A Fast and Numerically Stable Euclidean-like Algorithm for Detecting Relatively Prime Numerical Polynomials

In this paper we provide a fast, numerically stable algorithm to determine when two given polynomialsa and b are relatively prime and remain relatively prime even after small perturbations of their coefficients. Such a problem is important in many applications where input data are only available up to a certain precision. Our method—an extension of the Cabay–Meleshko algorithm for Padé approximation—is typically an order of magnitude faster than previously known stable methods. As such it may be used as an inexpensive test which may be applied before attempting to compute a “numerical GCD”, in general a much more difficult task. We prove that the algorithm is numerically stable and give experiments verifying the numerical behaviour. Finally, we discuss possible extensions of our approach that can be applied to the problem of actually computing a numerical GCD.