In the last decade or so there has been an out-growth of research in algebraic complexity of computations which showed how to derive algorithms systematically. One of the features of these derivation is their reliance on algebraic and symbolic constructions. I would like, in this paper, to survey some of the symbolic and algebraic constructions which are used, and then draw some conclusions on the implications of this development for symbol manipulation systems. Let me start by sketching some of these derivations.
{"title":"Algebraic constructions for algorithms (Extended Abstract)","authors":"S. Winograd","doi":"10.1145/800206.806385","DOIUrl":"https://doi.org/10.1145/800206.806385","url":null,"abstract":"In the last decade or so there has been an out-growth of research in algebraic complexity of computations which showed how to derive algorithms systematically. One of the features of these derivation is their reliance on algebraic and symbolic constructions. I would like, in this paper, to survey some of the symbolic and algebraic constructions which are used, and then draw some conclusions on the implications of this development for symbol manipulation systems. Let me start by sketching some of these derivations.","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":"114078002","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}
Kovacic [3] has given an algorithm for the closed form solution of differential equations of the form ay" + by' + cy &equil; 0, where a, b, and c are rational functions with complex coefficients of the independent variable x. The algorithm provides a Liouvillian solution (i.e. one that can be expressed in terms of integrals, exponentials and algebraic functions) or reports that no such solution exists. In this note a version of Kovacic's algorithm is described. This version has been implemented in MACSYMA and tested successfully on examples in Boyce and DiPrima [1], Kamke [2], and Kovacic [3]. Modifications to the algorithm have been made to minimize the amount of code needed and to avoid the complete factorization of a polynomial called for. In Section 2 these issues are discussed and in Section 3 the author's current version of the algorithm is described.
{"title":"An implementation of Kovacic's algorithm for solving second order linear homogeneous differential equations","authors":"B. D. Saunders","doi":"10.1145/800206.806378","DOIUrl":"https://doi.org/10.1145/800206.806378","url":null,"abstract":"Kovacic [3] has given an algorithm for the closed form solution of differential equations of the form ay\" + by' + cy &equil; 0, where a, b, and c are rational functions with complex coefficients of the independent variable x. The algorithm provides a Liouvillian solution (i.e. one that can be expressed in terms of integrals, exponentials and algebraic functions) or reports that no such solution exists.\u0000 In this note a version of Kovacic's algorithm is described. This version has been implemented in MACSYMA and tested successfully on examples in Boyce and DiPrima [1], Kamke [2], and Kovacic [3]. Modifications to the algorithm have been made to minimize the amount of code needed and to avoid the complete factorization of a polynomial called for. In Section 2 these issues are discussed and in Section 3 the author's current version of the algorithm is described.","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":"130260734","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}
So-called general purpose systems for algebraic computation such as ALTRAN, MACSYMA, SAC, SCRATCHPAD and REDUCE are almost exclusively concerned with what is usually known as “classical algebra”, that is, rings of real or complex polynomials and rings of real or complex functions. These systems have been designed to compute with elements in a fixed algebraic structure (usually the ring of real functions). Typical of the facilities provided are: the arithmetic operations of the ring, the calculation of polynomial gcd's, the location of the zeros of a polynomial; and some operations from calculus: differentiation, integration, the calculation of limits, and the analytic solution of certain classes of differential equations. For brevity, we shall refer to these systems as CA systems.
{"title":"The basis of a computer system for modern algebra","authors":"John J. Cannon","doi":"10.1145/800206.806362","DOIUrl":"https://doi.org/10.1145/800206.806362","url":null,"abstract":"So-called general purpose systems for algebraic computation such as ALTRAN, MACSYMA, SAC, SCRATCHPAD and REDUCE are almost exclusively concerned with what is usually known as “classical algebra”, that is, rings of real or complex polynomials and rings of real or complex functions. These systems have been designed to compute with elements in a fixed algebraic structure (usually the ring of real functions). Typical of the facilities provided are: the arithmetic operations of the ring, the calculation of polynomial gcd's, the location of the zeros of a polynomial; and some operations from calculus: differentiation, integration, the calculation of limits, and the analytic solution of certain classes of differential equations. For brevity, we shall refer to these systems as CA systems.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"21 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":"131806119","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 considers the problem of factoring polynomials over a variety of domains. We first describe the current methods of factoring polynomials over the integers, and extend them to the integers mod p. We then consider the problem of factoring over algebraic domains. Having produced several negative results, showing that, if the domain is not properly specified, then the problem is insoluble, we then show that, for a properly specified finitely generated extension of the rationals or the integers mod p, the problem is soluble. We conclude by discussing the problems of factoring over algebraic closures.
{"title":"Factorization over finitely generated fields","authors":"J. Davenport, B. Trager","doi":"10.1145/800206.806396","DOIUrl":"https://doi.org/10.1145/800206.806396","url":null,"abstract":"This paper considers the problem of factoring polynomials over a variety of domains. We first describe the current methods of factoring polynomials over the integers, and extend them to the integers mod p. We then consider the problem of factoring over algebraic domains. Having produced several negative results, showing that, if the domain is not properly specified, then the problem is insoluble, we then show that, for a properly specified finitely generated extension of the rationals or the integers mod p, the problem is soluble. We conclude by discussing the problems of factoring over algebraic closures.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"24 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":"133432986","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 attempts to list some optimising transformations for user programs for an Algebraic Manipulation System. To investigate optimisation of both computer time and space, a general purpose system REDUCE has been chosen for study. The optimising transformations may be applied manually. However, the authors hope to automate the process. Examples using various optimisations are included and clearly show the benefit of the process.
{"title":"The optimization of user programs for an Algebraic Manipulation System","authors":"P. D. Pearce, R. J. Hicks","doi":"10.1145/800206.806383","DOIUrl":"https://doi.org/10.1145/800206.806383","url":null,"abstract":"This paper attempts to list some optimising transformations for user programs for an Algebraic Manipulation System. To investigate optimisation of both computer time and space, a general purpose system REDUCE has been chosen for study. The optimising transformations may be applied manually. However, the authors hope to automate the process. Examples using various optimisations are included and clearly show the benefit of the process.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"48 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":"133865019","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 reviews issues in the design of special-purpose VLSI chips in general, and suggests VLSI designs for polynomial multiplication and division, which are basic functional modules in algebraic computation.
{"title":"Use of VLSI in algebraic computation: Some suggestions","authors":"H. T. Kung","doi":"10.1145/800206.806399","DOIUrl":"https://doi.org/10.1145/800206.806399","url":null,"abstract":"This paper reviews issues in the design of special-purpose VLSI chips in general, and suggests VLSI designs for polynomial multiplication and division, which are basic functional modules in algebraic computation.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"115 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":"133821636","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}
SMP is a new general-purpose symbolic manipulation computer program which has been developed during the past year by the authors, with help from G.C. Fox, J.M. Greif, E.D. Mjolsness, L.J. Romans, T. Shaw and A.E. Terrano. The primary motivation for the construction of the program was the necessity of performing very complicated algebraic manipulations in certain areas of theoretical physics. The need to deal with advanced mathematical constructs required the program to be of great generality. In addition, the size of the calculations anticipated demanded that the program should operate quickly and be capable of handling very large amounts of data. The resulting program is expected to be valuable in a wide variety of applications. In this paper, we describe some of the basic concepts and principles of SMP. The extensive capabilities of SMP are described, with examples, in the “SMP Handbook” (available on request from the authors).
{"title":"SMP - A Symbolic Manipulation Program","authors":"Chris A. Cole, S. Wolfram","doi":"10.1145/800206.806365","DOIUrl":"https://doi.org/10.1145/800206.806365","url":null,"abstract":"SMP is a new general-purpose symbolic manipulation computer program which has been developed during the past year by the authors, with help from G.C. Fox, J.M. Greif, E.D. Mjolsness, L.J. Romans, T. Shaw and A.E. Terrano. The primary motivation for the construction of the program was the necessity of performing very complicated algebraic manipulations in certain areas of theoretical physics. The need to deal with advanced mathematical constructs required the program to be of great generality. In addition, the size of the calculations anticipated demanded that the program should operate quickly and be capable of handling very large amounts of data. The resulting program is expected to be valuable in a wide variety of applications.\u0000 In this paper, we describe some of the basic concepts and principles of SMP. The extensive capabilities of SMP are described, with examples, in the “SMP Handbook” (available on request from the authors).","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"102 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":"134309930","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 are interested in computing the invariant subring (also referred to as fixed ring) which is obtained as a result of the action of a certain finite group of k-linear automorphisms on the polynomial ring in two variables with coefficients in a field of characteristic p.
{"title":"Computing an invariant subring of k[X,Y]","authors":"Rosalind Neuman","doi":"10.1145/800206.806392","DOIUrl":"https://doi.org/10.1145/800206.806392","url":null,"abstract":"We are interested in computing the invariant subring (also referred to as fixed ring) which is obtained as a result of the action of a certain finite group of k-linear automorphisms on the polynomial ring in two variables with coefficients in a field of characteristic p.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"35 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":"121090778","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 availability of new large-address-space computers has provided us an opportunity to examine techniques for transferring programming systems, and in particular, Lisp systems, to new computers. We contrast two approaches: designing and building a Virtual Machine implementation of Lisp, and (re)writing the system in a “portable” programming language ('C'). Our conclusion is that the latter approach may very well be better.
{"title":"Views on transportability of Lisp and Lisp-based systems","authors":"R. Fateman","doi":"10.1145/800206.806384","DOIUrl":"https://doi.org/10.1145/800206.806384","url":null,"abstract":"The availability of new large-address-space computers has provided us an opportunity to examine techniques for transferring programming systems, and in particular, Lisp systems, to new computers. We contrast two approaches: designing and building a Virtual Machine implementation of Lisp, and (re)writing the system in a “portable” programming language ('C'). Our conclusion is that the latter approach may very well be better.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"61 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":"125873311","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}
Polynomial remainder sequences are the basis of many important algorithms in symbolic and algebraic manipulation. In a number of these algorithms, the actual coefficients of the sequence are not required; rather, the method uses the signs of the coefficients. Present techniques, however, compute the exact coefficients (or a mixed radix representation of them), and then obtain the signs. This paper discusses a new approach in which interval arithmetic is used to obtain the signs of the coefficients without computing their exact values. Comparisons of this method with analogous standard techniques show empirical computing time reductions of two orders of magnitude for even relatively small cases.
{"title":"Interval arithmetic applied to polynomial remainder sequences","authors":"J. Pinkert","doi":"10.1145/800205.806337","DOIUrl":"https://doi.org/10.1145/800205.806337","url":null,"abstract":"Polynomial remainder sequences are the basis of many important algorithms in symbolic and algebraic manipulation. In a number of these algorithms, the actual coefficients of the sequence are not required; rather, the method uses the signs of the coefficients. Present techniques, however, compute the exact coefficients (or a mixed radix representation of them), and then obtain the signs. This paper discusses a new approach in which interval arithmetic is used to obtain the signs of the coefficients without computing their exact values. Comparisons of this method with analogous standard techniques show empirical computing time reductions of two orders of magnitude for even relatively small cases.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"358 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1976-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133666976","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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