Symbolic solutions of large sparse systems of linear equations, such as those encountered in several engineering disciplines (electrical engineering, biology, chemical engineering etc.) are often very lengthy, and received for this reason only occasional attention. This places the designer of a new and probably more successful symbolic solution method for the hard problem to find a representation which is suitable in the corresponding engineering areas, while still being neat and compact. It is believed that this problem has been solved to a great deal with the introduction of the new Factoring Recursive Minor Expansion algorithm with Memo, FDSLEM, presented in this paper. The FDSLEM algorithm has important properties which make the implementation of an algorithm which can generate the approximate solution of a perturbed system of equations relatively straight forward. The algorithms given can operate on arbitrary sparse matrices, but one obtains optimal profit of the properties of the algorithm if the matrices have a certain fundamental form, as is illustrated in the paper.
{"title":"A cancellation free algorithm, with factoring capabilities, for the efficient solution of large sparse sets of equations","authors":"J. Smit","doi":"10.1145/800206.806386","DOIUrl":"https://doi.org/10.1145/800206.806386","url":null,"abstract":"Symbolic solutions of large sparse systems of linear equations, such as those encountered in several engineering disciplines (electrical engineering, biology, chemical engineering etc.) are often very lengthy, and received for this reason only occasional attention. This places the designer of a new and probably more successful symbolic solution method for the hard problem to find a representation which is suitable in the corresponding engineering areas, while still being neat and compact. It is believed that this problem has been solved to a great deal with the introduction of the new Factoring Recursive Minor Expansion algorithm with Memo, FDSLEM, presented in this paper.\u0000 The FDSLEM algorithm has important properties which make the implementation of an algorithm which can generate the approximate solution of a perturbed system of equations relatively straight forward.\u0000 The algorithms given can operate on arbitrary sparse matrices, but one obtains optimal profit of the properties of the algorithm if the matrices have a certain fundamental form, as is illustrated in the paper.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"2013 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":"127435954","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 class of univariate polynomials is defined which make the Berlekamp-Hensel factorization algorithm take an exponential amount of time. The class contains as subclasses the Swinnerton-Dyer polynomials discussed by Berlekamp and a subset of the cyclotomic polynomials. Aside from shedding light on the complexity of polynomial factorization this class is also useful in testing implementations of the Berlekamp-Hensel and related algorithms.
{"title":"A generalized class of polynomials that are hard to factor","authors":"E. Kaltofen, D. Musser","doi":"10.1145/800206.806394","DOIUrl":"https://doi.org/10.1145/800206.806394","url":null,"abstract":"A class of univariate polynomials is defined which make the Berlekamp-Hensel factorization algorithm take an exponential amount of time. The class contains as subclasses the Swinnerton-Dyer polynomials discussed by Berlekamp and a subset of the cyclotomic polynomials. Aside from shedding light on the complexity of polynomial factorization this class is also useful in testing implementations of the Berlekamp-Hensel and related algorithms.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"44 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":"114142093","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 algebraic manipulation system Macsyma [Grou77, Fate80] has been running for over a year on Digital Equipment Corp. VAX-11 large-address-space medium-scale computers [Stre78]. In order to run Macsyma in this environment, a Lisp system for the VAX, FRANZ LISP[Fode50], was constructed at Berkeley. The goal of running Macsyma provided direction and motivation and is partially responsible for the rapid development of the Lisp system. Because Lisp is a high level language there are many decisions to be made about the internal framework of the system. Efforts to increase efficiency require that we be able to characterize the demands of a large, compiled, Lisp system. Fortunately, the VAX/UNIX operating system provides useful tools for determining such characteristics. This paper presents some of our data and related analysis.
{"title":"Characterization of VAX Macsyma","authors":"John K. Foderaro, R. Fateman","doi":"10.1145/800206.806364","DOIUrl":"https://doi.org/10.1145/800206.806364","url":null,"abstract":"The algebraic manipulation system Macsyma [Grou77, Fate80] has been running for over a year on Digital Equipment Corp. VAX-11 large-address-space medium-scale computers [Stre78]. In order to run Macsyma in this environment, a Lisp system for the VAX, FRANZ LISP[Fode50], was constructed at Berkeley. The goal of running Macsyma provided direction and motivation and is partially responsible for the rapid development of the Lisp system.\u0000 Because Lisp is a high level language there are many decisions to be made about the internal framework of the system. Efforts to increase efficiency require that we be able to characterize the demands of a large, compiled, Lisp system. Fortunately, the VAX/UNIX operating system provides useful tools for determining such characteristics. This paper presents some of our data and related analysis.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"19 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":"128387689","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 describe a programming environment which combines the Macsyma algebraic manipulation system with convenient and direct access to numeric Fortran run-time libraries. With this system it is also convenient to generate, compile, load, and invoke totally new Fortran programs which may have been produced by combining algebraically derived formulas and program “templates”. These facilities, available on VAX-11 computers, provide an environment for the generation and testing of advanced scientific software. Enhancements of Fortran for high-precision calculations, interval arithmetic, and other purposes are also supported.
{"title":"An algebraic front-end for the production and use of numeric programs","authors":"Douglas H. Lanam","doi":"10.1145/800206.806400","DOIUrl":"https://doi.org/10.1145/800206.806400","url":null,"abstract":"We describe a programming environment which combines the Macsyma algebraic manipulation system with convenient and direct access to numeric Fortran run-time libraries. With this system it is also convenient to generate, compile, load, and invoke totally new Fortran programs which may have been produced by combining algebraically derived formulas and program “templates”. These facilities, available on VAX-11 computers, provide an environment for the generation and testing of advanced scientific software. Enhancements of Fortran for high-precision calculations, interval arithmetic, and other purposes are also supported.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"16 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":"129931343","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 algorithmic approach to symbolic solution of 2nd order linear ODEs. The algorithm consists of two parts. The first part involves complete algorithms for hypergeometric equations and hypergeometric equations of confluent type. These algorithms are based on Riemann's P-functions and Hukuhara's P-functions respectively. Another part involves an algorithm for transforming a given equation to a hypergeometric equation or a hypergeometric equation of confluent type. The transformation is possible if a given equation satisfies certain conditions, otherwise it works only as one of heuristic methods. However our method can solve many equations which seem to be very difficult to solve by conventional methods.
{"title":"A technique for solving ordinary differential equations using Riemann's P-functions","authors":"S. Watanabe","doi":"10.1145/800206.806369","DOIUrl":"https://doi.org/10.1145/800206.806369","url":null,"abstract":"This paper presents an algorithmic approach to symbolic solution of 2nd order linear ODEs. The algorithm consists of two parts. The first part involves complete algorithms for hypergeometric equations and hypergeometric equations of confluent type. These algorithms are based on Riemann's P-functions and Hukuhara's P-functions respectively. Another part involves an algorithm for transforming a given equation to a hypergeometric equation or a hypergeometric equation of confluent type. The transformation is possible if a given equation satisfies certain conditions, otherwise it works only as one of heuristic methods. However our method can solve many equations which seem to be very difficult to solve by conventional methods.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"96 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":"126563523","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}
Powerful algebraic manipulation systems will be available at very low costs in this decade. We are thus close to achieving the goal many of us have been after for nearly two decades, namely, availability of our systems to the masses. In this talk we shall discuss the problems we still face in making our systems useful to the hordes of potential users.
{"title":"Algebraic computation for the masses","authors":"J. Moses","doi":"10.1145/800206.806389","DOIUrl":"https://doi.org/10.1145/800206.806389","url":null,"abstract":"Powerful algebraic manipulation systems will be available at very low costs in this decade. We are thus close to achieving the goal many of us have been after for nearly two decades, namely, availability of our systems to the masses. In this talk we shall discuss the problems we still face in making our systems useful to the hordes of potential users.","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":"128113419","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}
Algebraic manipulation systems such as MACSYMA include algorithms and heuristic procedures for indefinite and definite integration, yet these system facilities are not as generally useful as might be thought. Most isolated definite integration problems are more efficiently tackled with numerical programs. Unfortunately, the answers obtained are sometimes incorrect, in spite of assurances of accuracy; furthermore, large classes of problems can sometimes be solved more rapidly by preliminary algebraic transformations. In this paper we indicate various directions for improving the usefulness of integration programs given closed form integrands, via algebraic manipulation techniques. These include expansions in partial fractions or Taylor series, detection and removal of singularities and symmetries, and various approximation techniques for troublesome problems.
{"title":"Computer algebra and numerical integration","authors":"R. Fateman","doi":"10.1145/800206.806401","DOIUrl":"https://doi.org/10.1145/800206.806401","url":null,"abstract":"Algebraic manipulation systems such as MACSYMA include algorithms and heuristic procedures for indefinite and definite integration, yet these system facilities are not as generally useful as might be thought. Most isolated definite integration problems are more efficiently tackled with numerical programs. Unfortunately, the answers obtained are sometimes incorrect, in spite of assurances of accuracy; furthermore, large classes of problems can sometimes be solved more rapidly by preliminary algebraic transformations.\u0000 In this paper we indicate various directions for improving the usefulness of integration programs given closed form integrands, via algebraic manipulation techniques. These include expansions in partial fractions or Taylor series, detection and removal of singularities and symmetries, and various approximation techniques for troublesome problems.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"195 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":"132718162","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}
Partial fractions is an important algebraic operation with many applications in applied mathematics, physics and engineering. It is also an important operation in any computer symbolic and algebraic system. Among other things, it is used in the integration algorithm.
{"title":"A p-adic algorithm for univariate partial fractions","authors":"Paul S. Wang","doi":"10.1145/800206.806398","DOIUrl":"https://doi.org/10.1145/800206.806398","url":null,"abstract":"Partial fractions is an important algebraic operation with many applications in applied mathematics, physics and engineering. It is also an important operation in any computer symbolic and algebraic system. Among other things, it is used in the integration algorithm.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"33 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":"127858350","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 several software tools based on the algebraic manipulation system MACSYMA have been implemented which facilitate the design, analysis and construction of finite difference programs for the numerical solution of systems of partial differential equations. Two of them are described here. The FDIFF package converts scalar, non-linear partial differential equations into linear, finite difference approximations. It includes tools for discretization of the domain of the PDE's dependent variables, linearization of non-linear terms and conversion of derivative terms into finite difference expressions. A notation and algebra for building arbitrary finite difference operators is provided. The FSTAB package automatically performs local Fourier stability analyses on sets of finite difference equations by deriving amplification matrices.
{"title":"Automatic generation of finite difference equations and fourier stability analyses","authors":"M. Wirth","doi":"10.1145/800206.806373","DOIUrl":"https://doi.org/10.1145/800206.806373","url":null,"abstract":"Recently several software tools based on the algebraic manipulation system MACSYMA have been implemented which facilitate the design, analysis and construction of finite difference programs for the numerical solution of systems of partial differential equations. Two of them are described here. The FDIFF package converts scalar, non-linear partial differential equations into linear, finite difference approximations. It includes tools for discretization of the domain of the PDE's dependent variables, linearization of non-linear terms and conversion of derivative terms into finite difference expressions. A notation and algebra for building arbitrary finite difference operators is provided. The FSTAB package automatically performs local Fourier stability analyses on sets of finite difference equations by deriving amplification matrices.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"84 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":"114501038","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 a network, built from links whose conductances are given by some discrete statistical distribution, including a finite zero-conductance probability, and where links are assumed independent, the distribution of conductances for the network as a whole is of interest in the study of percolation on lattices. This quantity is computed by different methods for a set of test networks. It is found that the computation is more efficiently done by manipulating the networks themselves in a suitable representation rather than by computing with symbolic expressions for their conductance. In particular, with ordinary computer algebra systems there were severe limitations due to expression growth in this study.
{"title":"Computation for conductance distributions of percolation lattice cells","authors":"R. Fogelholm","doi":"10.1145/800206.806376","DOIUrl":"https://doi.org/10.1145/800206.806376","url":null,"abstract":"For a network, built from links whose conductances are given by some discrete statistical distribution, including a finite zero-conductance probability, and where links are assumed independent, the distribution of conductances for the network as a whole is of interest in the study of percolation on lattices. This quantity is computed by different methods for a set of test networks. It is found that the computation is more efficiently done by manipulating the networks themselves in a suitable representation rather than by computing with symbolic expressions for their conductance. In particular, with ordinary computer algebra systems there were severe limitations due to expression growth in this study.","PeriodicalId":314618,"journal":{"name":"Symposium on Symbolic and Algebraic Manipulation","volume":"91 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":"124943230","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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