Picard-Vessiot extensions for ordinary differential and difference equations are well known and are at the core of the associated Galois theories. In this paper, we construct fundamental matrices and Picard-Vessiot extensions for systems of linear partial functional equations having finite linear dimension. We then use those extensions to show that all the solutions of a factor of such a system can be completed to solutions of the original system.
{"title":"Picard--Vessiot extensions for linear functional systems","authors":"M. Bronstein, Ziming Li, Min Wu","doi":"10.1145/1073884.1073896","DOIUrl":"https://doi.org/10.1145/1073884.1073896","url":null,"abstract":"Picard-Vessiot extensions for ordinary differential and difference equations are well known and are at the core of the associated Galois theories. In this paper, we construct fundamental matrices and Picard-Vessiot extensions for systems of linear partial functional equations having finite linear dimension. We then use those extensions to show that all the solutions of a factor of such a system can be completed to solutions of the original system.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129029064","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 algorithms that outperform straightforward implementations of classical Taylor shift by 1. For input poly-nomials of low degrees a method of the SACLIB library is faster than straightforward implementations by a factor of at least 2; for higher degrees we develop a method that is faster than straightforward implementations by a factor of up to 7. Our Taylor shift algorithm requires more word additions than straightforward methods but it reduces the number of cycles per word addition by reducing memory traffic and the number of carry computations. The introduction of signed digits, suspended normalization, radix reduction, and delayed carry propagation enables our algorithm to take advantage of the technique of register tiling which is commonly used by optimizing compilers. While our algorithm is written in a high-level language, it depends on several parameters that can be tuned to the underlying architecture.
{"title":"Architecture-aware classical Taylor shift by 1","authors":"Jeremy R. Johnson, W. Krandick, A. Ruslanov","doi":"10.1145/1073884.1073913","DOIUrl":"https://doi.org/10.1145/1073884.1073913","url":null,"abstract":"We present algorithms that outperform straightforward implementations of classical Taylor shift by 1. For input poly-nomials of low degrees a method of the SACLIB library is faster than straightforward implementations by a factor of at least 2; for higher degrees we develop a method that is faster than straightforward implementations by a factor of up to 7. Our Taylor shift algorithm requires more word additions than straightforward methods but it reduces the number of cycles per word addition by reducing memory traffic and the number of carry computations. The introduction of signed digits, suspended normalization, radix reduction, and delayed carry propagation enables our algorithm to take advantage of the technique of register tiling which is commonly used by optimizing compilers. While our algorithm is written in a high-level language, it depends on several parameters that can be tuned to the underlying architecture.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"55 14","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132940176","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 prove that any admissible ordering on ordinary differential monomials in one differential indeterminate can be specified by a canonical set of matrices. The relations between some classes of these orderings are studied. We give criteria of finiteness of differential standard bases and propose an algorithm that computes such bases if they are finite.
{"title":"Admissible orderings and finiteness criteria for differential standard bases","authors":"A. Zobnin","doi":"10.1145/1073884.1073935","DOIUrl":"https://doi.org/10.1145/1073884.1073935","url":null,"abstract":"We prove that any admissible ordering on ordinary differential monomials in one differential indeterminate can be specified by a canonical set of matrices. The relations between some classes of these orderings are studied. We give criteria of finiteness of differential standard bases and propose an algorithm that computes such bases if they are finite.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129810110","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 present a method to compute or estimate the sum of roots with positive real parts (SORPRP) of a polynomial, which is related to a certain index of "average" stability in optimal control, without computing explicit numerical values of the roots. The method is based on symbolic and algebraic computations and enables us to deal with polynomials with parametric coefficients for their SORPRP. This leads to provide a novel systematic method to achieve optimal regulator design in control by combining the method with quantifier elimination. We also report some experimental result for a typical class of plants and confirm the effectiveness of the proposed method.
{"title":"Sum of roots with positive real parts","authors":"H. Anai, S. Hara, K. Yokoyama","doi":"10.1145/1073884.1073890","DOIUrl":"https://doi.org/10.1145/1073884.1073890","url":null,"abstract":"In this paper we present a method to compute or estimate the sum of roots with positive real parts (SORPRP) of a polynomial, which is related to a certain index of \"average\" stability in optimal control, without computing explicit numerical values of the roots. The method is based on symbolic and algebraic computations and enables us to deal with polynomials with parametric coefficients for their SORPRP. This leads to provide a novel systematic method to achieve optimal regulator design in control by combining the method with quantifier elimination. We also report some experimental result for a typical class of plants and confirm the effectiveness of the proposed method.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123518418","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}
ISSAC 2005 is a continuation of a well-established series of international conferences for the presentation of the latest advances in the field of Symbolic and Algebraic Computation. The first meeting of the series (1966) was held in Washington, DC, and sponsored by the Association for Computing Machinery (ACM). Since then, the abbreviated name of the meeting has evolved from SYMSAM, SYMSAC, EUROSAM, EUROCAL to finally settle on the present name ISSAC. This 30th meeting was hosted by the Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing, China from July 24 to July 27. The topics of the conference include, but are not limited to: Algorithmic mathematics. Algebraic, symbolic and symbolic-numeric algorithms. Simplification, function manipulation, equations, summation, integration, ODE/PDE, linear algebra, number theory, group and geometric computing. Computer Science. Theoretical and practical problems in symbolic computation. Systems, problem solving environments, user interfaces, software, libraries, parallel/distributed computing and programming languages for symbolic computation, concrete analysis, benchmarking, theoretical and practical complexity of computer algebra algorithms, automatic differentiation, code generation, mathematical data structures and exchange protocols. Applications. Problem treatments using algebraic, symbolic or symbolic-numeric computation in an essential or a novel way. Engineering, economics and finance, physical and biological sciences, computer science, logic, mathematics, statistics, education. Following tradition, ISSAC 2005 featured invited talks, contributed papers, tutorials, poster sessions, software exhibitions, and satellite workshops. This volume contains all the contributed papers which were presented at the meeting as well as the abstracts of the invited talks. The picture on the front cover shows a page from the classic Chinese math book bearing the title "Jade Mirrors of Four Elements" by Zhu Shijie, written in 1303 AD during the Yuan Dynasty. In this page, a system of equations of three unknowns and degree three is reduced to a univariate equation by eliminating variables.
{"title":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","authors":"X. Gao, G. Labahn","doi":"10.1145/1073884","DOIUrl":"https://doi.org/10.1145/1073884","url":null,"abstract":"ISSAC 2005 is a continuation of a well-established series of international conferences for the presentation of the latest advances in the field of Symbolic and Algebraic Computation. The first meeting of the series (1966) was held in Washington, DC, and sponsored by the Association for Computing Machinery (ACM). Since then, the abbreviated name of the meeting has evolved from SYMSAM, SYMSAC, EUROSAM, EUROCAL to finally settle on the present name ISSAC. This 30th meeting was hosted by the Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing, China from July 24 to July 27. \u0000 \u0000The topics of the conference include, but are not limited to: Algorithmic mathematics. Algebraic, symbolic and symbolic-numeric algorithms. Simplification, function manipulation, equations, summation, integration, ODE/PDE, linear algebra, number theory, group and geometric computing. Computer Science. Theoretical and practical problems in symbolic computation. Systems, problem solving environments, user interfaces, software, libraries, parallel/distributed computing and programming languages for symbolic computation, concrete analysis, benchmarking, theoretical and practical complexity of computer algebra algorithms, automatic differentiation, code generation, mathematical data structures and exchange protocols. Applications. Problem treatments using algebraic, symbolic or symbolic-numeric computation in an essential or a novel way. Engineering, economics and finance, physical and biological sciences, computer science, logic, mathematics, statistics, education. \u0000 \u0000Following tradition, ISSAC 2005 featured invited talks, contributed papers, tutorials, poster sessions, software exhibitions, and satellite workshops. This volume contains all the contributed papers which were presented at the meeting as well as the abstracts of the invited talks. \u0000 \u0000The picture on the front cover shows a page from the classic Chinese math book bearing the title \"Jade Mirrors of Four Elements\" by Zhu Shijie, written in 1303 AD during the Yuan Dynasty. In this page, a system of equations of three unknowns and degree three is reduced to a univariate equation by eliminating variables.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122555672","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}
Over the past few decades several variations on a "half GCD" algorithm for obtaining the pair of terms in the middle of a Euclidean sequence have been proposed. In the integer case algorithm design and proof of correctness are complicated by the effect of carries. This paper will demonstrate a variant with a relatively simple proof of correctness. We then apply this to the task of rational recovery for a linear algebra solver.
{"title":"Half-GCD and fast rational recovery","authors":"Daniel Lichtblau","doi":"10.1145/1073884.1073917","DOIUrl":"https://doi.org/10.1145/1073884.1073917","url":null,"abstract":"Over the past few decades several variations on a \"half GCD\" algorithm for obtaining the pair of terms in the middle of a Euclidean sequence have been proposed. In the integer case algorithm design and proof of correctness are complicated by the effect of carries. This paper will demonstrate a variant with a relatively simple proof of correctness. We then apply this to the task of rational recovery for a linear algebra solver.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131827530","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}
Let A and B be two polynomials in ℤ [x,y] and let R = resx(A,B) denote the resultant of A and B taken wrt x. In this paper we modify Collins' modular algorithm for computing R to make it output sensitive. The advantage of our algorithm is that it will be faster when the bounds needed by Collins' algorithm for the coefficients of R and for the degree of R are inaccurate. Our second contribution is an output sensitive modular algorithm for computing the monic resultant in ℚ[y]. The advantage of this algorithm is that it is faster still when the resultant has a large integer content. Both of our algorithms are necessarily probabilistic.The paper includes a number of resultant problems that motivate the need to consider such algorithms. We have implemented our algorithms in Maple. We have also implemented Collins' algorithm and the subresultant algorithm in Maple for comparison. The timings we obtain demonstrate that a good speedup is obtained.
{"title":"Probabilistic algorithms for computing resultants","authors":"M. Monagan","doi":"10.1145/1073884.1073919","DOIUrl":"https://doi.org/10.1145/1073884.1073919","url":null,"abstract":"Let A and B be two polynomials in ℤ [x,y] and let R = resx(A,B) denote the resultant of A and B taken wrt x. In this paper we modify Collins' modular algorithm for computing R to make it output sensitive. The advantage of our algorithm is that it will be faster when the bounds needed by Collins' algorithm for the coefficients of R and for the degree of R are inaccurate. Our second contribution is an output sensitive modular algorithm for computing the monic resultant in ℚ[y]. The advantage of this algorithm is that it is faster still when the resultant has a large integer content. Both of our algorithms are necessarily probabilistic.The paper includes a number of resultant problems that motivate the need to consider such algorithms. We have implemented our algorithms in Maple. We have also implemented Collins' algorithm and the subresultant algorithm in Maple for comparison. The timings we obtain demonstrate that a good speedup is obtained.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132947037","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 consider algebraic numbers defined by univariate polynomials over the rationals. In the syntax of Maple, such numbers are expressed using the RootOf function. This paper defines a canonical form for RootOf with respect to affine transformations. The affine shifts of monic irreducible polynomials form a group, and the orbits of the polynomials can be used to define a canonical form. The canonical form of the polynomials then defines a canonical form for the corresponding algebraic numbers. Reducing any RootOf to its canonical form has the advantage that affine relations between algebraic numbers are readily identified. More generally, the reduction minimizes the number of algebraic numbers appearing in a computation, and also allows the Maple indexed RootOf to be used more easily.
{"title":"Affine transformations of algebraic numbers","authors":"D. J. Jeffrey, Pratibha, K. Roach","doi":"10.1145/1073884.1073912","DOIUrl":"https://doi.org/10.1145/1073884.1073912","url":null,"abstract":"We consider algebraic numbers defined by univariate polynomials over the rationals. In the syntax of Maple, such numbers are expressed using the RootOf function. This paper defines a canonical form for RootOf with respect to affine transformations. The affine shifts of monic irreducible polynomials form a group, and the orbits of the polynomials can be used to define a canonical form. The canonical form of the polynomials then defines a canonical form for the corresponding algebraic numbers. Reducing any RootOf to its canonical form has the advantage that affine relations between algebraic numbers are readily identified. More generally, the reduction minimizes the number of algebraic numbers appearing in a computation, and also allows the Maple indexed RootOf to be used more easily.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126439791","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 a new hybrid symbolic-numeric method for the fast and accurate evaluation of definite integrals in multiple dimensions. This method is well-suited for two classes of problems: (1) analytic integrands over general regions in two dimensions, and (2) families of analytic integrands with special algebraic structure over hyperrectangular regions in higher dimensions.The algebraic theory of multivariate interpolation via natural tensor product series was developed in the doctoral thesis by Chapman, who named this broad new scheme of bilinear series expansions "Geddes series" in honour of his thesis supervisor. This paper describes an efficient adaptive algorithm for generating bilinear series of Geddes-Newton type and explores applications of this algorithm to multiple integration. We will present test results demonstrating that our new adaptive integration algorithm is effective both in high dimensions and with high accuracy. For example, our Maple implementation of the algorithm has successfully computed nontrivial integrals with hundreds of dimensions to 10-digit accuracy, each in under 3 minutes on a desktop computer.Current numerical multiple integration methods either become very slow or yield only low accuracy in high dimensions, due to the necessity to sample the integrand at a very large number of points. Our approach overcomes this difficulty by using a Geddes-Newton series with a modest number of terms to construct an accurate tensor-product approximation of the integrand. The partial separation of variables achieved in this way reduces the original integral to a manageable bilinear combination of integrals of essentially half the original dimension. We continue halving the dimensions recursively until obtaining one-dimensional integrals, which are then computed by standard numeric or symbolic techniques.
{"title":"Hybrid symbolic-numeric integration in multiple dimensions via tensor-product series","authors":"Orlando A. Carvajal, F. Chapman, K. Geddes","doi":"10.1145/1073884.1073898","DOIUrl":"https://doi.org/10.1145/1073884.1073898","url":null,"abstract":"We present a new hybrid symbolic-numeric method for the fast and accurate evaluation of definite integrals in multiple dimensions. This method is well-suited for two classes of problems: (1) analytic integrands over general regions in two dimensions, and (2) families of analytic integrands with special algebraic structure over hyperrectangular regions in higher dimensions.The algebraic theory of multivariate interpolation via natural tensor product series was developed in the doctoral thesis by Chapman, who named this broad new scheme of bilinear series expansions \"Geddes series\" in honour of his thesis supervisor. This paper describes an efficient adaptive algorithm for generating bilinear series of Geddes-Newton type and explores applications of this algorithm to multiple integration. We will present test results demonstrating that our new adaptive integration algorithm is effective both in high dimensions and with high accuracy. For example, our Maple implementation of the algorithm has successfully computed nontrivial integrals with hundreds of dimensions to 10-digit accuracy, each in under 3 minutes on a desktop computer.Current numerical multiple integration methods either become very slow or yield only low accuracy in high dimensions, due to the necessity to sample the integrand at a very large number of points. Our approach overcomes this difficulty by using a Geddes-Newton series with a modest number of terms to construct an accurate tensor-product approximation of the integrand. The partial separation of variables achieved in this way reduces the original integral to a manageable bilinear combination of integrals of essentially half the original dimension. We continue halving the dimensions recursively until obtaining one-dimensional integrals, which are then computed by standard numeric or symbolic techniques.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127789567","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 the design and implementation of two components in the LinBox library. The first is an implementation of black box matrix multiplication as a lazy matrix-times-matrix product. The implementation uses template meta-programming to set the intermediate vector type used during application of the matrix product. We also describe an interface mechanism that allows incorporation of external components with native memory management such as garbage collection into LinBox. An implementation of the interface based on SACLIB's field arithmetic procedures is presented.
{"title":"Generic matrix multiplication and memory management in linBox","authors":"E. Kaltofen, D. Morozov, George Yuhasz","doi":"10.1145/1073884.1073915","DOIUrl":"https://doi.org/10.1145/1073884.1073915","url":null,"abstract":"We describe the design and implementation of two components in the LinBox library. The first is an implementation of black box matrix multiplication as a lazy matrix-times-matrix product. The implementation uses template meta-programming to set the intermediate vector type used during application of the matrix product. We also describe an interface mechanism that allows incorporation of external components with native memory management such as garbage collection into LinBox. An implementation of the interface based on SACLIB's field arithmetic procedures is presented.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126334159","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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