We present an iterative algorithm for calculating approximate greatest common divisor (GCD) of univariate polynomials with the real or the complex coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. The problem of approximate GCD is transfered to a constrained minimization problem, then solved with the so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. We demonstrate that, in some test cases, our algorithm calculates approximate GCD with perturbations as small as those calculated by a method based on the structured total least norm (STLN) method and the UVGCD method, while our method runs significantly faster than theirs by approximately up to 30 or 10 times, respectively, compared with their implementation. We also show that our algorithm properly handles some ill-conditioned polynomials which have a GCD with small or large leading coefficient.

We present an iterative algorithm for calculating approximate greatest common divisor (GCD) of univariate polynomials with the real coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. The problem of approximate GCD is transfered to a constrained minimization problem, then solved with a so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. We demonstrate that our algorithm calculates approximate GCD with perturbations as small as those calculated by a method based on the structured total least norm (STLN) method, while our method runs significantly faster than theirs by approximately up to 30 times, compared with their implementation. We also show that our algorithm properly handles some ill-conditioned problems with GCD containing small or large leading coefficient.

Abstract. We present an extension of our GPGCD method, an iterative method for calculating approximate greatest common divisor (GCD) of univariate polynomials, to multiple polynomial inputs. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. In our GPGCD method, the problem of approximate GCD is transferred to a constrained minimization problem, then solved with the so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. In this paper, we extend our method to accept more than two polynomials with the real coefficients as an input.

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Let P(x) be a univariate polynomial over C, such that P(x) = Cn xn +... + cm+ lxm+ where max{]cn [,..., [cm+l I} = 1 and e = max{Iem_l [, [em_z[U2,..., leo[!/m} << 1. P(x) has m small roots around the origin so long as e << 1. In 1999, we derived a formula that if e < 1/9 then P(x) has m roots inside a disc Din of radius Rin and other n -m roots outside a disc Dout of radius Rout, located at the origin, where Rin(out) = [1 -(+)X/I -(16e)/(1 + 3e)Z]. (1 + 3e)/4. Note that Rin = Rout ife = 1/9. Our formula is essentially the same as that derived independently by Yakoubsohn at almost the same time. In this short article, we introduce the formula and check its sharpness on many polynomials generated randomly.