This research focuses on devising and analyzing practical numerical algorithms for the solution of large nonsymmetric eigenvalue problems. Such problems arise frequently in the numerical simulation or analysis of physical or man-made systems in many different disciplines. The algorithms are based upon Lanczos recursions, are not application specific but will be tested on matrices arising in various applications. Arnoldi algorithms will also be considered but only as they relate to Lanczos algorithms. The basic forms of both these types of methods do not explicitly transform the iteration matrix, and this is crucial for very large matrices. These algorithms are of interest because they appear to be able to compute other portions of the spectrum in addition to the magnitude dominant eigenvalues, and preliminary numerical experiments indicate that this is a very promising approach. There are however many open questions about both the applicability and viability of nonsymmetric Lanczos procedures. Therefore, the overall goals of this research are understanding the strengths and weaknesses of both of these types of methods, their applicability, and what is required to obtain practical robust Lanczos algorithms. This research will be a combination of analysis and extensive numerical experiments. Three types of interactive activities being carried out are: (1) A graduate course on practical numerical algorithms for large scale eigenvalue problems; (2) A 'field-directed' interdisciplinary seminar with a significant proportion of women speakers; and (3) A brown bag lunch series for women students only, designed to serve as counselling sessions on any topics proposed by the students.
