An algorithm to solve integer linear systems exactly using numerical methods

Abstract

In this paper, we present a new algorithm for the exact solutions of linear systems with integer coefficients using numerical methods. It terminates with the correct answer in well-conditioned cases or quickly aborts in ill-conditioned cases. Success of this algorithm on a linear equation requires that the linear system must be sufficiently well-conditioned for the numeric linear algebra method being used to compute a solution with sufficient accuracy. Our method is to find an initial approximate solution by using a numerical method, then amplify the approximate solution by a scalar, and adjust the amplified solution and corresponding residual to integers so that they can be computed without large integer arithmetic involved and can be stored exactly. Then we repeat these steps to refine the solution until sufficient accuracy is achieved, and finally reconstruct the rational solution. Our approximating, amplifying, and adjusting idea enables us to compute the solutions without involving high precision software floating point operations in the whole procedure or large integer arithmetic except at the final rational reconstruction step. We will expose the theoretical cost and show some experimental results.