6 research outputs found
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
The computation of multiple roots of a Bernstein basis polynomial
This paper describes the algorithms of Musser and Gauss for the computation of multiple roots of a theoretically exact Bernstein basis polynomial ˆ 5 f(y) when the coefficients of its
given form f(y) are corrupted by noise. The exact roots of f(y) can therefore be assumed to be simple, and thus the problem reduces to the calculation of multiple roots of a polynomial f˜(y) that is near f(y), such that the backward error is small. The algorithms require many greatest common divisor (GCD) computations and polynomial deconvolutions, both of which are implemented by a structure-preserving matrix method. The motivation of these algorithms arises from the unstructured and structured condition numbers of a multiple root of a polynomial. These condition numbers have an elegant interpretation in terms of the pejorative manifold of ˆ 12 f(y), which allows the geometric significance of the GCD computations and polynomial deconvolutions to be considered. A variant of the Sylvester resultant matrix is used for the GCD computations because it yields better results than the standard form of this matrix, and the polynomial deconvolutions can be computed in several different ways, sequentially or simultaneously, and with the inclusion or omission of the preservation of the structure of the coefficient matrix. It is shown that Gauss’ algorithm yields better results than Musser’s algorithm, and the reason for these superior results is explained