Date of Award


Degree Type


Degree Name

Doctor of Philosophy in Applied Mathematics


Mathematics and Applied Mathematical Sciences

First Advisor

James Baglama


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.



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