Using an Efficient Sparse Minor Expansion Algorithm to Compute Polynomial Subresultants and the Greatest Common Denominator (original) (raw)
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Using an efficient sparse minor expansion algorithm to compute polynomial subresultants and GCD
In this paper, the use of an efficient sparse minor expansion method to directly compute the subresultants needed for the greatest common denominator (GCD) of two polynomials is described. The sparse minor expansion method (applied either to Sylvester's or Bezout's matrix) naturally computes the coefficients of the subresultants in the order corresponding to a polynomial remainder sequence (PRS), avoiding wasteful recomputation as much as possible. It is suggested that this is an efficient method to compute the resultant and GCD of sparse polynomials.
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.
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Proceedings of the 2005 international symposium on Symbolic and algebraic computation - ISSAC '05, 2005
Let G = (4y 2 + 2z)x 2 + (10y 2 + 6z) be the greatest common divisor (gcd) of two polynomials A, B ∈ Z[x, y, z]. Because G is not monic in the main variable x, the sparse modular gcd algorithm of Richard Zippel cannot be applied directly as one is unable to scale univariate images of G in x consistently. We call this the normalization problem. We present two new sparse modular gcd algorithms which solve this problem without requiring any factorizations. The first, a modification of Zippel's algorithm, treats the scaling factors as unknowns to be solved for. This leads to a structured coupled linear system for which an efficient solution is still possible. The second algorithm reconstructs the monic gcd x 2 + (5y 2 + 3z)/(2y 2 + z) from monic univariate images using a sparse, variable at a time, rational function interpolation algorithm.
A sparse modular GCD algorithm for polynomials over algebraic function fields
Proceedings of the 2007 international symposium on Symbolic and algebraic computation - ISSAC '07, 2007
We present a first sparse modular algorithm for computing a greatest common divisor of two polynomials f1, f2 ∈ L[x] where L is an algebraic function field in k ≥ 0 parameters with r ≥ 0 field extensions. Our algorithm extends the dense algorithm of Monagan and van Hoeij from 2004 to support multiple field extensions and to be efficient when the gcd is sparse. Our algorithm is an output sensitive Las Vegas algorithm. We have implemented our algorithm in Maple. We provide timings demonstrating the efficiency of our algorithm compared to that of Monagan and van Hoeij and with a primitive fraction-free Euclidean algorithm for both dense and sparse gcd problems.
Matrix computation of subresultant polynomial remainder sequences in integral domains
Reliable Computing, 1995
We present an improved variant of the matrix-triangularization subresultant prs method [2] for the computation of a greatest common divisor of two polynomials A and B (of degrees m and n, respectively) along with their polynomial remainder sequence. It is improved in the sense that we obtain complete theoretical results, independent of Van Vleck's theorem [13] (which is not always true [1], [6]), and, instead of transforming a matrix of order 2•max(m, n) [2], we are now transforming a matrix of order m + n. An example is also included to clarify the concepts.
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...
A subdivision-based algorithm for the sparse resultant
2000
Abstract Multivariate resultants generalize the Sylvester resultant of two polynomials and characterize the solvability of a polynomial system. They also reduce the computation of all common roots to a problem in linear algebra. We propose a determinantal formula for the sparse resultant of an arbitrary system of n+ 1 polynomials in n variables. This resultant generalizes the classical one and has significantly lower degree for polynomials that are sparse in the sense that their mixed volume is lower than their B��zout number.
A practical method for the sparse resultant
1993
Abstract We propose an efficient method for computing the resultant, of a sparse polynomial system of n+ 1 equations in n unknowns. Our approach carries over from[(UE93] and constructs a matrix whose determinant is a nonzero multiple of the resultant, and from which the latter is easily extracted. For certain classes of syskms, it attains optimality by expressing the resultant, as a sillgle determinant. An illll> lenlelltatioll of the algorithm is described and empirical results presented and conlpared with those from[CE93] and [SZ].
Sparse Polynomial Pseudo Division Using a Heap
We present a new algorithm for pseudo-division of sparse multivariate polynomials with integer coefficients. It uses a heap of pointers to simultaneously merge the dividend and partial products, sorting the terms efficiently and delaying all coefficient arithmetic to produce good complexity. The algorithm uses very little memory and we expect it to run in the processor cache. We give benchmarks comparing our implementation to existing computer algebra systems.
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.