On the Remainders Obtained in Finding the Greatest Common Divisor of Two Polynomials (original) (raw)
Related papers
2015
In 1917 Pell 1 and Gordon used sylvester2, Sylvester's little known and hardly ever used matrix of 1853, to compute 2 the coefficients of a Sturmian remainder-obtained in applying in É[x], Sturm's algorithm on two polynomials f, g ∈ [x] of degree n-in terms of the determinants 3 of the corresponding submatrices of sylvester2. Thus, they solved a problem that had eluded both J. J. Sylvester, in 1853, and E. B. Van Vleck, in 1900. 4 In this paper we extend the work by Pell and Gordon and show how to compute 2 the coefficients of an Euclidean remainder-obtained in finding in É[x], the greatest common divisor of f, g ∈ [x] of degree n-in terms of the determinants 5 of the corresponding submatrices of sylvester1, Sylvester's widely known and used matrix of 1840. See the link http://en.wikipedia.org/wiki/Anna\_Johnson\_Pell\_Wheeler for her biography. 2 Both for complete and incomplete sequences, as defined in the sequel. 3 Also known as modified subresultants. 4 Using determinan...
Computing the Greatest Common Divisor of Polynomials Using the Comrade Matrix
Lecture Notes in Computer Science, 2008
The comrade matrix of a polynomial is an analogue of the companion matrix when the matrix is expressed in terms of a general basis such that the basis is a set of orthogonal polynomials satisfying the three-term recurrence relation. We present the algorithms for computing the comrade matrix, and the coefficient matrix of the corresponding linear systems derived from the recurrence relation. The computing times of these algorithms are analyzed. The computing time bounds, which dominate these times, are obtained as functions of the degree and length of the integers that represent the rational number coefficients of the input polynomials. The ultimate aim is to apply these computing time bounds in the analysis of the performance of the generalized polynomial greatest common divisor algorithms.
Four New Algorithms for Multivariate Polynomial GCD
2016
Four new algorithms for multivariate polynomial GCD (greatest common divisor) are given. The first is a simple improvement of PRS (polynomial remainder sequence) algorithms. The second is to calculate a Groebner basis with a certain term ordering. The third is to calculate subresultant by treating the coefficients as truncated power series. The fourth is to calculate PRS by treating the coefficients as truncated power series. The first and second algorithms are not important practically, but the third and fourth ones are quite efficient and seem to be useful practi-cally. 1.
Three new algorithms for multivariate polynomial GCD
Journal of Symbolic Computation, 1992
Three new algorithms for multivariate polynomial GCD (greatest common divisor) are given. The first is to calculate a GrSbner basis with a certain term ordering. The second is to calculate the subresultant by treating the coefficients w.r.t, the main variable as truncated power series. The third is to calculate a PRS (polynomial remainder sequence) by treating the coefficients as truncated power series. The first algorithm is not important praetioaUy, but the second and third ones are efficient and seem to be useful practically. The third algorithm has been implemented naively and compared with the trial-division PRS algorithm and the EZGCD algorithm. Although it is too early to derive a definite conclusion, the PRS method with power series coefficients is very efficient for calculating low degree GCD of high degree non-sparse polynomials.
On the computation of the GCD (LCM) of 2-d polynomials
2007 European Control Conference (ECC), 2007
The main contribution of this work is to provide an algorithm for the computation of the GCD and LCM of 2-d polynomials, based on the DFT techniques. The whole theory is implemented via illustrative examples.
Resultant properties of gcd of many polynomials and a factorization representation of gcd
International Journal of Control, 2003
The properties of the greatest common divisor (gcd) of a set of polynomials of R½s may be investigated using the Sylvester resultant. New properties of the Sylvester resultant linked to gcd are established and these lead to canonical factorizations of resultants expressing the extraction of common divisors from the elements of the original set. These results lead to a new representation of the gcd introduced in terms of a canonical factorization of the Sylvester resultant into a reduced Sylvester resultant and a Toeplitz matrix representing the gcd. The use of the Sylvester resultant allows a simplification of the ERES and matrix pencils computational procedures for gcd computation and provides the means for formulating simpler and robust computational procedures.
Linear Algebra and Its Applications, 2006
Barnett's method through Bezoutians is a purely linear algebra method allowing to compute the degree of the greatest common divisor of several univariate polynomials in a very compact way. Two different uses of this method in computer algebra are introduced here. Firstly, we describe an algorithm for parameterizing the greatest common divisor of several polynomials in K[x, y], being x a parameter taking values in an real field K. Secondly, we consider the problem of computing the approximate greatest common divisor with limited accuracy for several univariate polynomials following Corless et al. [R.M. Corless, P.M. Gianni, B.M. Trager, S. Watt, The singular value decomposition for polynomial systems, in: ACM International Symposium on Symbolic and Algebraic Computation, 1995, pp. 195-207]. Given a family of polynomials whose coefficients are imperfectly known, we describe an algorithm for computing their approximate greatest common divisor by using, as main tools, Barnett's method and singular value decomposition computations. Furthermore, we show how to use this algorithm in order to obtain the approximate squarefree decomposition of a given polynomial with imperfectly known coefficients.
A sympy/sage Module for Computing Polynomial Remainder Sequences: [preprint]
2017
The Modified Euclidean prs is obtained by modifying the sign of the remainder of each polynomial division performed for the computation of the Euclidean prs. Analogously, the Modified Subresultant prs is obtained by modifying the matrix from which the Subresultant prs is obtained. Even though prs’s (c) and (d) are computed by evaluating sub-determinants of given matrices, our objective is to compute all four prs’s using the same type of polynomial divisions over the ring Z[x]. Our objective is not at all trivial and has eluded the efforts of great mathematicians, as our brief review below indicates. Initially, Collins, Brown and Traub [8], [9], [11], [12] used the so called prem pseudo-remainder function defined by LC(g) · f = q · g + h, (1)
On the computation of the GCD of 2-D polynomials
2007
The main contribution of this work is to provide an algorithm for the computation of the GCD of 2-D polynomials, based on DFT techniques. The whole theory is implemented via illustrative examples.
A simple algorithm for GCD of polynomials
Annals of mathematics and physics, 2022
Based on the Bezout approach we propose a simple algorithm to determine the gcd of two polynomials which doesn't need division, like the Euclidean algorithm, or determinant calculations, like the Sylvester matrix algorithm. The algorithm needs only n steps for polynomials of degree n. Formal manipulations give the discriminant or the resultant for any degree without needing division nor determinant calculation.