The Schur Aggregation for Linear Systems and Determinants

According to our previous theoretical and experimental study, additive preconditioners can be readily computed for ill conditioned matrices, but it is not straightforward how such preconditioners can help us to facilitate the solution of linear systems of equations, computation of determinants, and other fundamental matrix computations. We develop some nontrivial techniques for this task. By applying the Sherman–Morrison–Woodbury formula and its new variations, we confine the original numerical problems to the computation of the Schur aggregates of smaller sizes. We overcome these problems by extending the classical algorithm for iterative refinement and applying advanced double-precision algorithms for high precision computation of sums and products. 2000 Math. Subject Classification: 65A12, 65F22, 65F05, 65F40