Double cosets are an important concept of group theory. Although the desirability of algorithms to compute double cosets has been recognized, there has not appeared any algorithm in the literature. The algorithm which we present is a variant of Dimino's algorithm for computing a list of elements of a small group. (By “small” we mean groups of order less than 104, whose list of elements we can explicitly store.) The paper focusses on the problem of searching a small group for elements with a given property. For the record we present Dimino's algorithm and a general algorithm for searching a small group. These two algorithms are not original. We analyse the search algorithm and discuss the role of double cosets in searching. The use of double cosets in the search algorithm does not appear to lead to an improvement over the use of right cosets.
{"title":"Double cosets and searching small groups","authors":"G. Butler","doi":"10.1145/800206.806393","DOIUrl":"https://doi.org/10.1145/800206.806393","url":null,"abstract":"Double cosets are an important concept of group theory. Although the desirability of algorithms to compute double cosets has been recognized, there has not appeared any algorithm in the literature. The algorithm which we present is a variant of Dimino's algorithm for computing a list of elements of a small group. (By “small” we mean groups of order less than 104, whose list of elements we can explicitly store.)\u0000 The paper focusses on the problem of searching a small group for elements with a given property. For the record we present Dimino's algorithm and a general algorithm for searching a small group. These two algorithms are not original. We analyse the search algorithm and discuss the role of double cosets in searching. The use of double cosets in the search algorithm does not appear to lead to an improvement over the use of right cosets.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130353413","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}
For many computational problems it is not known whether verification of a result can be done faster than its computation. For instance, it is unknown whether the verification of the validity of the integer equality x*y&equil;z needs fewer bit operations than a computation of the product x*y. It is sometimes much easier, however, to speed up the computation probabilistically if just the verification of the result is involved. In this paper we present linear probabilistic algorithms for verification of the validity of the integer equality f1(x1,...,xN)&equil;f2(x1,...,xN) for rational functions f1 and f2, which can be of the form of a rational combination of rational functions.
{"title":"Note on probabilistic algorithms in integer and polynomial arithmetic","authors":"M. Kaminski","doi":"10.1145/800206.806380","DOIUrl":"https://doi.org/10.1145/800206.806380","url":null,"abstract":"For many computational problems it is not known whether verification of a result can be done faster than its computation. For instance, it is unknown whether the verification of the validity of the integer equality x*y&equil;z needs fewer bit operations than a computation of the product x*y. It is sometimes much easier, however, to speed up the computation probabilistically if just the verification of the result is involved.\u0000 In this paper we present linear probabilistic algorithms for verification of the validity of the integer equality f<subscrpt>1</subscrpt>(x<subscrpt>1</subscrpt>,...,x<subscrpt>N</subscrpt>)&equil;f<subscrpt>2</subscrpt>(x<subscrpt>1</subscrpt>,...,x<subscrpt>N</subscrpt>) for rational functions f<subscrpt>1</subscrpt> and f<subscrpt>2</subscrpt>, which can be of the form of a rational combination of rational functions.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"4 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113932524","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 presents an organization of the p-adic lifting (or Hensel) algorithm that differs from the organization previously presented by Zassenhaus [Zas69] and currently used in algebraic manipulation circles [Mos73, Yun74, Wan75, Mus75]. Our organization is somewhat more general than the earlier one and admits the improvements that yielded the “sparse modular” algorithm [Zip79] more easily than the Zassenhaus algorithm. From a pedagogical point of view, the relationship between Newton's iteration and the p-adic algorithms is clearer in our formulation than with the Zassenhaus algorithm.
{"title":"Newton's iteration and the sparse Hensel algorithm (Extended Abstract)","authors":"R. Zippel","doi":"10.1145/800206.806372","DOIUrl":"https://doi.org/10.1145/800206.806372","url":null,"abstract":"This paper presents an organization of the p-adic lifting (or Hensel) algorithm that differs from the organization previously presented by Zassenhaus [Zas69] and currently used in algebraic manipulation circles [Mos73, Yun74, Wan75, Mus75]. Our organization is somewhat more general than the earlier one and admits the improvements that yielded the “sparse modular” algorithm [Zip79] more easily than the Zassenhaus algorithm. From a pedagogical point of view, the relationship between Newton's iteration and the p-adic algorithms is clearer in our formulation than with the Zassenhaus algorithm.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121143197","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}
Parallel execution of algebraic computation is discussed in the first half of this paper. It is argued that, although a high efficiency is obtained by parallel execution of divide-and-conquer algorithms, the ratio of the throughput to the number of processors is still small. Parallel processing will be most successful for the modular algorithms and many algorithms in linear algebra. In the second half of this paper, parallel algorithms for symbolic determinants and linear equations are proposed. The algorithms manifest a very high efficiency in a simple parallel processing scheme. These algorithms are well usable in also the serial processing scheme.
{"title":"Parallelism in algebraic computation and parallel algorithms for symbolic linear systems","authors":"Tateaki Sasaki, Y. Kanada","doi":"10.1145/800206.806388","DOIUrl":"https://doi.org/10.1145/800206.806388","url":null,"abstract":"Parallel execution of algebraic computation is discussed in the first half of this paper. It is argued that, although a high efficiency is obtained by parallel execution of divide-and-conquer algorithms, the ratio of the throughput to the number of processors is still small. Parallel processing will be most successful for the modular algorithms and many algorithms in linear algebra. In the second half of this paper, parallel algorithms for symbolic determinants and linear equations are proposed. The algorithms manifest a very high efficiency in a simple parallel processing scheme. These algorithms are well usable in also the serial processing scheme.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127506556","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}
It follows from [1] that every n-dimensional nilpotent Lie algebra is a central extension of a lower dimensional nilpotent Lie algebra. This paper develops algorithms to handle two problems: (1) the decomposition of a given nilpotent Lie algebra @@@@ as a finite sequence of central extensions of lower dimensional nilpotent Lie algebras and (2) the construction of all n-dimensional nilpotent Lie algebras as central extensions of lower dimensional nilpotent Lie algebras.
{"title":"Algorithms for central extensions of Lie algebras","authors":"R. Beck, B. Kolman","doi":"10.1145/800206.806391","DOIUrl":"https://doi.org/10.1145/800206.806391","url":null,"abstract":"It follows from [1] that every n-dimensional nilpotent Lie algebra is a central extension of a lower dimensional nilpotent Lie algebra. This paper develops algorithms to handle two problems: (1) the decomposition of a given nilpotent Lie algebra @@@@ as a finite sequence of central extensions of lower dimensional nilpotent Lie algebras and (2) the construction of all n-dimensional nilpotent Lie algebras as central extensions of lower dimensional nilpotent Lie algebras.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121012975","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}
Efficient Gaussian elimination method for symbolic determinants and linear systems
符号行列式和线性系统的有效高斯消去方法
{"title":"Efficient Gaussian elimination method for symbolic determinants and linear systems (Extended Abstract)","authors":"Tateaki Sasaki, H. Murao","doi":"10.1145/800206.806387","DOIUrl":"https://doi.org/10.1145/800206.806387","url":null,"abstract":"Efficient Gaussian elimination method for symbolic determinants and linear systems","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"77 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126002054","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 report is made on the present state of development of a project to construct a tracing aid for users of symbolic computing systems that are written in LISP (or, in principle, any similar high-level language). The traces in question are intended to provide information which is primarily in terms that are natural for a user, e.g. on patterns of actions performed on his data, or patterns occurring in the data themselves during the operation of his program. Patterns are described in a syntax which is inspired by SNOBOL.
{"title":"Tracing occurrences of patterns in symbolic computations","authors":"F. Gardin, J. Campbell","doi":"10.1145/800206.806402","DOIUrl":"https://doi.org/10.1145/800206.806402","url":null,"abstract":"A report is made on the present state of development of a project to construct a tracing aid for users of symbolic computing systems that are written in LISP (or, in principle, any similar high-level language). The traces in question are intended to provide information which is primarily in terms that are natural for a user, e.g. on patterns of actions performed on his data, or patterns occurring in the data themselves during the operation of his program. Patterns are described in a syntax which is inspired by SNOBOL.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"191 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124922387","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 presents an algorithm for building a fundamental system of formal solutions in the neighbourhood of every singularity of a linear homogenous differential operator.
本文给出了在线性齐次微分算子的每一个奇点的邻域中构造基本形式解系统的一种算法。
{"title":"Formal solutions of differential equations in the neighborhood of singular points (Regular and Irregular)","authors":"J. Dora, E. Tournier","doi":"10.1145/800206.806367","DOIUrl":"https://doi.org/10.1145/800206.806367","url":null,"abstract":"This paper presents an algorithm for building a fundamental system of formal solutions in the neighbourhood of every singularity of a linear homogenous differential operator.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131334021","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 common problem in electrical technology is to design a current carrying coil that will produce a given magnetic field. For over a hundred years an equation, the Biot-Savart law, has been available that defines precisely the magnetic field at any point as a line integral along the path of the electric currents that are the sources of the field. In principle then, the design problem is straightforward - it is merely necessary to invert the Biot-Savart law and find a path whose line integral has the given values at the specified field points. However, the actual solution is not trivial and there is a continuing need for improved computational methods for relating magnetic fields to their sources. Two forms of series expansion will be considered here - the ordinary Taylor series and the expansion in spherical harmonics. Other possible expansions - such as Bessel function methods for cylindrical coordinate problems - usually involve integrals over some eigenparameter rather than discrete sums and are not directly competitive with the methods discussed here.
{"title":"Formulation of design rules for NMR imaging coil by using symbolic manipulation","authors":"J. Schenck, M. Hussain","doi":"10.1145/800206.806375","DOIUrl":"https://doi.org/10.1145/800206.806375","url":null,"abstract":"A common problem in electrical technology is to design a current carrying coil that will produce a given magnetic field. For over a hundred years an equation, the Biot-Savart law, has been available that defines precisely the magnetic field at any point as a line integral along the path of the electric currents that are the sources of the field. In principle then, the design problem is straightforward - it is merely necessary to invert the Biot-Savart law and find a path whose line integral has the given values at the specified field points. However, the actual solution is not trivial and there is a continuing need for improved computational methods for relating magnetic fields to their sources.\u0000 Two forms of series expansion will be considered here - the ordinary Taylor series and the expansion in spherical harmonics. Other possible expansions - such as Bessel function methods for cylindrical coordinate problems - usually involve integrals over some eigenparameter rather than discrete sums and are not directly competitive with the methods discussed here.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126045188","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}
It is shown that a simple program, written in the “C” programming language, can be interfaced to a Lisp algebraic manipulation system with substantial performance improvement as a result.
结果表明，用C语言编写的简单程序可以与Lisp代数操作系统接口，从而大大提高了性能。
{"title":"A case study in interlanguage communication: Fast LISP polynomial operations written in 'C'","authors":"R. Fateman","doi":"10.1145/800206.806381","DOIUrl":"https://doi.org/10.1145/800206.806381","url":null,"abstract":"It is shown that a simple program, written in the “C” programming language, can be interfaced to a Lisp algebraic manipulation system with substantial performance improvement as a result.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1981-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129163717","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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