Solving an ill conditioned linear system  accurately is a difficult task in numerical analysis. Our method of solving an ill conditioned linear system is done in three steps. We first use the concept of the additive preconditioning  for preconditioner  of a smaller rank r to decrease the conditioned number of A. Next we use the Sherman-Morrison-Woodbury (SMW) formula  to avoid the direct computation of  This step reduces the computation of  to the computation of the Schur aggregate  Last we compute S with high precision using the technique of extended rative refinement/improvement. This paper aims to prove that under a certain condition the matrix  is nonsingular and

where  and  is the perturbation to the matrix C.

error analysis, ill conditioned linear system, Sherman-Morrison- Woodbury (SMW) formula, preconditioning, Schur aggregation, iterative refinement or improvement, singular value decomposition, algorithms, matrix.