3,934 Hits in 4.6 sec

On the genericity of the modular polynomial GCD algorithm

Erich Kaltofen, Michael B. Monagan
1999 Proceedings of the 1999 international symposium on Symbolic and algebraic computation - ISSAC '99  
In this paper we study the generic setting of the modular GCD algorithm. We develop the algorithm for multivariate polynomials over Euclidean domains which have a special kind of remainder function.  ...  Applying this generic algorithm to a GCD problem in Z/(p)[t] [x] where p is small yields an improved asymptotic performance over the usual approach, and a very practical algorithm for polynomials over  ...  Barry Trager shared the Axiom code for the modular GCD algorithm with us for comparison.  ... 
doi:10.1145/309831.309861 dblp:conf/issac/KaltofenM99 fatcat:b3d3tgpkpneybig62duxhzgp2a

On square-free factorization of multivariate polynomials over a finite field

Laurent Bernardin
1997 Theoretical Computer Science  
In this paper we present a new deterministic algorithm for computing the square-free decomposition of multivariate polynomials with coefficients from a finite field.  ...  The new algorithm is also simpler to implement and it can rely on any existing GCD algorithm without having to worry about choosing "good" evaluation points.  ...  The resulting algorithm will, in general, be much more efficient than algorithms based on Musser's as the GCD computations involve polynomials of smaller degree.  ... 
doi:10.1016/s0304-3975(97)00059-5 fatcat:3zucsxzip5dchdpeayu5sijx2m

High-performance polynomial GCD computations on graphics processors

Pavel Emeliyanenko
2011 2011 International Conference on High Performance Computing & Simulation  
Our approach exhibits block structure to distribute the computation of a single modular GCD over several thread blocks, and thus to remove any hardware limitations on the maximal size of polynomials that  ...  We propose an algorithm to compute a greatest common divisor (GCD) of univariate polynomials with large integer coefficients on Graphics Processing Units (GPUs).  ...  IMPLEMENTATION We begin with a general framework of the algorithm. Next, we briefly discuss the modular arithmetic on the GPU.  ... 
doi:10.1109/hpcsim.2011.5999827 dblp:conf/ieeehpcs/Emeliyanenko11 fatcat:saz6fddhqjbgtdbslwpmvojdyi

On Degrees of Modular Common Divisors and the Big PrimegcdAlgorithm

Vahagn Mikaelian
2016 International Journal of Mathematics and Mathematical Sciences  
Our modifications are based on bounds of degrees of modular common divisors of polynomials, on estimates of the number of prime divisors of a resultant, and on finding preliminary bounds on degrees of  ...  To illustrate the ideas we apply the constructed algorithms on certain polynomials, in particular on polynomials from Knuth's example of intermediate expression swell.  ...  Acknowledgments The author was supported in part by joint grant 15RF-054 of RFBR and SCS MES RA (in frames of joint research projects SCS and RFBR) and by 15T-1A258 grant of SCS MES RA.  ... 
doi:10.1155/2016/3262450 fatcat:emgpqyzbf5amjpy6xvc2eod5tm

A Modular Algorithm for Computing Polynomial GCDs over Number Fields presented with Multiple Extensions [article]

Mark van Hoeij, Michael Monagan
2016 arXiv   pre-print
Encarnacion also showed how to use rational number to make the algorithm for Q(alpha) output sensitive, that is, the number of primes used depends on the size of the integers in the gcd and not on bounds  ...  Our fourth contribution is an implementation of the modular GCD algorithm in Maple and in Magma.  ...  We also acknowlege Mark Moreno Maza for providing details of his implementation of the fraction free algorithm in [15] .  ... 
arXiv:1601.01038v1 fatcat:svnzapkrbzb23nhfs7m6gkt62i

On degrees of modular common divisors and the Big prime gcd algorithm [article]

Vahagn H. Mikaelian
2014 arXiv   pre-print
We consider a few modifications of the Big prime modular algorithm for polynomials in [x].  ...  Our modifications are based on bounds of degrees of modular common divisors of polynomials, on estimates of the number of prime divisors of a resultant and on finding preliminary bounds on degrees of common  ...  The Big prime modular gcd algorithm is one of the first and most popular algorithms of computer algebra.  ... 
arXiv:1407.5898v1 fatcat:2t7blorqmzcxvizienlfktb7t4

Algebraic algorithms in GF(q)

J. Calmet
1985 Discrete Mathematics  
Karlsruhe) on algebraic algorithms for computing in large Galois Fields GF(q) with q = p" where p is the characteristic of the field and may be arbitrarily large.  ...  This work is materialized by a module of algorithms implemented in the ALDES/SACZ computer algebra system, which will be available with the next release of this system.  ...  Acknowledgment The author is very grateful for the hospitality at the University of Karlsruhe where this work was completed.  ... 
doi:10.1016/0012-365x(85)90017-2 fatcat:cwm6z3l5h5hfxglqxitvdwzq5u

The computation of polynomial greatest common divisors over an algebraic number field

Lars Langemyr, Scott McCallum
1989 Journal of symbolic computation  
A modular algorithm independently developed by Brown (1971) and Collins (1972) eliminates tile problem of coefficient growth in polynomial gcd computation over the rational integers Z.  ...  We present a modular algorithm for computing the greatest common divisor of two polynomials over an algebraic number field. Our algorithm is an application of ideas of Brown and Collins.  ...  From ~he modular representation of this subresultant PRS the degree of the gcd over Q(a) is determined, using a partially modular method (a modular algorithm for the gcd of univariate integral polynomials  ... 
doi:10.1016/s0747-7171(89)80053-7 fatcat:zvignluswjfb5bqa3efsbiha3q

Binary GCD algorithm for computing error locator polynomials in Reed-Solomon decoding

F. Argüello
2005 Electronics Letters  
The binary GCD algorithm, discovered by Stein, is an alternative to the Euclidean algorithm for computing the greatest common divisor of two integers.  ...  In this work, the binary GCD algorithm is applied to Reed-Solomon decoding and a novel iterative algorithm for computing error locator polynomials is proposed.  ...  Evaluation: Compared to the computation of error locator polynomials based on Euclid's algorithm [1, 2] , the binary GCD-based one requires a similar number of AND and XOR operations for updating r(x)  ... 
doi:10.1049/el:20050769 fatcat:dzklqgw3jrh6xbg2bpiwag5rrm

Book reports

2005 Computers and Mathematics with Applications  
Book Reports section is a regular feature of Computers ~ Mathematics with Applications.  ...  Available online at computers & .o,=.o. mathematics with applications Computers and Mathematics with Applications 49 (2005) 953-962 BOOK REPORTS The  ...  The resultant and gcd computation. 6.1 Coefficient growth in the Euclidean Algorithm. 6.2 Gaul3' lemma. 6.3 The resultant. 6.4 Modular gcd algorithms. 6.5 Modular gcd algorithm in F[x,y]. 6.6 Mignotte's  ... 
doi:10.1016/j.camwa.2005.02.001 fatcat:twqxvijfkrdx7gnr26gmsbuzvu

Probabilistic algorithms for sparse polynomials [chapter]

Richard Zippel
1979 Lecture Notes in Computer Science  
The probabilistic algorithm presen[ed here will be a variation of the modular GCD algorit, hm.  ...  In [ii], we present a formulation of Hensel's lemma that is somewhat more general than the one in current use and our probabilistic analogue to it. Here, we shall only present the modular algorithm.  ...  The polynomials that were fed to the various GCD routines were dill and dig~. 2. 1 . 1 Overview of Sparse Modular Algorithm.  ... 
doi:10.1007/3-540-09519-5_73 fatcat:krms52qt4bbypk5qqo5lnvczbq

Evaluation of the heuristic polynomial GCD

Hsin-Chao Liao, Richard J. Fateman
1995 Proceedings of the 1995 international symposium on Symbolic and algebraic computation - ISSAC '95  
The Heuristic Polynomial GCD procedure (GCDHEU) is Richard Zippel, Effective Polynomial Computation, Kluwer Academic Publishers, Boston 1993.  ...  Acknowledgments We thank Daniel Lichtblau of Wolfram Research Inc. for his information on the GCD methods used in Mathematical.  ...  We also thank Keith Geddes for his encouragement and explanations of the fine details in Maple.  ... 
doi:10.1145/220346.220376 dblp:conf/issac/LiaoF95 fatcat:aox7tsovordg7ky6v3l5bt4zcm

Polynomial factorization over ${\mathbb F}_2$

Joachim von zur Gathen, Jürgen Gerhard
2002 Mathematics of Computation  
They allow polynomials of degree up to 250 000 to be factored in about one day of CPU time, distributing the work on two processors.  ...  We describe algorithms for polynomial factorization over the binary field F 2 , and their implementation.  ...  However, for q = 2, modular squaring is cheaper than a general modular multiplication. • D(n), the cost for one division with remainder of two polynomials of degree at most n. • G(n), the cost for one  ... 
doi:10.1090/s0025-5718-02-01421-7 fatcat:qoigim4n6nbjpi2hsdxsuomjpe

Toward high-performance polynomial system solvers based on triangular decompositions

Xin Li
2010 ACM SIGSAM Bulletin  
Important observation of this operation. ◮ Modular multiplication is efficiency-critical to many other operations (GCD, inversion, Hensel Lifting), which are themselves the major sub-algorithm of polynomial  ...  scale. ◮ We have reported new algorithms, i.e. modular multiplication, regular GCD, and regularity test. ◮ In this research, we have focused on algorithms modulo regular chains in dimension-zero.  ...  PART V / Regular chain and GCD ◮ Let T ⊂ k[x 1 < · · · < x n ] \ k be a triangular set, hence the polynomials of T have pairwise distinct main variables. ◮ One can compute T 1 , . . . , T e and G 1 , .  ... 
doi:10.1145/1823931.1823956 fatcat:xqix6emmrfgndf7rqvsspfywpu


L. H. ZHI, M. -T. NODA
2000 Computer Mathematics  
We describe algorithms for computing the greatest common divisor of two multivariate polynomials with inexactly known coefficients.  ...  We focus on extending standard exact EZ-GCD algorithm to an efficient and stable algorithm in approximate case.  ...  Now the coefficients of the GCDs we computed will be polynomials in all variables and can be interpolated one by one.  ... 
doi:10.1142/9789812791962_0002 fatcat:3up2ac2i7vgspjz2g2igdwhpx4
« Previous Showing results 1 — 15 out of 3,934 results