Numerical Solution of Eigenvalue Problems and Related Least Squares Problems

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This research seeks to improve the accuracy of methods to solve some particular eigenvalue 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 necessary and sufficient condition for a matrix to be ``well-behaved'' is being investigated. A number of related issues have to be considered, including: analysis of methods for updating indefinite eigenvalue problems; investigation of methods for updating the ULV and ULLV approximations to the singular value decomposition and generalized singular value decomposition; and development of analysis of methods for computing the sparse generalized singular value decomposition. The last two issues are extremely important in the solution of ill-posed least squares problems and total least squares problems. The implications of derived results for these problems are being investigated.