Accurate and efficient algorithm for solving ill-conditioned linear systems by preconditioning methods

Abstract

In this paper, we develop an accurate and efficient algorithm for solving ill-conditioned linear systems. For this purpose, we propose two preconditioning methods that are based on LU factorization. One is the method using the inverse of an LU factor. The other is the method using the residual of an LU factorization. The latter method requires less computational cost than the former one. Using an LU factorization with the iterative refinement, we can accurately solve a linear system Ax =b for $\kappa(A) \lesssim u^{-1}$, where u is the relative rounding error unit in working precision. If we use the proposed algorithm with accurate dot product, we can obtain an accurate approximate solution for ill-conditioned linear systems beyond the limit of the working precision. Results of numerical experiments show that the proposed algorithm can work for $\kappa(A) \lesssim u^{-2}$ in reasonable computing time.