We consider a general linear dynamical system and want to control its behavior. The goal is to reach a given target by minimizing a cost function. We provide a new generic algorithm with together exact, symbolic and numerical modules. In particular new efficient methods computing a block Kalman canonical exact decomposition and the optimal solutions are presented. We also propose a new numerical algorithm under-approximating the controllable domain in view of its analytical resolution in the context of singular sub-arcs.
{"title":"Algorithms for symbolic/numeric control of affine dynamical systems","authors":"A. Rondepierre, J. Dumas","doi":"10.1145/1073884.1073923","DOIUrl":"https://doi.org/10.1145/1073884.1073923","url":null,"abstract":"We consider a general linear dynamical system and want to control its behavior. The goal is to reach a given target by minimizing a cost function. We provide a new generic algorithm with together exact, symbolic and numerical modules. In particular new efficient methods computing a block Kalman canonical exact decomposition and the optimal solutions are presented. We also propose a new numerical algorithm under-approximating the controllable domain in view of its analytical resolution in the context of singular sub-arcs.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"98 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":"134432519","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}
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}
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}
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}
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}
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}
Algorithms for solving linear systems of equations over the integers are designed and implemented. The implementations are based on the highly optimized and portable ATLAS/BLAS library for numerical linear algebra and the GNU Multiple Precision library (GMP) for large integer arithmetic.
{"title":"A BLAS based C library for exact linear algebra on integer matrices","authors":"Zhuliang Chen, A. Storjohann","doi":"10.1145/1073884.1073899","DOIUrl":"https://doi.org/10.1145/1073884.1073899","url":null,"abstract":"Algorithms for solving linear systems of equations over the integers are designed and implemented. The implementations are based on the highly optimized and portable ATLAS/BLAS library for numerical linear algebra and the GNU Multiple Precision library (GMP) for large integer arithmetic.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"56 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":"122554471","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 G = (4y2+2z)x2 + (10y2+6z) be the greatest common divisor (Gcd) of two polynomials A, B ∈ ℤ[x,y,z]. Because G is not monic in the main variable x, the sparse modular Gcd algorithm of Richard Zippel cannot be applied directly as one is unable to scale univariate images of G in x consistently. We call this the normalization problem.We present two new sparse modular Gcd algorithms which solve this problem without requiring any factorizations. The first, a modification of Zippel's algorithm, treats the scaling factors as unknowns to be solved for. This leads to a structured coupled linear system for which an efficient solution is still possible. The second algorithm reconstructs the monic Gcd x2 + (5y2+3z)/(2y2+z) from monic univariate images using a sparse, variable at a time, rational function interpolation algorithm.
{"title":"Algorithms for the non-monic case of the sparse modular GCD algorithm","authors":"Jennifer de Kleine, M. Monagan, A. Wittkopf","doi":"10.1145/1073884.1073903","DOIUrl":"https://doi.org/10.1145/1073884.1073903","url":null,"abstract":"Let G = (4y2+2z)x2 + (10y2+6z) be the greatest common divisor (Gcd) of two polynomials A, B ∈ ℤ[x,y,z]. Because G is not monic in the main variable x, the sparse modular Gcd algorithm of Richard Zippel cannot be applied directly as one is unable to scale univariate images of G in x consistently. We call this the normalization problem.We present two new sparse modular Gcd algorithms which solve this problem without requiring any factorizations. The first, a modification of Zippel's algorithm, treats the scaling factors as unknowns to be solved for. This leads to a structured coupled linear system for which an efficient solution is still possible. The second algorithm reconstructs the monic Gcd x2 + (5y2+3z)/(2y2+z) from monic univariate images using a sparse, variable at a time, rational function interpolation algorithm.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"97 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":"125559087","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：