Numerical computation of the least common multiple of a set of polynomials

Abstract

The paper presents a number of properties of the least common multiple (LCM) m(s) of a given set of polynomials P. These results lead to the formulation of a new procedure for computing the LCM that avoids the computation of roots. This procedure involves the computation of the greatest common divisor (GCD) z(s) of a set of polynomials T derived from P, and the factorisation of the product of the original set P, p(s) as p(s) = m(s)·z(s). The symbolic procedure leads to a numerical one, where robust methods for the computation of GCD are first used. In this numerical method the approximate factorisation of polynomials is an important part of the overall algorithm. The latter problem is handled by studying two associated problems: evaluation of order of approximation and the optimal completion problem. The new method provides a robust procedure for the computation of LCM and enables the computation of approximate values, when the original data are inaccurate.