Large Sparse Stable Matrix Computations

Abstract

Abstract : The project proposal discussed two problem areas: (1) The solution of large sparse of linear equations; and (2) The solution of sparse least squares problems. We report significant progress in both of these areas and in a third area, the solution of the algebraic eigenvalue problem. The progress in solving systems of linear equations included an algorithm for computing ordering for efficiently factoring sparse symmetric, positive definite systems in parallel. We also made important progress in computing the ordering itself in parallel. Other progress included a method for handling singular blocks in a one-way dissection ordering and an error analysis of Gaussian elimination in unnormalized arithmetic. For linear least squares problems we developed an efficient reliable method for detecting the rank of a sparse matrix without column exhanges. The method used a static data structure. We also analyzed and compared methods for computing sparse and dense QR factorizations on message passing architectures. On the algebraic eigenvalue problem, we participated in resolving long standing open questions on relative perturbation bounds on certain diagonally dominant eigenvalue problems. (KR)