Numerical Verification Method for Dense Linear Systems with Arbitrarily Ill-conditioned Matrices

Abstract—This paper is concerned with the problem of verifying an accuracy of a computed solution of linear sys-tems with an arbitrarily ill-conditioned coefficient matrix. In this paper, a method of obtaining an accurate computed solution of such linear systems and its verified error bound is proposed. The proposed method is based on the accurate computation of dot product and IEEE standard 754 arith-metic. A verified and accurate computed solution with a desired tolerance can be obtained by the proposed method with iterative refinement. Numerical results are presented for illustrating the effectiveness of the proposed method. 1