The schur aggregation for solving linear systems of equations

Abstract

According to our previous theoretical and experimental study, additive preconditioners can be readily computed for ill conditioned matrices, but application of such preconditioners to facilitating matrix computations is not straight-forward. In the present paper we develop some nontrivial techniques for this task.They enabled us to con ne the original numerical problems to the computation of the Schur aggregates of smaller sizes. We overcome these problems by extending the Wilkinson's iterative re nement and applying some advanced semi-symbolic algorithms for multiplication and summation.In particular with these techniques we control precision throughout our computations.