This is the home of the research of Anthony Kellems and Dr. Mark Embree, Rice University, Summer 2003.
Our research consisted of looking at two main methods for solving eigenvalue
problems. The first method, Rayleigh Quotient Iteration, was studied to
discover how choices of initial starting vectors affected convergence. We
developed some visual tools to let us analyze these problems in a very
attractive manner for the 3-by-3 cases, and thus our attention was focused for
problems of that order. Other visual tools let us examine certain aspects of
RQI for matrices in higher dimensions, but the results, although somewhat
visually interesting, were not very substantial.
The second method was Arnoldi's Method, an iterative scheme used to find a
small number of eigenvalues for a very large matrix. Although it's main
applications are for matrices of dimension 10,000 or greater, we found it
beneficial to understanding the dynamics of this process by studying the 3-by-3
case and utilizing the visual tools we previously developed for RQI.
The following are links to some of the images, graphs, and results of our
research.
Alpha Pics--A gallery of RQI convergence images
organized by matrix class. Movie files can also be found here which show the
evolution of the unit sphere and pseudospectra for each class.
A Visual Analysis of RQI--PDF file of the final report
for our RQI research. It is entitled "A Visual Analysis of Convergence of
Iterative Eigenvalue Methods". August 1, 2003.
Relevant Files--PDF file indicating and briefly
annotating the important MATLAB scripts, graphs, and images produced during our
research.
MATLAB Goodies--Critical MATLAB m-files written
for this research. This code can be used to verify our findings or to explore
RQI and Arnoldi's Method further.