Many scientific problems require mathematical modeling and symbolic derivation to produce formulas which are evaluated numerically. Computer algebra systems can be used effectively in the symbolic derivation phase and subsequently to generate numerical code. This paper describes the design of the GENTRAN code generation facility for REDUCE.
{"title":"A numerical code generation facility for REDUCE","authors":"Barbara L. Gates","doi":"10.1145/32439.32459","DOIUrl":"https://doi.org/10.1145/32439.32459","url":null,"abstract":"Many scientific problems require mathematical modeling and symbolic derivation to produce formulas which are evaluated numerically. Computer algebra systems can be used effectively in the symbolic derivation phase and subsequently to generate numerical code. This paper describes the design of the GENTRAN code generation facility for REDUCE.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"117 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":"123771914","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 deals with the system of n linear differential equations (*) y'(x) = A(x)y where A(x) is a matrix with formal series coefficients. A sequence of formal invariants related to (*) is defined. An algorithm which reduces (*) by means of meromorphic transformations to a “super-irreducible” form is given. The computation of these invariants follows directly from this form. This algorithm is implemented in Reduce.
{"title":"Algorithm for computing formal invariants of linear differential systems","authors":"A. Hilali, A. Wazner","doi":"10.1145/32439.32478","DOIUrl":"https://doi.org/10.1145/32439.32478","url":null,"abstract":"This paper deals with the system of n linear differential equations (*) y'(x) = A(x)y where A(x) is a matrix with formal series coefficients. A sequence of formal invariants related to (*) is defined. An algorithm which reduces (*) by means of meromorphic transformations to a “super-irreducible” form is given. The computation of these invariants follows directly from this form. This algorithm is implemented in Reduce.","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":"131227857","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. IntnniuctIon The first time that a student is exposed to formal mathematical proofs normally happens during a high school geometry course. it is well known that this topic causes significant difficulties for both teachers and students. According to [Hoffer 811, geometry is usually rated as the least liked of all subjects. Although proofs are all but welcomed in high school mathematics, it is generally recognized that an understanding of the basic principles underlying mathematical proofs is essential for the development of higher mathematical and logical skiHs.
{"title":"An application of knowledge-base technology in education: a geometry theorem prover","authors":"M. Hadzikadic, F. Lichtenberger, D. Yun","doi":"10.1145/32439.32468","DOIUrl":"https://doi.org/10.1145/32439.32468","url":null,"abstract":"1. IntnniuctIon The first time that a student is exposed to formal mathematical proofs normally happens during a high school geometry course. it is well known that this topic causes significant difficulties for both teachers and students. According to [Hoffer 811, geometry is usually rated as the least liked of all subjects. Although proofs are all but welcomed in high school mathematics, it is generally recognized that an understanding of the basic principles underlying mathematical proofs is essential for the development of higher mathematical and logical skiHs.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"45 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":"133361570","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 this paper we propose a design and implementation of operators and their associated functionality for symbolic algebra systems. We believe that operators should blend harmoniously (syntactically and semantically) with the underlying language, in such a way that users will find them convenient and appealing to use. It is “vox populi” that operators are needed in a symbolic algebra system, although there is little consensus on what these should be, what the semantics should be, allowable operations, syntax, etc. All of these ideas, and examples, have been implemented and work as described in our current version of Maple Cha85. During the first Maple retreat83 we established a basic design for operators. The implementation of this design was delayed until some remaining crucial details were finally solved during the 1985 Maple retreat (Sept 1985). In this sense, this paper is the result of the collective work of all the participants of these two retreats. What is an operator? We would like to define an operator to be an abstract data type which describes (at various possible degrees: totally, partially or minimally) an operation to be performed on its arguments. This abstract data type is closely associated with the operations of application and composition, but will also allow most (or all) of the other algebraic operations. We it found useful to have some “witness” examples that we want to solve in an elegant and general form. The two main examples were: (a) How to represent the first derivative of ƒ(x) at 0, i.e. ƒ′(0) (the above really boils down to an effective representation of the differentiation operator) (b) How to represent and to operate with a non-communative multiplication operator, for example matrix multiplication. Of course many systems solve the above problems, but in some cases (in particular for the first example) as an ad-hoc solution. By an ad-hoc solution we mean that, for the differentiation example, this operator cannot be written in terms of the primitives given by the language. It is important to note that there are three issues to resolve:a purely representational/syntactic argument: how to input/output these operators. a purely functional argument: how to perform all the operations we want performed. an integrational argument: how to join operators harmoniously with a symbolic algebra system.
{"title":"An implementation of operators for symbolic algebra systems","authors":"G. Gonnet","doi":"10.1145/32439.32487","DOIUrl":"https://doi.org/10.1145/32439.32487","url":null,"abstract":"In this paper we propose a design and implementation of operators and their associated functionality for symbolic algebra systems. We believe that operators should blend harmoniously (syntactically and semantically) with the underlying language, in such a way that users will find them convenient and appealing to use. It is “vox populi” that operators are needed in a symbolic algebra system, although there is little consensus on what these should be, what the semantics should be, allowable operations, syntax, etc. All of these ideas, and examples, have been implemented and work as described in our current version of Maple Cha85.\u0000During the first Maple retreat83 we established a basic design for operators. The implementation of this design was delayed until some remaining crucial details were finally solved during the 1985 Maple retreat (Sept 1985). In this sense, this paper is the result of the collective work of all the participants of these two retreats.\u0000What is an operator? We would like to define an operator to be an abstract data type which describes (at various possible degrees: totally, partially or minimally) an operation to be performed on its arguments. This abstract data type is closely associated with the operations of application and composition, but will also allow most (or all) of the other algebraic operations.\u0000We it found useful to have some “witness” examples that we want to solve in an elegant and general form. The two main examples were:\u0000(a) How to represent the first derivative of ƒ(x) at 0, i.e. ƒ′(0) (the above really boils down to an effective representation of the differentiation operator)\u0000(b) How to represent and to operate with a non-communative multiplication operator, for example matrix multiplication.\u0000Of course many systems solve the above problems, but in some cases (in particular for the first example) as an ad-hoc solution. By an ad-hoc solution we mean that, for the differentiation example, this operator cannot be written in terms of the primitives given by the language.\u0000It is important to note that there are three issues to resolve:a purely representational/syntactic argument: how to input/output these operators.\u0000a purely functional argument: how to perform all the operations we want performed.\u0000an integrational argument: how to join operators harmoniously with a symbolic algebra system.\u0000","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":"132725709","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 discuss the design, implementation, and benchmarking of a system that can manipulate symbolic expressions represented by their straight-line computations. Our system is capable of performing rational arithmetic, evaluating, differentiating, taking greatest common divisors of, and factoring polynomials in straight-line format. The straight-line results can also be converted to standard sparse format. We show by example that our system can handle problems for which conventional methods lead to excessive intermediate expression swell.
{"title":"A system for manipulating polynomials given by straight-line programs","authors":"T. S. Freeman, Gregory M. Imirzian, E. Kaltofen","doi":"10.1145/32439.32473","DOIUrl":"https://doi.org/10.1145/32439.32473","url":null,"abstract":"We discuss the design, implementation, and benchmarking of a system that can manipulate symbolic expressions represented by their straight-line computations. Our system is capable of performing rational arithmetic, evaluating, differentiating, taking greatest common divisors of, and factoring polynomials in straight-line format. The straight-line results can also be converted to standard sparse format. We show by example that our system can handle problems for which conventional methods lead to excessive intermediate expression swell.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"23 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":"120958620","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 [l] a systematic effort to carry out the structural analysis of a Lie algebra L on a computer was begun. The effort concentrated on the calculation of the various radicals of L , the various series of L , a Cartan subalgebra of L , and the derivation algebra of L. From these computations one can determine whether L is nilpotent, solvable, or semisimple. The computer implementations of this analysis were done using APL. This paper describes the extension of this work to the structural analysis of a Lie algebra L whose multiplication table is defined symbolically rather than numerically. The expanded context of this analysis requires a symbol manipulation language for the computer implementations; we have chosen to use MACSYMA.
{"title":"Symbolic algorithms for Lie algebra computation","authors":"R. Beck, B. Kolman","doi":"10.1145/32439.32456","DOIUrl":"https://doi.org/10.1145/32439.32456","url":null,"abstract":"In [l] a systematic effort to carry out the structural analysis of a Lie algebra L on a computer was begun. The effort concentrated on the calculation of the various radicals of L , the various series of L , a Cartan subalgebra of L , and the derivation algebra of L. From these computations one can determine whether L is nilpotent, solvable, or semisimple. The computer implementations of this analysis were done using APL. This paper describes the extension of this work to the structural analysis of a Lie algebra L whose multiplication table is defined symbolically rather than numerically. The expanded context of this analysis requires a symbol manipulation language for the computer implementations; we have chosen to use MACSYMA.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"4 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":"132370027","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 communication between a user @nd Q computer can be realized in the dialogue or package mode. In numer ical calculations the package mode is quite acceptable, but it is not so good for analytic calcul ations because of their specific char acter. In analytic cal cul a-tions an active search for solution i s, as a r ul e, done which is based, to a large extent , on the tr ial .-and-err or method. Al 1 moder n systems of analytic calculations use the dialogue mode. When organizing a dial ague, psychological factor s must, natu-r ally, be taken into account. 'T'he neglect of these factors decreases the efficiency in the use of the system. When devel oping the dialogue mode of the REDUCE system we did our best to include the r esults of the psy-chologic exper iment /I/. Among the universal systems the REDUCE system of computer al gebra /2/ is the most popular. The system is equipped with sufficiently powerful mathematics incl uciing, along with the polynomial algebr a, the oper ations on matr ices, differ entiation, integr ation, algebra of Dir ac r matr ices,etc. The po-pul ar ity of the system is expl aineci, large1 y, by the convenient ALGOL-like language which possesses all the necessary constructions fr om the standpoint of moder n sty1 e of pr og-r amming. 2. Dialogue mode in the REDUCE system. The dialogue mode of the REDUCE system is based on a special method of processing Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specfic permission. error s. For example, if an err or is revealed in the package mode the pr ogr am execution is inter r upted for the syntactic parsing of the remaining text. In the dialogue mode, the program execution is inter r upted, if an err or occur es, but, as opposed to the package mode, the control is tr ansfer red to a user. The expression r eponsible for an er r-Or is neglected in both the cases. In the dialogue mode a user …
{"title":"Dialogue in REDUCE: experience and development","authors":"A. Kryukov","doi":"10.1145/32439.32461","DOIUrl":"https://doi.org/10.1145/32439.32461","url":null,"abstract":"The communication between a user @nd Q computer can be realized in the dialogue or package mode. In numer ical calculations the package mode is quite acceptable, but it is not so good for analytic calcul ations because of their specific char acter. In analytic cal cul a-tions an active search for solution i s, as a r ul e, done which is based, to a large extent , on the tr ial .-and-err or method. Al 1 moder n systems of analytic calculations use the dialogue mode. When organizing a dial ague, psychological factor s must, natu-r ally, be taken into account. 'T'he neglect of these factors decreases the efficiency in the use of the system. When devel oping the dialogue mode of the REDUCE system we did our best to include the r esults of the psy-chologic exper iment /I/. Among the universal systems the REDUCE system of computer al gebra /2/ is the most popular. The system is equipped with sufficiently powerful mathematics incl uciing, along with the polynomial algebr a, the oper ations on matr ices, differ entiation, integr ation, algebra of Dir ac r matr ices,etc. The po-pul ar ity of the system is expl aineci, large1 y, by the convenient ALGOL-like language which possesses all the necessary constructions fr om the standpoint of moder n sty1 e of pr og-r amming. 2. Dialogue mode in the REDUCE system. The dialogue mode of the REDUCE system is based on a special method of processing Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specfic permission. error s. For example, if an err or is revealed in the package mode the pr ogr am execution is inter r upted for the syntactic parsing of the remaining text. In the dialogue mode, the program execution is inter r upted, if an err or occur es, but, as opposed to the package mode, the control is tr ansfer red to a user. The expression r eponsible for an er r-Or is neglected in both the cases. In the dialogue mode a user …","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":"128675656","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}
Recently, geometry theorem proving has become an important topic of research in symbolic computation. In this paper we present a new approach to automated geometry theorem proving that is based on Buchberger's Gröbner bases method, one of the most important general purpose methods in computer algebra. The goal is to automatically prove geometry theorems whose hypotheses and conjecture can be expressed algebraically, i.e. in form of polynomial equations. After shortly reviewing the basic notions of Gröbner bases and discussing some new aspects on confirming theorems, we describe two different methods for applying Buchberger's algorithm to geometry theorem proving, each of them being more efficient than the other on a certain class of problems. The second method requires a new notion of reduction, which we call pseudoreduction. This pseudoreduction yields results on polynomials over some rational function field by computations that are done merely over the rationals and, therefore, is of general interest. Finally, a computing time statistics on about 40 non-trivial examples is given, based on an implementation of the methods in the computer algebra system SAC-2 on an IBM 4341.
{"title":"Automated geometry theorem proving using Buchberger's algorithm","authors":"B. Kutzler, S. Stifter","doi":"10.1145/32439.32480","DOIUrl":"https://doi.org/10.1145/32439.32480","url":null,"abstract":"Recently, geometry theorem proving has become an important topic of research in symbolic computation. In this paper we present a new approach to automated geometry theorem proving that is based on Buchberger's Gröbner bases method, one of the most important general purpose methods in computer algebra. The goal is to automatically prove geometry theorems whose hypotheses and conjecture can be expressed algebraically, i.e. in form of polynomial equations. After shortly reviewing the basic notions of Gröbner bases and discussing some new aspects on confirming theorems, we describe two different methods for applying Buchberger's algorithm to geometry theorem proving, each of them being more efficient than the other on a certain class of problems. The second method requires a new notion of reduction, which we call pseudoreduction. This pseudoreduction yields results on polynomials over some rational function field by computations that are done merely over the rationals and, therefore, is of general interest. Finally, a computing time statistics on about 40 non-trivial examples is given, based on an implementation of the methods in the computer algebra system SAC-2 on an IBM 4341.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"74 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120847257","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 here various ways of generalizing the Padé approximation to multivariate functions. To compute them, we use a computer algebra system: REDUCE.
{"title":"How to compute multivariate Pade approximants","authors":"C. Chaffy","doi":"10.1145/32439.32450","DOIUrl":"https://doi.org/10.1145/32439.32450","url":null,"abstract":"We present here various ways of generalizing the Padé approximation to multivariate functions. To compute them, we use a computer algebra system: REDUCE.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"58 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":"125796576","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 artificial intelligence system that learns by observing its users perform symbolic mathematical problem solving is presented. This fully-implemented system is being evaluated as a problem solver in the domain of classical physics. Using its mathematical and physical knowledge, the system determines why a human-provided solution to a specific problem suffices to solve the problem, and then extends the solution technique to more general situations, thereby improving its own problem-solving performance. This research illustrates a need for symbolic mathematics systems to produce explanations of their problem-solving steps, as these explanations guide learning. Although physics problem solving is currently being investigated, the results obtained are relevant to other mathematically-based domains. This work also has implications for intelligent computer-aided instruction in domains of this type.
{"title":"Computer understanding and generalization of symbolic mathematical calculations: a case study in physics problem solving","authors":"J. Shavlik, G. DeJong","doi":"10.1145/32439.32469","DOIUrl":"https://doi.org/10.1145/32439.32469","url":null,"abstract":"An artificial intelligence system that learns by observing its users perform symbolic mathematical problem solving is presented. This fully-implemented system is being evaluated as a problem solver in the domain of classical physics. Using its mathematical and physical knowledge, the system determines why a human-provided solution to a specific problem suffices to solve the problem, and then extends the solution technique to more general situations, thereby improving its own problem-solving performance. This research illustrates a need for symbolic mathematics systems to produce explanations of their problem-solving steps, as these explanations guide learning. Although physics problem solving is currently being investigated, the results obtained are relevant to other mathematically-based domains. This work also has implications for intelligent computer-aided instruction in domains of this type.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128023482","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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