A matrix pencil based numerical method for the computation of the GCD of polynomials

摘要

The paper presents a new numerical method for the computation of the greatest common divisor (GCD) of an m-set of polynomials of R[s], P/sub m,d/, of maximal degree d. It is based on a previously proposed theoretical procedure (Karcanias, 1989) that characterizes the GCD of P/sub m,d/ as the output decoupling zero polynomial of a linear system S(A/spl circ/,C/spl circ/) that may be associated with P/sub m,d/. The computation of the GCD is thus reduced to finding the finite zeros of the pencil sW-AW, where W is the unobservable subspace of S(A/spl circ/,C/spl circ/). If k=dim W, the GCD is determined as any nonzero entry of the kth compound C/sub k/(sW-A/spl circ/W). The method defines the exact degree of GCD, works satisfactorily with any number of polynomials and evaluates successfully approximate solutions.