Backward error analysis of the extended iterative refinement or improvement algorithm for solving ill conditioned linear system

Abstract

Solving an ill conditioned linear system \(Ax = b\) is a central problem in matrix computations. We solved the ill conditioned linear system \(Ax = b\) using the preconditioning method and the Schur aggregation approach. The Schur aggregation is a process of transforming the linear system \(Ax = b\) into better conditioned linear systems of small sizes, with well conditioned matrices \(V^HC^{-1}\), \(C^{-1}U\), and \(S = I_{r} - V^HC^{-1}U\) using the Sherman–Morrison–Woodbury formula \(A^{-1}=(C-UV^H)^{-1}=C^{-1}+C^{-1}U(I_r-V^HC^{-1}U)^{-1}V^HC^{-1}\). We used the technique of extended iterative refinement or improvement algorithm to compute the Schur aggregate \(S = I_{r} - V^HC^{-1}U\) with high precision. In this paper we provide an extensive backward error analysis of the algorithm also we calculate a bound of the backward error.