Techniques for numerical integration within a symbolic computation environment are discussed. The goal is to develop a fully automated numerical integration code that handles infinite intervals of integration and that handles various types of integrand singularities. Such a code should also be able to compute to arbitrarily high precision. For the case of an analytic integrand on a finite interval, a Clenshaw-Curtis quadrature routine is used. A concept of general (non-Taylor) series expansions forms the basis of techniques for identifying transformations that may yield an analytic integrand. For the case when no transformation is successful, the general series expansion is used to represent the integrand and it is directly integrated to move beyond the singular point. The latter technique relies on a powerful symbolic integrator that can express integrals in terms of special functions.
{"title":"Numerical integration in a symbolic context","authors":"K. Geddes","doi":"10.1145/32439.32476","DOIUrl":"https://doi.org/10.1145/32439.32476","url":null,"abstract":"Techniques for numerical integration within a symbolic computation environment are discussed. The goal is to develop a fully automated numerical integration code that handles infinite intervals of integration and that handles various types of integrand singularities. Such a code should also be able to compute to arbitrarily high precision. For the case of an analytic integrand on a finite interval, a Clenshaw-Curtis quadrature routine is used. A concept of general (non-Taylor) series expansions forms the basis of techniques for identifying transformations that may yield an analytic integrand. For the case when no transformation is successful, the general series expansion is used to represent the integrand and it is directly integrated to move beyond the singular point. The latter technique relies on a powerful symbolic integrator that can express integrals in terms of special functions.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122229086","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 semantic matcher for computer algebra","authors":"G. Cooperman","doi":"10.1145/32439.32466","DOIUrl":"https://doi.org/10.1145/32439.32466","url":null,"abstract":"","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114856523","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 theory of elementary algebra and elementary geometry was shown to be decidable by Tarski using a quantifier elimination technique in the 1930’s [26]. Subsquently, Tarski’s decision algorithm was improved by others notably among them Seidenberg [25], Monk [23], and Collins [12], and recently by Ben-Or et al [4]. These methods are algebraic and are based on translating geometry statements into first-order formulae using the operations 0, 1, -1, +, *, 2, = of an ordered field with variables rangmg over real numbers. Among these decision procedures, Collins’s method based on cylinderical algebraic decomposition technique is, to our knowledge, the only decision procedure implemented so far; see [2, 31 for details.
{"title":"Geometry theorem proving using Hilbert's Nullstellensatz","authors":"D. Kapur","doi":"10.1145/32439.32479","DOIUrl":"https://doi.org/10.1145/32439.32479","url":null,"abstract":"The theory of elementary algebra and elementary geometry was shown to be decidable by Tarski using a quantifier elimination technique in the 1930’s [26]. Subsquently, Tarski’s decision algorithm was improved by others notably among them Seidenberg [25], Monk [23], and Collins [12], and recently by Ben-Or et al [4]. These methods are algebraic and are based on translating geometry statements into first-order formulae using the operations 0, 1, -1, +, *, 2, = of an ordered field with variables rangmg over real numbers. Among these decision procedures, Collins’s method based on cylinderical algebraic decomposition technique is, to our knowledge, the only decision procedure implemented so far; see [2, 31 for details.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128799557","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 Fundamental Theorem of Arithmetic (uniqueness of the prime factorisation of positive integers) allows us to represent multivariate polynomials by LISP lists of ordered pairs of numbers. In this representation one can perform all the elementary polynomial arithmetic operations of adding, negating, subtracting and multiplying multivariate polynomials or raising them to non-negative integer powers. The scheme involves the use of an isomorphic image of the ring of polynomials in n variables with rational coefficients. It has the speed and space advantages of Kronecker's trick to transform multivariate polynomials to univariate polynomials. Additional advantages are that the exponents cannot overflow and that the scheme can accommodate terms with negative integer powers.
{"title":"A sparse distributed representation using prime numbers","authors":"C. Mawata","doi":"10.1145/32439.32462","DOIUrl":"https://doi.org/10.1145/32439.32462","url":null,"abstract":"The Fundamental Theorem of Arithmetic (uniqueness of the prime factorisation of positive integers) allows us to represent multivariate polynomials by LISP lists of ordered pairs of numbers. In this representation one can perform all the elementary polynomial arithmetic operations of adding, negating, subtracting and multiplying multivariate polynomials or raising them to non-negative integer powers. The scheme involves the use of an isomorphic image of the ring of polynomials in n variables with rational coefficients. It has the speed and space advantages of Kronecker's trick to transform multivariate polynomials to univariate polynomials. Additional advantages are that the exponents cannot overflow and that the scheme can accommodate terms with negative integer powers.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128823663","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}
In extending Buchberger's theory[1.2] of Gröbner basis of polynomial ideals, Gröbner basis (standard basis in the notion of Hironaka[3]) of ideals containing power series has been discussed by several authors: Galligo[4] discussed reduction procedure of power series w.r.t. a given Gröbner basis, and Mora[5] derived a construction procedure of Gröbner basis in a local ring. In this paper, we formulate the Gröbner basis theory of convergent power series via truncated power series. In this formulation, finiteness and construction of Gröbner basis is proved quite simply. However, the termination of construction procedure remains an open problem although we have several results on this problem.
{"title":"Grobner bases of ideals of convergent power series","authors":"H. Kobayashi, A. Furukawa, T. Sasaki","doi":"10.1145/32439.32484","DOIUrl":"https://doi.org/10.1145/32439.32484","url":null,"abstract":"In extending Buchberger's theory[1.2] of Gröbner basis of polynomial ideals, Gröbner basis (standard basis in the notion of Hironaka[3]) of ideals containing power series has been discussed by several authors: Galligo[4] discussed reduction procedure of power series w.r.t. a given Gröbner basis, and Mora[5] derived a construction procedure of Gröbner basis in a local ring. In this paper, we formulate the Gröbner basis theory of convergent power series via truncated power series. In this formulation, finiteness and construction of Gröbner basis is proved quite simply. However, the termination of construction procedure remains an open problem although we have several results on this problem.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"213 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117324853","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}
In previous papers by the current authors various attempts that have been made in the automatic derivation of periodic solutions to weakly nonlinear differential equations have been reported. The equations in question are all perturbations of y′+y = 0, related to the equations that occur in the study of celestial mechanics. In the current paper the analysis of these equations has been taken further, with the automatic system, named Alkahest III, being able to determine the solution to equations even when the user's initial conditions are invalid. The system can produce an approximate solution itself, and there are facilities to write an algebra program for REDUCE or CAMAL to generate higher order solutions.
{"title":"Alkahest III: automatic analysis of periodic weakly nonlinear ODEs","authors":"J. ffitch, A. Norman, M. .. Moore","doi":"10.1145/32439.32446","DOIUrl":"https://doi.org/10.1145/32439.32446","url":null,"abstract":"In previous papers by the current authors various attempts that have been made in the automatic derivation of periodic solutions to weakly nonlinear differential equations have been reported. The equations in question are all perturbations of y′+y = 0, related to the equations that occur in the study of celestial mechanics. In the current paper the analysis of these equations has been taken further, with the automatic system, named Alkahest III, being able to determine the solution to equations even when the user's initial conditions are invalid. The system can produce an approximate solution itself, and there are facilities to write an algebra program for REDUCE or CAMAL to generate higher order solutions.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127347401","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 paradigm of divide-and-conquer appears in five guises in computational group theory: Subgroup Restriction, Subgroup Lifting, Extension, Normal Lifting, and Homomorphic Lifting. The building blocks for these strategies are algorithms for subgroup construction, coset enumeration, and homomorphisms.
{"title":"Divide-and-conquer in computational group theory","authors":"G. Butler","doi":"10.1145/32439.32451","DOIUrl":"https://doi.org/10.1145/32439.32451","url":null,"abstract":"The paradigm of divide-and-conquer appears in five guises in computational group theory: <italic>Subgroup Restriction, Subgroup Lifting, Extension, Normal Lifting,</italic> and <italic>Homomorphic Lifting</italic>. The building blocks for these strategies are algorithms for <italic>subgroup construction, coset enumeration,</italic> and <italic>homomorphisms</italic>.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115185554","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}
A new package for factoring polynomials with integer coefficients is described which yields significant improvements over previous implementations in both time and space requirements. For multivariate problems, the package features an inexpensive method for early detection and correction of spurious factors. This essentially solves the multivariate extraneous factor problem and eliminates the need to factor more than one univariate image, except in rare cases. Also included is an improved technique for coefficient prediction which is successful more frequently than prior versions at short-circuiting the expensive multivariate Hensel lifting stage. In addition some new approaches are discussed for the univariate case as well as for the problem of finding good integer substitution values. The package has been implemented both in Scratchpad II and in an experimental version of muMATH.
{"title":"A fast implementation of polynomial factorization","authors":"M. Lucks","doi":"10.1145/32439.32485","DOIUrl":"https://doi.org/10.1145/32439.32485","url":null,"abstract":"A new package for factoring polynomials with integer coefficients is described which yields significant improvements over previous implementations in both time and space requirements. For multivariate problems, the package features an inexpensive method for early detection and correction of spurious factors. This essentially solves the multivariate extraneous factor problem and eliminates the need to factor more than one univariate image, except in rare cases. Also included is an improved technique for coefficient prediction which is successful more frequently than prior versions at short-circuiting the expensive multivariate Hensel lifting stage. In addition some new approaches are discussed for the univariate case as well as for the problem of finding good integer substitution values. The package has been implemented both in Scratchpad II and in an experimental version of muMATH.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127203616","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}
We use simple arguments from Galois theory to prove the impossibility of exact algorithms for problems under various models of computation. In particular we show that there exist applied computational problems for which there are no closed from solutions over models such as Q(+, -, *, /, √), Q(+, -, *, /, k√), and Q(+, -, *, /, k√, q(x)), where Q is the field of rationals and q(x)ε Q[x] are polynomials with non-solvable Galois groups.
{"title":"Limitations to algorithm solvability: Galois methods and models of computation","authors":"C. Bajaj","doi":"10.1145/32439.32453","DOIUrl":"https://doi.org/10.1145/32439.32453","url":null,"abstract":"We use simple arguments from Galois theory to prove the impossibility of exact algorithms for problems under various models of computation. In particular we show that there exist applied computational problems for which there are no closed from solutions over models such as <italic>Q</italic>(+, -, *, /, √), <italic>Q</italic>(+, -, *, /, k√), and <italic>Q</italic>(+, -, *, /, k√, q(x)), where <italic>Q</italic> is the field of rationals and <italic>q</italic>(<italic>x</italic>)ε <italic>Q</italic>[<italic>x</italic>] are polynomials with non-solvable Galois groups.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126331879","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}
This paper describes a package implemented in REDUCE 3.2 for the manipulation of algebraic numbers. The package regards algebraic numbers as elements of abstract extensions of the rational numbers, not as particular real or complex numbers. We describe in this paper the various design choices that were made, and the current state of the package, as well as future possibilities for enhancement.
{"title":"The Bath algebraic number package","authors":"J. Abbott, R. Bradford, J. Davenport","doi":"10.1145/32439.32490","DOIUrl":"https://doi.org/10.1145/32439.32490","url":null,"abstract":"This paper describes a package implemented in REDUCE 3.2 for the manipulation of algebraic numbers. The package regards algebraic numbers as elements of abstract extensions of the rational numbers, not as particular real or complex numbers. We describe in this paper the various design choices that were made, and the current state of the package, as well as future possibilities for enhancement.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"5 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125753157","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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