Exact solutions to linear systems of equations using output sensitive lifting

Abstract

Many methods have been developed to symbolically solve systems of linear equations over the rational numbers. A common approach is to use p-adic lifting or iterative refinement to build a modular or approximate solution, then apply rational number reconstruction. An upper bound can be computed on the number of iterations these algorithms must perform before applying rational reconstruction. In practice such bounds can be conservative. Output sensitive lifting is the technique of performing rational reconstruction at intermediate steps of the algorithm and verifying correctness which allows the possibility of early termination when the solution size is small. In this paper we show how using an appropriate output sensitive lifting strategy can improve several algorithms. We show this procedure to be computationally effective and introduce a variant of the iterative-refinement method that incorporates warm starts into the rational reconstruction procedure.