RICE UNIVERSITY Ritz Values and Arnoldi Convergence for Nonsymmetric Matrices

Ritz Values and Arnoldi Convergence for Nonsymmetric Matrices by Russell Carden The restarted Arnoldi method, useful for determining a few desired eigenvalues of a matrix, employs shifts to refine eigenvalue estimates. In the symmetric case, using selected Ritz values as shifts produces convergence due to interlacing. For nonsymmetric matrices the behavior of Ritz values is insufficiently understood, and hence no satisfactory general convergence theory exists. Towards developing such a theory, this work demonstrates that Ritz values of nonsymmetric matrices can obey certain geometric constraints, as illustrated through careful analysis of Jordan blocks. By constructing conditions for localizing the Ritz values of a matrix with one simple normal wanted eigenvalue, this work develops sufficient conditions that guarantee convergence of the restarted Arnoldi method with exact shifts. As Ritz values are the basis for many iterative methods for determining eigenvalues and solving linear systems, an understanding of Ritz value behavior for nonsymmetric matrices has the potential to inform a broad range of analysis. Acknowledgments First, I would like to thank the members of my committee for their encouragement. I thank my adviser Dr. Embree whose skills as an applied mathematician, professor and writer are inspiring. I thank the instructors of the thesis writing class, especially Dr. Hewitt, and Dr. Sorensen who have taken the time to show students the demands of writing, particularly mathematical writing, as well as the demands of a career as an applied mathematician. I thank Josef Sifuentes for helping me prepare for my defense. I thank my officemate Nabor Reyna for keeping me on task. I thank Dr. Richard Tapia and Dr. Pablo Tarazaga, without whom I would not have considered applying to Rice. I thank Dr. Stephen Sedory for sparking my interest in linear algebra. Finally, I thank my family, for supporting me in all my endeavors.