Abstract not found

Abstract not found

Abstract not found

We study the approximate GCD of two univariate polynomials given with limited accuracy or, equivalently, the exact GCD of the perturbed polynomials within some prescribed tolerance. A perturbed polynomial is regarded as a family of polynomials in a classification space, which leads to an accurate analysis of the computation. Considering only the Sylvester matrix singular values, as is frequently suggested in the literature, does not suffice to solve the problem completely, even when the extended euclidean algorithm is also used. We provide a counterexample that illustrates this claim and indicates the problem's hardness. SVD computations on subresultant matrices lead to upper bounds on the degree of the approximate GCD. Further use of the subresultant matrices singular values yields an approximate syzygy of the given polynomials, which is used to establish a gap theorem on certain singular values that certifies the maximum-degree approximate GCD. This approach leads directly to an algorithm for computing the approximate GCD polynomial. Lastly, we suggest the use of weighted norms in order to sharpen the theorem's conditions in a more intrinsic context.

We describe algorithms for computing the greatest common divisor (GCD) of two univariate polynomials with inexactlyknown coefficients. Assuming that an estimate for the GCD degree is available (e.g., using an SVD-based algorithm), we formulate and solve a nonlinear optimization problem in order to determine the coefficients of the "best" GCD. We discuss various issues related to the implementation of the algorithms and present some preliminary test results. 1 Introduction There are many applications in which it is necessary to compute the greatest common divisor (GCD) of two or more polynomials. For example, symbolic computation programs must be able to simplify rational functions, such as (x 2 + 4x + 4)=(x + 2). Sometimes, the coefficients may be inexact, due to the accumulation of floating-point errors or to imprecise input (e.g., the coefficients come from physical measurements). This situation can cause great difficulties in GCD computation. Suppose we have p(x) = x 2 +3:999x...

A new algorithm is presented for factoring bivariate approximate polynomials over C [x, y]. Given a particular polynomial, the method constructs a nearby composite polynomial, if one exists, and its irreducible factors. Subject to a conjecture, the time to produce the factors is polynomial in the degree of the problem. This method has been implemented in Maple, and has been demonstrated to be efficient and numerically robust.

In the first part of this paper, we dene approximate polynomial gcds (greatest common divisors) and extended gcds provided that approximations to the zeros of the input polynomials are available. We relate our novel definition to the older and weaker ones, based on perturbation of the coefficients of the input polynomials, we demonstrate some deficiency of the latter definitions (which our denition avoids), and we propose new effective sequential and parallel (RNC and NC) algorithms for computing approximate gcds and extended gcds. Our stronger results are obtained with no increase of the asymptotic bounds on the computational cost. This is partly due to application of our recent nearly optimal algorithms for approximating polynomial zeros. In the second part of our paper, working under the older and more customary definition of approximate gcds, we modify and develop an alternative approach, which was previously based on the computation of the Singular Value Decomposition (SVD) of the associat...

We propose a fast algorithm for computing approximate GCD of univariate polynomials with coefficients that are given only to a finite accuracy. The algorithm is based on a stabilized version of the generalized Schur algorithm for Sylvester matrix and its embedding. All computations can be done in O(n 2) operations, where n is the sum of the degrees of polynomials. The stability of the algorithm is also discussed. 1.

Let a and b be two polynomials having numerical coefficients. We consider the question: when are a and b relatively prime? Since the coefficients of a and b are approximant, the question is the same as: when are two polynomials relatively prime, even after small perturbations of the coefficients? In this paper we provide a numeric parameter for determining that two polynomials are prime, even under small perturbations of the coefficients. Our methods rely on an inversion formula for Sylvester matrices to establish an effective criterion for relative primeness. The inversion formula can also be used to approximate the condition number of a Sylvester matrix. 1

In this paper we provide a fast, numerically stable algorithm to determine when two given polynomials a 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 is only available up to a certain precision. Our method -- an extension of the Cabay--Meleshko algorithm for Pad'e 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. 1 Introduction Let a; b 2 C[z] be (univariate) polynomials with real or complex coefficients a(z) = a 0 + a 1 z + : : ...