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.
{"title":"Sparse polynomials in FLINT","authors":"A. Groves, Daniel S. Roche","doi":"10.1145/3015306.3015314","DOIUrl":"https://doi.org/10.1145/3015306.3015314","url":null,"abstract":"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.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"11 3 1","pages":"105-108"},"PeriodicalIF":0.0,"publicationDate":"2016-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78585970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Polynomial homotopy continuation in Macaulay2","authors":"A. Leykin","doi":"10.1145/3015306.3015316","DOIUrl":"https://doi.org/10.1145/3015306.3015316","url":null,"abstract":"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.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"33 1","pages":"113-116"},"PeriodicalIF":0.0,"publicationDate":"2016-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78791963","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Decomposing and solving quasilinear second-order differential equations","authors":"F. Schwarz","doi":"10.1145/3015306.3015307","DOIUrl":"https://doi.org/10.1145/3015306.3015307","url":null,"abstract":"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.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"21 1","pages":"73-82"},"PeriodicalIF":0.0,"publicationDate":"2016-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72680322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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。
{"title":"CGSQE/SyNRAC: a real quantifier elimination package based on the computation of comprehensive Gröbner systems","authors":"Ryoya Fukasaku, Hidenao Iwane, Yosuke Sato","doi":"10.1145/3015306.3015313","DOIUrl":"https://doi.org/10.1145/3015306.3015313","url":null,"abstract":"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.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"6 1","pages":"101-104"},"PeriodicalIF":0.0,"publicationDate":"2016-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84676861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Computing limits with the regularchains and powerseries libraries: from rational functions to Zariski closure","authors":"P. Alvandi, Mahsa Kazemi, M. M. Maza","doi":"10.1145/3015306.3015311","DOIUrl":"https://doi.org/10.1145/3015306.3015311","url":null,"abstract":"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.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"118 1","pages":"93-96"},"PeriodicalIF":0.0,"publicationDate":"2016-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73637737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Fleshing out the generalized Lambert W function","authors":"A. Maignan, Tony C. Scott","doi":"10.1145/2992274.2992275","DOIUrl":"https://doi.org/10.1145/2992274.2992275","url":null,"abstract":"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.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"130 1","pages":"45-60"},"PeriodicalIF":0.0,"publicationDate":"2016-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80427228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"The conference \"computer algebra\" in Moscow","authors":"S. Abramov, L. Sevastianov","doi":"10.1145/2992274.2992276","DOIUrl":"https://doi.org/10.1145/2992274.2992276","url":null,"abstract":"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.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"19 1","pages":"61-68"},"PeriodicalIF":0.0,"publicationDate":"2016-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76027593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Memories on Professor Matu-tarow Noda","authors":"Tateaki Sasaki, H. Kai","doi":"10.1145/2992274.2992277","DOIUrl":"https://doi.org/10.1145/2992274.2992277","url":null,"abstract":"","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"140 1","pages":"69"},"PeriodicalIF":0.0,"publicationDate":"2016-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91503532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Satisfiability checking and symbolic computation","authors":"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","doi":"10.1145/3055282.3055285","DOIUrl":"https://doi.org/10.1145/3055282.3055285","url":null,"abstract":"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.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"92 1","pages":"145-147"},"PeriodicalIF":0.0,"publicationDate":"2016-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85680018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Black box linear algebra: extending wiedemann's analysis of a sparse matrix preconditioner for computations over small fields","authors":"W. Eberly","doi":"10.1145/3055282.3055291","DOIUrl":"https://doi.org/10.1145/3055282.3055291","url":null,"abstract":"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.\u0000 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.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"36 1","pages":"164-166"},"PeriodicalIF":0.0,"publicationDate":"2016-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75129077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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