System theoretic based characterisation and computation of the least common multiple of a set of polynomials

AbstractThe paper provides a system theoretic characterisation of the least common multiple (LCM) m(s) of a given set of polynomials P which leads to an efficient numerical procedure for the computation of LCM that avoids root finding and use of greatest common divisor (GCD) procedures. The procedure that is presented also leads to the computation of the associated set of multipliers of P with respect to LCM. The basis of the new characterisation and computational procedure are the controllability properties of a natural realization S(A,b,C) associated with the set P. It is shown, that the coefficients of the LCM are defined by the properties of the controllable subspace of the pair (A,b), which also leads to the characterisation of associated multipliers. An algorithmic procedure that exploits the companion structure of A is formulated and its numerical properties are investigated. The new method provides a robust procedure for the computation of LCM and enables the computation of approximate values, when the original data are innacurate