GPGCD: An iterative method for calculating approximate GCD of univariate polynomials

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.Comment: Preliminary versions have been presented as doi:10.1145/1576702.1576750 and arXiv:1007.183

Symbolic-numeric algorithms for univariate polynomials

Thesis (Ph. D. in Science)--University of Tsukuba, (B), no. 2485, 2010.3.25 Includes bibliographical referencesNote to the re-typeset version: This is re-typeset version of the original dissertation. While I have maintained the original contents without changing any words and/or formulas in the main body, I have added the following information: 1. Copyright notice of corresponding articles in each chapter; 2. Digital Object Identifiers (DOI) or URLs of references as many as possible.Please note that the number of pages is slightly increased in the present edition from that of the original edition, possibly by changes of page style parameters.200

Over-constrained Weierstrass iteration and the nearest consistent system

We propose a generalization of the Weierstrass iteration for over-constrained systems of equations and we prove that the proposed method is the Gauss-Newton iteration to find the nearest system which has at least $k$ common roots and which is obtained via a perturbation of prescribed structure. In the univariate case we show the connection of our method to the optimization problem formulated by Karmarkar and Lakshman for the nearest GCD. In the multivariate case we generalize the expressions of Karmarkar and Lakshman, and give explicitly several iteration functions to compute the optimum. The arithmetic complexity of the iterations is detailed

Structured total least norm and approximate GCDs of inexact polynomials

The determination of an approximate greatest common divisor (GCD) of two inexact polynomials f=f(y) and g=g(y) arises in several applications, including signal processing and control. This approximate GCD can be obtained by computing a structured low rank approximation S*(f,g) of the Sylvester resultant matrix S(f,g). In this paper, the method of structured total least norm (STLN) is used to compute a low rank approximation of S(f,g), and it is shown that important issues that have a considerable effect on the approximate GCD have not been considered. For example, the established works only yield one matrix S*(f,g), and therefore one approximate GCD, but it is shown in this paper that a family of structured low rank approximations can be computed, each member of which yields a different approximate GCD. Examples that illustrate the importance of these and other issues are presented

Study on Calculating Power-series Roots of Multivariate Algebraic Equations

課題番号：1970000

Matrix representation of the shifting operation and numerical properties of the ERES method for computing the greatest common divisor of sets of many polynomials

The Extended-Row-Equivalence and Shifting (ERES) method is a matrix-based method developed for the computation of the greatest common divisor (GCD) of sets of many polynomials. In this paper we present the formulation of the shifting operation as a matrix product which allows us to study the fundamental theoretical and numerical properties of the ERES method by introducing its complete algebraic representation. Then, we analyse in depth its overall numerical stability in finite precision arithmetic. Numerical examples and comparison with other methods are also presented

A Subdivision Method for Computing Nearest Gcd with Certification

International audienceA new subdivision method for computing the nearest univariate gcd is described and analyzed. It is based on an exclusion test and an inclusion test. The xclusion test in a cell exploits Taylor expansion of the polynomial at the center of the cell. The inclusion test uses Smale's alpha-theorems to certify the existence and unicity of a solution in a cell. Under the condition of simple roots for the distance minimization problem, we analyze the complexity of the algorithm in terms of a condition number, which is the inverse of the distance to the set of degenerate systems. We report on some experimentation on representative examples to illustrate the behavior of the algorithm