Numerical Solution of Eigenvalue Problems

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This research concerns the improvement of the accuracy of methods to solve two particular eignenvalue problems. The first is the classical symmetric eigenvalue problem. Recent results have shown that there is a large class of symmetric eigenvalue problems for which small structured perturbations result in small changes in the eigenvalues in the relative sense. This class of matrices is called "well- behaved". The problem of finding a neccessary and sufficent condition for a matrix to be "well-behaved" will be investigated. Another concern is that only a few algorithms have been shown to achieve relative accuracy on a subset of the set of "well- behaved" matrices. The investigation will try to discover if that set of algorithms can be expanded. The divide and conquer approaches of Cuppen, Dongarra, Jessup, Sorensen, and Tang are considered. Good absolute error bounds on these algorithms have already been established and it remains to show tighter relative error bounds for diagonal dominant problems and for the singular value decomposition. The other large problem area concerns the eigenvector problem for non-symmetric matrices. The problem arises in the numerical solution of Markov modeling problems. Accurate general bounds on Gaussian elimination have been proven. These allow for the straighforward application of standard sparse matrix techniques to this problem. However, better structured perturbation bounds need to be proved for the nearly uncoupled case.