The ill-conditioned cases we consider are 1) the small leading-coefficient case, 2) small leading-coefficient GCD case, 3) big leading coefficient case, and 4) approximately singular leading-coefficient case. We propose three new algorithms for computing the approximate GCDs in these cases. The first one is to stabilize the univariate PRS by avoiding the… (More)
Given polynomials with floating-point number coefficients, one can now compute the approximate GCD stably, except in ill-conditioned cases where the GCD has small or large leading coefficient/constant term. The cost is <i>O</i>(<i>m</i><sup>2</sup>), where <i>m</i> is the maximum of degrees of given polynomials. On the other hand, for polynomial with… (More)
We present algorithms for multivariate GCD and approximate GCD by modifying Barnett's theorem, which is based on the LU-decomposition of Bézout matrix. Our method is suited for multivariate polynomials with large degrees. Also, we analyze ill-conditioned cases of our method. We show our method is stabler and faster than many other methods.