Fraction-free RNS algorithms for solving linear systems

Abstract

This paper is concerned with overcoming the arithmetic problems which arise in the solution of linear systems with integer coefficients. Specifically, solving each systems using (integer) Gauss elimination or its variants usually results in severe growth in the dynamic range of the integers that must be represented. To alleviate this problem, a residue number system (RNS) can be utilized so that large integers can be represented by a vector of residues which require only short wordlengths. RNS arithmetic, however, cannot easily handle any divisions that are needed in the solution process. This paper presents fraction-free algorithms for the solution of integer systems. This does involve divisions, but only divisions where the result is known to be an exact integer. The other principal contribution of this paper is the presentation of an RNS division algorithm for exact integer division which does not require any conversion to standard binary form. It uses entirely modular arithmetic, perhaps including a step equivalent to RNS base extension.