SIGSAM Bull.最新文献

{"title":"CoCoA: a laboratory for computations in commutative algebra","authors":"J. Abbott","doi":"10.1145/980175.980182","DOIUrl":"https://doi.org/10.1145/980175.980182","url":null,"abstract":"The CoCoA program together with Singular and Macaulay 2 form an elite group of highly specialized systems having as their main forte the capability to calculate Gröbner bases. Although a number of general purpose symbolic computation systems (e.g. REDUCE and Maple) do offer the possibility to compute Gröbner bases, their non-specialist nature implies a number of severe compromises which make them far less suitable to act as a laboratory---most notably: relatively poor execution speed and limited control over the algorithm parameters. In contrast, CoCoA does not offer facilities for calculus, floating point computation, or any other area not closely related to commutative polynomial algebra.Development in CoCoA is often spurred on by symbiosis with those using it, not only in algebraic geometry research but also in other fields such as mathematical analysis, and statistics. In fact, CoCoA has become widely used in many countries both for research and teaching, and its didactic role means that ease of use is a high priority. Indeed, the mathematically natural way of instructing CoCoA to do computations enhances the worth of its technical skills.CoCoA 5 represents an important new phase in the project: the software is being completely rewritten in C++. CoCoA 5 will be available as an interactive system, as a C++ library, and as a server (using an OpenMath-like interface). We have already achieved significant improvements in flexibility and speed, and note that ease of use remains a high priority while also offering the skilled practitioner a high degree of control. Ultimately CoCoA 5 will extend the range of specialization of the current public release.A big challenge in the design of CoCoA 5 was to reconcile two traditionally conflicting goals: flexibility and efficiency. The inheritance mechanism of C++ plays a crucial role here. Our use of inheritance is exemplified by the way in which rings and their elements are implemented, inspired by the tenet to allow arbitrary (commutative) rings wherever possible. The run-time penalty for this additional abstraction is virtually negligible.In a few cases, to meet the run-time performance targets, the CoCoA library foundations do use \"quick and dirty hackery\". Here credit is due to the design of C++ which allows such desperate code to be written where necessary, and which also permits it to fit seamlessly into a well-structured program.Undoubtedly the principal feature of CoCoA 5 is its code for computing Gröbner bases. In the case of Buchberger's algorithm there is an unusually wide gulf between the neat elegance of the theory and the complex engineering within a refined implementation. In CoCoA 5 a carefully structured design, which comprises clearly separated components with well defined roles, will facilitate the inclusion of future improvements in an implementation already including many modern ideas: e.g. binary divisibility mask, geobuckets, fraction free representation, sophisticated reducer select","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133747514","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":"A Matlab package computing polynomial roots and multiplicities","authors":"Zhonggang Zeng","doi":"10.1145/980175.980189","DOIUrl":"https://doi.org/10.1145/980175.980189","url":null,"abstract":"MULTROOT is a collection of Matlab modules for accurate computation of polynomial roots, especially roots with high multiplicities, using standard machine precision. As a blackbox-type software, MULTROOT requires the polynomial coefficients as the only input, and outputs the computed roots, multiplicities, backward error, estimated forward error, as well as the pejorative condition number.There are two common limitations for standard numerical root-finding software when multiple roots are present. Namely, those methods suffer from severe loss of accuracy and lack the capacity of multiplicity identification. Symbolic polynomial factorization requires exact rational coefficients. In contrast, the most significant features of MULTROOT are the multiplicity identification capability and the remarkable accuracy on multiple roots without using the multiprecision arithmetic, even if the polynomial coefficients are inexact.There is a so-called \"attainable accuracy\" for conventional root-finders: the attainable number of corrected digits of a computed root is limited by the data/machine precision divided by the multiplicity. For roots with high multiplicities, this accuracy barrier suggests that multiprecision arithmetic is required in addition to exact coefficients. Using a novel approach based on the pejorative manifold theory, MULTROOT achieves high accuracy even if the polynomial is perturbed and the machine precision is not extended. A stable numerical polynomial GCD-finder is also developed as an essential component that determines the multiplicity structure.MULTROOT is a combination of two programs that can be used independently. GCDROOT calculates the multiplicity structure and initial root approximation. PEJROOT refines the root approximations by projecting the polynomial onto a prescribed pejorative manifold. A comprehensive test suit of polynomials that are collected from the literature is included for numerical experiments and performance comparison.The detailed algorithm and analysis is presented in this conference [1].","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"78 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126247093","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":"Using Fermat to solve large polynomial and matrix problems","authors":"Robert H. Lewis","doi":"10.1145/980175.980188","DOIUrl":"https://doi.org/10.1145/980175.980188","url":null,"abstract":"<i>Fermat</i> is an interactive system for mathematical experimentation. It is a super calculator - computer algebra system, in which the basic items being computed can be rational numbers, modular numbers, finite fields, multivariable polynomials, rational functions, or polynomials modulo other polynomials.In <i>Fermat</i> the default \"ground ring\" <i>F</i> is the field of rational numbers. One may choose to work modulo a specified integer <i>m</i>, thereby changing the ground ring <i>F</i> from <b><i>Q</i></b> to <b><i>Z</i></b>/m. On top of this may be attached any number of symbolic variables <i>t</i>₁, <i>t</i>₂, . . ., <i>t</i>_n, thereby creating the polynomial ring <i>F</i>[<i>t</i>₁, <i>t</i>₂, . . ., <i>t</i>_n] and its quotient field, the field of rational functions, whose elements are called <i>quolynomials</i>, Further, polynomials <i>p</i>, <i>q</i>, . . . can be chosen to mod out with, creating the quotient ring <i>F</i>(<i>t</i>₁, <i>t</i>₂, . . .) / < <i>p</i>, <i>q</i>, . . .>, whose elements are called <i>polymods</i>. If this is done correctly, finite fields result. Finally, it is possible to allow <i>Laurent polynomials</i>, those with negative as well as positive exponents. Once the computational ring is established in this way, all computations are of elements of this ring.<i>Fermat</i> has extensive built-in primitives for array and matrix manipulations, such as submatrix, sparse matrix, determinant, minors, normalize, column reduce, reduced row echelon, matrix inverse, Smith normal form, and characteristic polynomial. It is consistently faster than some well known computer algebra systems - orders of magnitude faster in some cases.<i>Fermat</i> is a complete programming language. Programs and data can be saved to an ordinary text file that can be read during a later session or read by some other software system.<i>Fermat</i> has solved real problems that other computer algebra systems could not. It is more efficient in both time and space. These problems have come from algebraic topology, group theory, image processing, computational geometry, decision theory, and signal processing.Most recent applications involve solving systems of polynomial equations with the <b>Dixon Resultant</b> technique. I will demonstrate this method at the conference, in particular how I attack the <i>spurious factor problem</i>.<i>Fermat</i> is available for Windows95/98/NT/etc and Mac OS. Fermat for Linux is ready for beta testing.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"428 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116568497","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":"QEPCAD B: a system for computing with semi-algebraic sets via cylindrical algebraic decomposition","authors":"Christopher W. Brown","doi":"10.1145/980175.980185","DOIUrl":"https://doi.org/10.1145/980175.980185","url":null,"abstract":"QEPCAD B² is a system for computing with semi-algebraic sets. a <i>semi-algebraic set</i> is a subset of ℝ^<i>n</i> that can be defined as the set of points satisfying a boolean formula combining polynomial equalities and inequalities in the variables <i>x</i>₁,...,<i>x</i>_<i>n</i>. So, for example, the upper-right quadrant of the unit disk is a semi-algebraic set, since it has the defining formula[see pdf for formula]Many important problems in mathematics, science and engineering boil down to questions about semi-algebraic sets. QEPCAD B allows its users to compute with semi-algebraic sets specified by defining formulae. Computation is exact and symbolic, results being returned in the same language of defining formulae. The basic operations the system supports are <i>formula simplification</i> and <i>quantifier elimination</i><b>Quantifier Elimination:</b> Adding quantifiers to a defining formula is, in a sense, asking a question. For example, ∃<i>x</i>[<i>x</i>²+<i>bx</i>+<i>c</i>=0] is the question \"when does <i>x</i>²+<i>bx</i>+<i>c</i> have a real root?\" The well-known answer \"when <i>b</i>²−4<i>c</i>≥0\" is an equivalent formula from which the quantified variable has been eliminated. Quantifier elimination algorithms, which produce such equivalent formulae, can be seen as providing \"answers\" to \"questions\" about semi-algebraic sets.<b>Formula Simplification:</b> Many procedures in mathematics, performed both manually and mechanically, produce \"answers\" in the form of defining formulae. These defining formulae are often not particularly nice characterizations of the sets they define--hence the need for formula simplification. For example, QEPCAD B determines that the formula<i>F</i>:=1+<i>b</i>²−<i>c</i>²≥<i>b</i>∧−<i>c</i>(<i>b</i>²−<i>c</i>²)³+3<i>b</i>²<i>c</i>(<i>b</i>²−<i>c</i>²)∨<i>b</i>²−<i>c</i>²<<i>b</i>under the assumption <i>b</i> > 0 ∧ <i>c</i> > 0 ∧ 1 < <i>b</i> + <i>c</i> ∧ <i>b</i> < 1 + <i>c</i> ∧ <i>c</i> < 1 = <i>b</i> is equivalent to <i>F</i>′ := <i>c</i>² − <i>bc</i> − 1 > 0. Obviously, <i>F</i>′ was a considerably better characterization for subsequent computations in the application from which this arose.<b>Cylindrical Algebraic Decomposition (CAD):</b> A CAD is essentially a data-structure providing an explicit representation of a semi-algebraic set. This representation is expensive to compute, but it contains so much information about the set it represents that quantifier elimination and simplification are easily accomplished, which is why CAD is the basis for these operations in QEPCAD B. A little insight into what CAD is and how it is used is provided by the following figures, produced by QEPCAD B, which show the CAD representation for the formula <i>F</i> from the simplification example, followed by the CAD representing <","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126596755","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":"MR: Macaulay Resultant package for Maple","authors":"Manfred Minimair","doi":"10.1145/980175.980187","DOIUrl":"https://doi.org/10.1145/980175.980187","url":null,"abstract":"The software package MR for Maple, Versions 7 and 8, contains an implementation of Macaulay's algorithm for computing the dense multi-variable resultant ([MC93], [Min02]) of a list of multivariate polynomials. An extended list of some of the features is:• Designed independently from the coefficient ring of the input polynomials.• Optimized for polynomials with coefficients being integer polynomials.• Optionally computes generalized characteristic polynomials.• Optionally self-profiling (timing, tracing).• Well integrated with Maple's LinearAlgebra package.• Designed to emulate overloading, according to the type of the input, of the main sub-functions, while not sacrificing efficiency.It important to point out that there are other resultant packages available for Maple (see e.g. [Tri02], [WME98], [BM02]). However, they do not provide the full range of features of the current package, MR, and they are not designed for the most recent versions of Maple. Therefore the author believes that the current package is of great interest to the symbolic computation community.Usage: We briefly describe how the package MR is to be used in Maple.The main function, exported by the package MR, is called asMR: MResultant (plist, vlist, options);The input argument \"plist\" is a list of not necessarily homogeneous polynomials. The variables of these polynomials are contained in the list \"vlist\". The symbol \"options\" stands for a sequence of optional arguments. These optional arguments are described in detail in http://minimair.org/MR.mpl. If called without optional arguments, the main function returns the Macaulay (multi-variable) resultant of the polynomials in \"plist\" with respect to the variables in \"vlist\".","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130119443","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":"Greedy algorithms for optimizing multivariate Horner schemes","authors":"M. Ceberio, V. Kreinovich","doi":"10.1145/980175.980179","DOIUrl":"https://doi.org/10.1145/980175.980179","url":null,"abstract":"For univariate polynomials <i>f</i>(<i>x</i>₁), Horner's scheme provides the fastest way to compute a value. For multivariate polynomials, several different version of Horner's scheme are possible; it is not clear which of them is optimal. In this paper, we propose a greedy algorithm, which it is hoped will lead to good computation times.The univariate Horner scheme has another advantage: if the value <i>x</i>₁ is known with uncertainty, and we are interested in the resulting uncertainty in <i>f</i>(<i>x</i>₁), then Horner scheme leads to a better estimate for this uncertainty that many other ways of computing <i>f</i>(<i>x</i>₁). The second greedy algorithm that we propose tries to find the multivariate Horner scheme that leads to the best estimate for the uncertainty in <i>f</i>(<i>x</i>₁,...,<i>x</i>_n).","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130462356","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":"GNU TeXmacs","authors":"Joris van der Hoeven","doi":"10.1145/980175.980186","DOIUrl":"https://doi.org/10.1145/980175.980186","url":null,"abstract":"GNU TeXmacs [vdH01, vdH02, Gro01] is a free software, which can both be used as a scientific text editor and as a front-end for computer algebra systems. The editor allows you to write structured documents via a wysiwyg (what-you-see-is-what-you-get) and user friendly interface. New styles may be created by the user. The program implements high-quality typesetting algorithms and TEX fonts, which allow the user to produce professionally looking documents.The high typesetting quality still goes through for automatically generated formulae, which makes TeXmacs suitable as an interface for computer algebra systems, or other types of \"plugins\". Currently, there are interfaces with Axiom, Giac, GNUplot, Graphviz, Gtybalt, Macaulay2, Maxima, Mupad, GNU Octave, Pari, Qcl, GNU R, Reduce, Scilab and Yacas. TeXmacs also supports the Guile/Scheme extension language, so that you may customize the interface and write your own extensions to the editor.In our demonstration, we plan to show briefly how to edit mathematical texts with TeXmacs, how to use computer algebra systems, and how to add interfaces with new systems in a very efficient way.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121491267","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":"MATCONT: a Matlab package for numerical bifurcation analysis of ODEs","authors":"Annick Dhooge, W. Govaerts, Y. Kuznetsov","doi":"10.1145/980175.980184","DOIUrl":"https://doi.org/10.1145/980175.980184","url":null,"abstract":"We consider generic parameterized autonomous ODEs of the form <i>dx</i>/<i>dt</i> ≡ ẋ = f(x, α), where x ∈ ℝ^<i>n</i> is the vector of <i>state variables</i>, α ∈ ℝ^<i>m</i> represents <i>parameters</i>, and f(x, α) ∈ ℝ^<i>n</i>. There are several interactive software packages for analysis of dynamical systems defined by ODEs. The most widely used are AUTO86/97[1], CONTENT[2] and XPPAUT.The Matlab software package MATCONT provides an interactive environment for the continuation and normal form analysis of dynamical systems. This analysis is complementary to the simulation of the systems which is also included in the package and can be used in their identification, control, and optimization. MATCONT is designed to exploit the power of Matlab. It is developed in parallel with the continuation toolbox CL_MATCONT, a package of Matlab routines that can be used from the command line.We consider the following model of an autonomous electronic circuit where <i>x, y</i> and <i>z</i> are state variables and β,γ,ν,<i>r</i>,<i>a</i>₃,<i>b</i>₃ are parameters: [see pdf for formula]We compute a branch of equilibria with free parameter ν stating from the trivial solution <i>x</i> = 0.00125, <i>y</i> = -0.001, <i>z</i> = 0.00052502 at <i>β</i> = 0.5, γ = -0.6, <i>r</i> = -0.6, <i>a</i>₃ = 0.32858, <i>b</i>₃ = 0.93358, ν = -0.9, ε = 0.001. We start a curve of periodic orbits from a Hopf point on this curve choosing ν as the free parameter. We detect a torus bifurcation point at ν = -0.59575. We continue the torus bifurcation in two parameters ν, ε and find that it shrinks to a single point for decreasing values of ν (Figure 2).","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134398355","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":"A comparison of methods for accurate summation","authors":"J. McNamee","doi":"10.1145/980175.980177","DOIUrl":"https://doi.org/10.1145/980175.980177","url":null,"abstract":"The summation of large sets of numbers is prone to serious rounding errors. Several methods of controlling these errors are compared, with respect to both speed and accuracy. It is found that the method of \"Cascading Accumulators\" is the fastest of several methods. The Double Compensation method (in both single and double precision versions) is also perfectly accurate in all the tests performed. Although slower than the Cascade method, it is recommended when double precision accuracy is required. C programs that implement both these methods are available in the BULLETIN online repository.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117090471","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":"Distribution of the error in estimated numbers of fixed points of the discrete logarithm","authors":"Joshua Holden","doi":"10.1145/1060328.1060329","DOIUrl":"https://doi.org/10.1145/1060328.1060329","url":null,"abstract":"Brizolis asked the question: does every prime p have a pair (g, h) such that h is a fixed point for the discrete logarithm with base g? This author and Pieter Moree, building on work of Zhang, Cobeli, and Zaharescu, gave heuristics for estimating the number of such pairs and proved bounds on the error in the estimates. These bounds are not descriptive of the true situation, however, and this paper is a first attempt to collect and analyze some data on the distribution of the actual error in the estimates.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115352380","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}