Exactly Solve Integer Linear Systems Using Numerical Methods

We present a new fast way to exactly solve non-singular linear systems with integer coefficients using numerical methods. This leads to high performace in practice and also leads to asymptoticly fast algorithms for some special family of linear systems. Our method is to find an initial approximate solution by using a fast numerical method, then amplify the approximate solution by a scalar, adjust the amplified solution and corresponding residual so that they can be stored exactly, and repeat these steps to refine the solution until high enough accuracy is achieved, and finally reconstruct the rational solution. We will expose the theoretical cost and show some experimental results.