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Sparse polynomials in FLINT FLINT中的稀疏多项式
Pub Date : 2016-11-04 DOI: 10.1145/3015306.3015314
A. Groves, Daniel S. Roche
We have implemented a high-performance C library for sparse polynomials, which is provided as an add-on module to the open-source computation library FLINT [7]. Our implementation incorporates a number of recent theoretical advances in supersparse polynomial arithmetic, most notably recent algorithms for sparse interpolation and multiplication. We provide a summary of the provided functionality, a selection of key implementation decisions, and some preliminary timing data.
我们已经实现了一个用于稀疏多项式的高性能C库,它作为一个附加模块提供给开源计算库FLINT[7]。我们的实现结合了超稀疏多项式算法的许多最新理论进展,最值得注意的是稀疏插值和乘法的最新算法。我们提供了所提供功能的摘要、关键实现决策的选择以及一些初步的时序数据。
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引用次数: 5
Polynomial homotopy continuation in Macaulay2 Macaulay2中的多项式同伦延拓
Pub Date : 2016-11-04 DOI: 10.1145/3015306.3015316
A. Leykin
While primarily symbolic, the computer algebra system Macaulay2 has acquired a range of numerical tools in the recent years. We describe the design changes in its NumericalAlgebraicGeometry package and the corresponding changes in the core of the system. We also discuss support packages and other packages that depend on methods provided by homotopy continuation.
虽然主要是象征性的,但计算机代数系统Macaulay2近年来获得了一系列数值工具。我们描述了其NumericalAlgebraicGeometry包的设计变化以及系统核心的相应变化。我们还讨论了支持包和其他依赖于同伦延拓提供的方法的包。
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引用次数: 0
Decomposing and solving quasilinear second-order differential equations 拟线性二阶微分方程的分解与求解
Pub Date : 2016-11-04 DOI: 10.1145/3015306.3015307
F. Schwarz
Decompositions of linear ordinary differential equations (ode's) into components of lower order have successfully been employed for determining its solutions. Here this method is generalized to certain classes of quasilinear equations of second order, i.e. equations that are linear w.r.t. the second derivative, and rational otherwise. Often it leads to simple expressions for the general solution that hardly can be obtained otherwise, i.e. it is a genuine extension of Lie's symmetry analysis. Due to its efficiency it is suggested that it is applied always as a first step in an ode solver.
线性常微分方程(ode’s)分解成低阶分量已成功地用于确定其解。本文将该方法推广到一类二阶拟线性方程,即二阶导数为线性,二阶导数为有理的拟线性方程。通常,它会导致一般解的简单表达式,否则很难得到,也就是说,它是李氏对称分析的真正扩展。由于它的有效性,建议它总是作为求解算法的第一步来应用。
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引用次数: 0
CGSQE/SyNRAC: a real quantifier elimination package based on the computation of comprehensive Gröbner systems CGSQE/SyNRAC:一个基于综合Gröbner系统计算的实量词消去包
Pub Date : 2016-11-04 DOI: 10.1145/3015306.3015313
Ryoya Fukasaku, Hidenao Iwane, Yosuke Sato
CGSQE is a Maple package for real quantifier elimination (QE) we are developing. It works cooperating with SyNRAC which is also a Maple package for real QE one of the authors is developing. For a given first order formula, CGSQE eliminates all possible quantifiers using the underlying equational constraints by the computation of comprehensive Gröbner systems (CGSs). In case all quantifiers are not removable, it transforms the given formula into a formula which contains only strict inequalities of quantified variables, then uses a cylindrical algebraic decomposition based real QE program of SyNRAC to remove the remaining quantifiers. The core algorithm of CGSQE is a CGS real QE algorithm which was first introduced by Weispfenning in 1998 and further improved by us in 2015 so that we can make a satisfactorily practical implementation. CGSQE is superior to other real QE implementations for many examples which contain many equational constraints. In the software presentation, we would like to show high-performance computation of CGSQE.
CGSQE是我们正在开发的一个用于实量词消去(QE)的Maple包。它可以与SyNRAC协同工作,SyNRAC也是一位作者正在开发的用于真正QE的Maple包。对于给定的一阶公式,CGSQE通过综合Gröbner系统(CGSs)的计算,利用底层方程约束消除所有可能的量词。在不能去除所有量词的情况下,将给定公式转化为只包含量化变量严格不等式的公式,然后使用SyNRAC的基于圆柱代数分解的实QE程序去除剩余的量词。CGSQE的核心算法是CGS real QE算法,该算法由Weispfenning于1998年首次提出,我们在2015年对其进行了进一步改进,使我们能够做出令人满意的实际实现。对于包含许多方程约束的实例,CGSQE优于其他实际的QE实现。在软件演示中,我们想展示CGSQE的高性能计算。
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引用次数: 1
Computing limits with the regularchains and powerseries libraries: from rational functions to Zariski closure 用正则函数和幂级数库计算极限:从有理函数到Zariski闭包
Pub Date : 2016-11-04 DOI: 10.1145/3015306.3015311
P. Alvandi, Mahsa Kazemi, M. M. Maza
Many fundamental concepts in mathematics are defined in terms of limits and it is desirable for computer algebra systems to be able to compute them. However, limits of functions, limits of secants or topological closures are, by essence, hard to compute in an algorithmic fashion, say by doing finitely many rational operations on polynomials or matrices over the usual coefficient fields of symbolic computation. This is why a computer algebra system like Maple is not capable of computing limits of rational functions in more than two variables while it can perform highly sophisticated algebraic computations like solving (formally) a system of partial differential equations.
数学中的许多基本概念都是用极限来定义的,计算机代数系统需要能够计算它们。然而,函数的极限、割线的极限或拓扑闭包,从本质上讲,很难用算法的方式来计算,比如在符号计算的通常系数域上对多项式或矩阵进行有限次有理运算。这就是为什么像Maple这样的计算机代数系统不能计算两个以上变量的有理函数的极限,而它可以执行高度复杂的代数计算,比如(正式地)求解偏微分方程系统。
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引用次数: 3
Fleshing out the generalized Lambert W function 充实了广义朗伯特W函数
Pub Date : 2016-08-25 DOI: 10.1145/2992274.2992275
A. Maignan, Tony C. Scott
Herein, we use Hardy's notion of the "false derivative" to obtain exact multiple roots in closed form of the transcendental-algebraic equations representing the generalized Lambert W function. In this fashion, we flesh out the generalized Lambert W function by complementing previous developments to produce a more complete and integrated body of work. Finally, we demonstrate the usefulness of this special function with some applications.
本文利用Hardy的“假导数”概念,得到了表示广义Lambert W函数的超越代数方程的闭形式的精确重根。以这种方式,我们通过补充以前的开发来充实广义Lambert W函数,以产生更完整和集成的工作主体。最后,我们在一些应用程序中演示了这个特殊函数的有用性。
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引用次数: 11
The conference "computer algebra" in Moscow 莫斯科的“计算机代数”会议
Pub Date : 2016-08-25 DOI: 10.1145/2992274.2992276
S. Abramov, L. Sevastianov
The International Conference "Computer Algebra" was held in Moscow, Russia from June 29 till July 2, 2016. The conference web-site is http://www.ccas.ru/ca/conference. Co-organized by the Dorodnicyn Computing Centre (Federal Research Center "Computer Science and Control") of Russian Academy of Sciences and the Peoples' Friendship University of Russia. It was devoted to computer algebra and related topics. The conference was supported by the Russian Foundation for Basic Research under Grant No. 16-01-20379.
2016年6月29日至7月2日,“计算机代数”国际会议在俄罗斯莫斯科召开。会议的网址是http://www.ccas.ru/ca/conference。本次会议由俄罗斯科学院多罗德尼琴计算中心(联邦计算机科学与控制研究中心)和俄罗斯人民友谊大学共同主办。它致力于计算机代数和相关主题。会议由俄罗斯基础研究基金会资助,资助号:16-01-20379。
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引用次数: 1
Memories on Professor Matu-tarow Noda
Pub Date : 2016-08-25 DOI: 10.1145/2992274.2992277
Tateaki Sasaki, H. Kai
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引用次数: 0
Satisfiability checking and symbolic computation 可满足性检验与符号计算
Pub Date : 2016-07-23 DOI: 10.1145/3055282.3055285
E. Ábrahám, J. Abbott, B. Becker, Anna Maria Bigatti, M. Brain, B. Buchberger, A. Cimatti, J. Davenport, M. England, P. Fontaine, S. Forrest, A. Griggio, D. Kroening, W. Seiler, T. Sturm
Symbolic Computation and Satisfiability Checking are viewed as individual research areas, but they share common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite these commonalities, the two communities are currently only weakly connected. We introduce a new project SC2 to build a joint community in this area, supported by a newly accepted EU (H2020-FETOPEN-CSA) project of the same name. We aim to strengthen the connection between these communities by creating common platforms, initiating interaction and exchange, identifying common challenges, and developing a common roadmap. This abstract and accompanying poster describes the motivation and aims for the project, and reports on the first activities.
符号计算和可满足性检验被视为独立的研究领域,但它们在算术理论决策过程的发展、实现和应用方面有着共同的兴趣。尽管有这些共同点,但这两个社区目前的联系很弱。我们引入了一个新的项目SC2,在这个领域建立一个联合社区,并由一个新接受的同名EU (H2020-FETOPEN-CSA)项目提供支持。我们的目标是通过创建共同的平台,发起互动和交流,确定共同的挑战,制定共同的路线图,加强这些社区之间的联系。这个摘要和附带的海报描述了项目的动机和目标,并报告了第一次活动。
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引用次数: 5
Black box linear algebra: extending wiedemann's analysis of a sparse matrix preconditioner for computations over small fields 黑箱线性代数:扩展wiedemann对小域计算的稀疏矩阵预条件的分析
Pub Date : 2016-07-15 DOI: 10.1145/3055282.3055291
W. Eberly
Wiedemann's paper, introducing his algorithm for sparse and structured matrix computations over arbitrary fields, also presented a pair of matrix preconditioners for computations over small fields. The analysis of the second of these is extended here in order to provide more explicit statements of the expected number of nonzero entries in the matrices obtained as well as bounds on the probability that the matrices being considered have maximal rank. It is hoped that this will make Wiedemann's second preconditioner of more practical use. This is part of ongoing work to establish that this matrix preconditioner can be used to bound the number of nontrivial nilpotent blocks in the Jordan normal form of a preconditioned matrix, in such a way that one can also sample uniformly from the null space of the originally given matrix. If successful this will result in a black box algorithm for the type of matrix computation required when using the number field sieve for integer factorization that is provably reliable (unlike some heuristics, presently in use) and --- by a small factor --- asymptotically more efficient than alternative provably reliable techniques that make use of other matrix preconditioners or require computations over field extensions.
Wiedemann的论文介绍了他在任意域上的稀疏和结构化矩阵计算的算法,并给出了小域上计算的一对矩阵前置条件。本文对第二种方法的分析进行了扩展,以便更明确地说明所得到的矩阵中非零条目的期望数目,以及所考虑的矩阵具有最大秩的概率的界限。希望这将使Wiedemann的第二预调节器具有更实际的用途。这是正在进行的工作的一部分,目的是建立这个矩阵预条件可以用来约束一个预条件矩阵的约旦范式中的非平凡幂零块的数量,这样就可以从原始给定矩阵的零空间中均匀采样。如果成功,这将导致在使用数字字段筛进行整数分解时所需的矩阵计算类型的黑盒算法,该算法可证明是可靠的(与目前使用的一些启发式方法不同),并且-通过一个小因素-渐近地比使用其他矩阵前置条件或需要在字段扩展上进行计算的可证明可靠的技术更有效。
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引用次数: 0
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ACM Commun. Comput. Algebra
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