we developed improves the refined values/vectors by using a scheme, where we replace the approximate eigenvalue/singular value in the original refined scheme with the latest computed refined Ritz value until convergence. The thick-restarting schemes are superior in reference to efficiency to other restarted schemes, but are not available when using refined or iterative refined Ritz vectors. Therefore, we developed hybrid restarted methods that switch between the efficient thick-restarted scheme and restarting with a linear combination of “the better approximating” iterative refined Ritz vectors. Our developed methods have shown to be very effective on small subspaces, i.e., when memory is limited. We provide many theoretical results and numerical examples.

Share

COinS
 
 

To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.