COMPUTATION OF THE GCD OF POLYNOMIALS USING GAUSSIAN TRANSFORMATIONS AND SHIFTING

A new numerical method for the computation of the greatest common divi sor (GCD) of an m-set of polynomials of R[s], P(m,d) of maximal degree d, is presented. This method is based on a recently developed theoret ical algorithm (Karcanias 1987) that uses elementary transformations a nd shifting operations; the present algorithm takes into account the n on-generic nature of GCD and thus uses steps, which minimize the intro duction of additional errors and defines the GCD in an approximate sen se. For a given set P(m,d), with a basis matrix P(m), the method defin es first, the most orthogonal uncorrupted base P(r) from the rows of P (m), where r = rank (P(m)) less-than-or-equal-to m. By applying succes sively gaussian transformations and shifting, on the basis matrix P(r) is-an-element-of R(rx(d+1)), we produce each time a new basis matrix P(z) with z = rank (P(z)) < r. The method terminates when the rank of P(z) is approximately equal to 1; the coefficient vector of the GCD is then defined as a row of the unit rank matrix P(z). The method define s the exact degree of the GCD, successfully evaluates an approximate s olution and works satisfactorily with large numbers of polynomials of any fixed degree.