Two Resultant Based Methods Computing the Greatest Common Divisor of Two Polynomials

Abstract

In this paper we develop two resultant based methods for the computation of the Greatest Common Divisor (GCD) of two polynomials. Let S be the resultant Sylvester matrix of the two polynomials. We modified matrix S to S ^*, such that the rows with non-zero elements under the main diagonal, at every column, to be gathered together. We constructed modified versions of the LU and QR procedures which require only the $\frac{1}{3}$ of floating point operations than the operations performed in the general LU and QR algorithms. Finally, we give a bound for the error matrix which arises if we perform Gaussian elimination with partial pivoting to S ^*. Both methods are tested for several sets of polynomials and tables summarizing all the achieved results are given.