Numerical Solution of Eigenvalue and Singular Value Problems with Applications

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This research concerns the development and analysis of numerical algorithms for the solution of eigenvalue and singular value problems, with applications to least squares and total least squares problems. Much of the work grows out of perturbation theory for generalized singular value problems. This theory is also applicable to certain generalized eigenvalue problems. The theory will thus be applied to algorithm development for various singular value and least squares problems. Specific problems to be addressed are 1. Development and analysis of methods for computing the sparse generalized singular value decomposition. 2. A more accurate procedure for computing a bidiagonal reduction of a matrix. This is the first step in the most efficient stable methods for computing the singular value decomposition. 3. An algorithm for the eigenvectors of symmetric arrow matrices that requires less communication on distributed architectures. 4. An algorithm for the solution of recursive total least squares problems using rank--revealing decompositions.