In this paper, we discuss a decision procedure for the indefinite integration of transcendental Liouvillian functions in terms of elementary functions and logarithmic integrals. We also discuss a decision procedure for the integration of a large class of transcendental Liouvillian functions in terms of elementary functions and error-functions.
{"title":"Integration of Liouvillian functions with special functions","authors":"P. H. Knowles","doi":"10.1145/32439.32475","DOIUrl":"https://doi.org/10.1145/32439.32475","url":null,"abstract":"In this paper, we discuss a decision procedure for the indefinite integration of transcendental Liouvillian functions in terms of elementary functions and logarithmic integrals. We also discuss a decision procedure for the integration of a large class of transcendental Liouvillian functions in terms of elementary functions and error-functions.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"147 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":"123078263","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}
An algorithm which finds the definite sum of many series involving binomial coefficients is presented. The method examines the ratio of two consecutive terms of the series in an attempt to express the sum as an ordinary hypergeometric function. A closed form for the infinite sum may be found by comparing the resulting function with known summation theorems. It may also be possible to identify ranges of the summation index for which summing to a finite upper limit is the same as summing to infinity.
{"title":"Summation of binomial coefficients using hypergeometric functions","authors":"M. B. Hayden, E. A. Lamagna","doi":"10.1145/32439.32454","DOIUrl":"https://doi.org/10.1145/32439.32454","url":null,"abstract":"An algorithm which finds the definite sum of many series involving binomial coefficients is presented. The method examines the ratio of two consecutive terms of the series in an attempt to express the sum as an ordinary hypergeometric function. A closed form for the infinite sum may be found by comparing the resulting function with known summation theorems. It may also be possible to identify ranges of the summation index for which summing to a finite upper limit is the same as summing to infinity.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"35 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":"116896296","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 method of parameterizing an object that is represented by constructive solid geometry (CSG) is provided. A method is developed for generating the constraint equations on the parameters which provides a sufficient condition so that the object remains geometrically similar as the parameters are varied. A canonical form for objects represented by CSG is developed. These are applied to the problem of geometric optimization which is part of Computer Aided Engineering.
{"title":"Constructive solid geometry: a symbolic computation approach","authors":"L. Leff, D. Yun","doi":"10.1145/32439.32464","DOIUrl":"https://doi.org/10.1145/32439.32464","url":null,"abstract":"A method of parameterizing an object that is represented by constructive solid geometry (CSG) is provided. A method is developed for generating the constraint equations on the parameters which provides a sufficient condition so that the object remains geometrically similar as the parameters are varied. A canonical form for objects represented by CSG is developed. These are applied to the problem of geometric optimization which is part of Computer Aided Engineering.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"92 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":"129256839","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 computations relating polynomial ideals are reduced to calculating polynomial solutions of a system of linear equations with polynomial coefficients[1]. Zacharias[2] pointed out that Buchberger's algorithm[3] for Gröbner basis can be applied to solving such a linear equation. From the computational viewpoint, Zacharias' method seems to be much better than the previous methods. Hence, we have generalized his method to solve a system of equations directly. After completing the paper, we knew that similar works had been done by several authors[4,5]. This paper describes our method briefly.
{"title":"The Grobner basis of a module over KUX1,...,Xne and polynomial solutions of a system of linear equations","authors":"A. Furukawa, T. Sasaki, H. Kobayashi","doi":"10.1145/32439.32483","DOIUrl":"https://doi.org/10.1145/32439.32483","url":null,"abstract":"Many computations relating polynomial ideals are reduced to calculating polynomial solutions of a system of linear equations with polynomial coefficients[1]. Zacharias[2] pointed out that Buchberger's algorithm[3] for Gröbner basis can be applied to solving such a linear equation. From the computational viewpoint, Zacharias' method seems to be much better than the previous methods. Hence, we have generalized his method to solve a system of equations directly. After completing the paper, we knew that similar works had been done by several authors[4,5]. This paper describes our method briefly.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"10 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":"124673175","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 MathScribe, a powerful user interface for computer algebra systems. The interface makes use of a bitmapped display, windows, menus, and a mouse. Significant new features of MathScribe are its display of both input and output in two-dimensional form, its ability to select previous expressions, and its computationally efficient manner of displaying large expressions.
{"title":"MathScribe: a user interface for computer algebra systems","authors":"C. J. Smith, N. Soiffer","doi":"10.1145/32439.32441","DOIUrl":"https://doi.org/10.1145/32439.32441","url":null,"abstract":"This paper describes MathScribe, a powerful user interface for computer algebra systems. The interface makes use of a bitmapped display, windows, menus, and a mouse. Significant new features of MathScribe are its display of both input and output in two-dimensional form, its ability to select previous expressions, and its computationally efficient manner of displaying large expressions.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"85 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":"127631884","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 new computer algebra system design based on the object-oriented style of programming and an implementation of this design, called Views, written in Smalltalk-80. The design is similar in goals to other 'new' generation computer algebra systems, by allowing the runtime creation of computational domains and providing a way to view these domains as members of categories such as 'group', 'ring' or 'field'. However, Views introduces several unique features. The most notable is the strong distinction made between a domain and its view as a member of a particular category. This distinction between the implementation of a domain and its adherence to a set of algebraic laws allows a great degree of flexibility when choosing the algebraic structures that are to be active during a computation. It also allows for a richer variety of interrelationships among categories than exhibited in other systems.
{"title":"An object-oriented approach to algebra system design","authors":"S. Abdali, Guy W. Cherry, N. Soiffer","doi":"10.1145/32439.32444","DOIUrl":"https://doi.org/10.1145/32439.32444","url":null,"abstract":"This paper describes a new computer algebra system design based on the object-oriented style of programming and an implementation of this design, called Views, written in Smalltalk-80. The design is similar in goals to other 'new' generation computer algebra systems, by allowing the runtime creation of computational domains and providing a way to view these domains as members of categories such as 'group', 'ring' or 'field'. However, Views introduces several unique features. The most notable is the strong distinction made between a domain and its view as a member of a particular category. This distinction between the implementation of a domain and its adherence to a set of algebraic laws allows a great degree of flexibility when choosing the algebraic structures that are to be active during a computation. It also allows for a richer variety of interrelationships among categories than exhibited in other systems.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"10 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":"127915484","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}
1. In recent years the rational approximations have been widely used to solve physical and computational problems /1,2/. When a real function f(x) is repeatedly calculated on a ≤ × ≤ b, it is reasonable to replace it by a polynomial or rational approximation on [a,b]. For example, if f(x) is a composite combination of elementary and special functions any of which can be calculated by means of the corresponding standard program, the f(x) values are obtainable by these programs. This method, however, involves unjustified losses in the computing time and often provides a too high accuracy inadequate to the problem in question. In this case it is more convenient to use the corresponding approximation. There exist iteration algorithms which ensure the best (in the sense of absolute or relative error) rational approximations based on P.L. Chebyshev theory /2,3/. Unfortunately, these algorithms are cumbersome and do not guarantee the convergence if the choice of the initial approximation is unsuccessful, see ref./2/. The present paper treats simple algorithms (Padé-Chebyshev approximation /1/ and Paszkowski algorithm /4/) providing approximations similar to the best ones with a relatively moderate computer resources required. In this case the calculation of the approximation coefficients reduces to the solution of a system (generally speaking, ill-conditioned) of linear algebraic equations. The errors of the Padé-Chebyshev approximations and the corresponding best approximations are compared in the paper /5/ where one of the methods of computation of the Padé-Chebyshev approximations is described. 2. The Analytic Computations System Reduce is a rather convenient tool of realization of algorithms of the rational approximation construction. This system saves one the trouble of inventing an effective algorithm of approximated-function computation if this function can be given in an analytic form or if the terms in the Taylor series expansion are known or determined analytically by the differential equation. The possibility of using the rational arithmetic (without round-off errors) is essential because the coefficients of rational approximations are not stable with respect to the perturbations of initial data and to the round-off errors. Specifically, the error is minimized which arises in solving the ill-conditioned systems of linear equations and when converting a power series into a series of the Chebyshev polynomials and vice versa. The ALGOL - like input language and the convenient tools of solving the problems of linear algebra ensure the simplicity and compactness of programs. For example, the program of computation of the Padé-Chebyshev coefficients occupies sixty two cards. 3. We compute the Padé-Chebyshev approximations by the standard “cross multiplied scheme”/1/. By means of the change of variable x → [(b-a)x+a+b]/2 the approximation on an arbitrary finite range [a,b] is reduced to the approximation on [-1, 1]. We shall, therefore, restrict our
{"title":"Construction of rational approximations by means of REDUCE","authors":"A. Kryukov, A. Rodionov, G. Litvinov","doi":"10.1145/32439.32445","DOIUrl":"https://doi.org/10.1145/32439.32445","url":null,"abstract":"1. In recent years the rational approximations have been widely used to solve physical and computational problems /1,2/. When a real function f(x) is repeatedly calculated on a ≤ × ≤ b, it is reasonable to replace it by a polynomial or rational approximation on [a,b]. For example, if f(x) is a composite combination of elementary and special functions any of which can be calculated by means of the corresponding standard program, the f(x) values are obtainable by these programs. This method, however, involves unjustified losses in the computing time and often provides a too high accuracy inadequate to the problem in question. In this case it is more convenient to use the corresponding approximation.\u0000There exist iteration algorithms which ensure the best (in the sense of absolute or relative error) rational approximations based on P.L. Chebyshev theory /2,3/. Unfortunately, these algorithms are cumbersome and do not guarantee the convergence if the choice of the initial approximation is unsuccessful, see ref./2/. The present paper treats simple algorithms (Padé-Chebyshev approximation /1/ and Paszkowski algorithm /4/) providing approximations similar to the best ones with a relatively moderate computer resources required. In this case the calculation of the approximation coefficients reduces to the solution of a system (generally speaking, ill-conditioned) of linear algebraic equations. The errors of the Padé-Chebyshev approximations and the corresponding best approximations are compared in the paper /5/ where one of the methods of computation of the Padé-Chebyshev approximations is described.\u00002. The Analytic Computations System Reduce is a rather convenient tool of realization of algorithms of the rational approximation construction. This system saves one the trouble of inventing an effective algorithm of approximated-function computation if this function can be given in an analytic form or if the terms in the Taylor series expansion are known or determined analytically by the differential equation. The possibility of using the rational arithmetic (without round-off errors) is essential because the coefficients of rational approximations are not stable with respect to the perturbations of initial data and to the round-off errors. Specifically, the error is minimized which arises in solving the ill-conditioned systems of linear equations and when converting a power series into a series of the Chebyshev polynomials and vice versa. The ALGOL - like input language and the convenient tools of solving the problems of linear algebra ensure the simplicity and compactness of programs. For example, the program of computation of the Padé-Chebyshev coefficients occupies sixty two cards.\u00003. We compute the Padé-Chebyshev approximations by the standard “cross multiplied scheme”/1/. By means of the change of variable x → [(b-a)x+a+b]/2 the approximation on an arbitrary finite range [a,b] is reduced to the approximation on [-1, 1]. We shall, therefore, restrict our","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"13 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":"115012205","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":"Investigating the structure of a Lie algebra","authors":"D. Rand, P. Winternitz","doi":"10.1145/32439.32447","DOIUrl":"https://doi.org/10.1145/32439.32447","url":null,"abstract":"","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"25 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":"122057165","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 most time consuming part is the S-polynomial reduction. Consequently Buchbergcr developed criteria for predicting that certain reductions lead to the zero polynomial, hence allowing the elimination of these reductions [Bu79]. A new interpretation of these criteria and an efficient implcmcntation of them is given in [GM861 and installed in Reduce 3.2 and in Scratchpad II. However there arc still many instances of s~~pcrfluous zero reductions not covered by these criteria.
{"title":"Buchberger's algorithm and staggered linear bases","authors":"Rudiger Gebauer","doi":"10.1145/32439.32482","DOIUrl":"https://doi.org/10.1145/32439.32482","url":null,"abstract":"The most time consuming part is the S-polynomial reduction. Consequently Buchbergcr developed criteria for predicting that certain reductions lead to the zero polynomial, hence allowing the elimination of these reductions [Bu79]. A new interpretation of these criteria and an efficient implcmcntation of them is given in [GM861 and installed in Reduce 3.2 and in Scratchpad II. However there arc still many instances of s~~pcrfluous zero reductions not covered by these criteria.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"28 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":"133081962","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 present the design of a user interface program that can be used with Maple and other symbolic algebra packages. Through the use of a standard communications protocol to such a program, symbolic algebra packages can shed the bulk of code not directly related to algebraic manipulations but can still use the facilities of a powerful user interface. This interface program is designed to be used on a variety of workstations in a consistent fashion.
{"title":"Iris: design of an user interface program for symbolic algebra","authors":"Benton L. Leong","doi":"10.1145/32439.32440","DOIUrl":"https://doi.org/10.1145/32439.32440","url":null,"abstract":"We present the design of a user interface program that can be used with Maple and other symbolic algebra packages. Through the use of a standard communications protocol to such a program, symbolic algebra packages can shed the bulk of code not directly related to algebraic manipulations but can still use the facilities of a powerful user interface. This interface program is designed to be used on a variety of workstations in a consistent fashion.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"49 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":"132277496","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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