We investigate polynomial solutions of homogeneous linear differential equations with coefficients that are polynomials with integer coefficients. The problems we consider are the existence of nonzero polynomial solutions, the determination of the dimension of the vector space of polynomial solutions, the computation of a basis of this space. Previous algorithms have a bit complexity that is at least quadratic in the largest integer valuation N of formal Laurent series solutions at infinity, even for merely detecting the existence of nonzero polynomial solutions. We give a deterministic algorithm that computes a compact representation of a basis of polynomial solutions in O(Nlog3N) bit operations. We also give a probabilistic algorithm that computes the dimension of the space of polynomial solutions in O(√Nlog2N) bit operations. In general, the integer N is not polynomially bounded in the bit size of the input differential equation. We isolate a class of equations for which detecting nonzero polynomial solutions can be performed in polynomial complexity. We discuss implementation issues and possible extensions.
{"title":"Fast algorithms for polynomial solutions of linear differential equations","authors":"A. Bostan, T. Cluzeau, B. Salvy","doi":"10.1145/1073884.1073893","DOIUrl":"https://doi.org/10.1145/1073884.1073893","url":null,"abstract":"We investigate polynomial solutions of homogeneous linear differential equations with coefficients that are polynomials with integer coefficients. The problems we consider are the existence of nonzero polynomial solutions, the determination of the dimension of the vector space of polynomial solutions, the computation of a basis of this space. Previous algorithms have a bit complexity that is at least quadratic in the largest integer valuation N of formal Laurent series solutions at infinity, even for merely detecting the existence of nonzero polynomial solutions. We give a deterministic algorithm that computes a compact representation of a basis of polynomial solutions in O(Nlog3N) bit operations. We also give a probabilistic algorithm that computes the dimension of the space of polynomial solutions in O(√Nlog2N) bit operations. In general, the integer N is not polynomially bounded in the bit size of the input differential equation. We isolate a class of equations for which detecting nonzero polynomial solutions can be performed in polynomial complexity. We discuss implementation issues and possible extensions.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"48 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":"115163051","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 define a class of special function inequalities that contains many classical examples, such as the Cauchy-Schwarz inequality, and introduce a proving procedure based on induction and Cylindrical Algebraic Decomposition. We present an array of non-trivial examples that can be done by our method. Most of them have not been proven automatically before. Some difficult well-known inequalities such as the Askey-Gasper inequality and Vietoris's inequality lie in our class as well, but we do not know if our proving procedure terminates for them.
{"title":"A procedure for proving special function inequalities involving a discrete parameter","authors":"S. Gerhold, Manuel Kauers","doi":"10.1145/1073884.1073907","DOIUrl":"https://doi.org/10.1145/1073884.1073907","url":null,"abstract":"We define a class of special function inequalities that contains many classical examples, such as the Cauchy-Schwarz inequality, and introduce a proving procedure based on induction and Cylindrical Algebraic Decomposition. We present an array of non-trivial examples that can be done by our method. Most of them have not been proven automatically before. Some difficult well-known inequalities such as the Askey-Gasper inequality and Vietoris's inequality lie in our class as well, but we do not know if our proving procedure terminates for them.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"43 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":"114666563","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}
X. Dahan, M. M. Maza, É. Schost, Wenyuan Wu, Yuzhen Xie
We present lifting techniques for triangular decompositions of zero-dimensional varieties, that extend the range of the previous methods. We discuss complexity aspects, and report on a preliminary implementation. Our theoretical results are comforted by these experiments.
{"title":"Lifting techniques for triangular decompositions","authors":"X. Dahan, M. M. Maza, É. Schost, Wenyuan Wu, Yuzhen Xie","doi":"10.1145/1073884.1073901","DOIUrl":"https://doi.org/10.1145/1073884.1073901","url":null,"abstract":"We present lifting techniques for triangular decompositions of zero-dimensional varieties, that extend the range of the previous methods. We discuss complexity aspects, and report on a preliminary implementation. Our theoretical results are comforted by these experiments.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"2 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":"122154642","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}
Two ideas are combined to construct a hybrid symbolic-numeric differential-elimination method for identifying and including missing constraints arising in differential systems. First we exploit the fact that a system once differentiated becomes linear in its highest derivatives. Then we apply diagonal homotopies to incrementally process new constraints, one at a time. The method is illustrated on several examples, combining symbolic differential elimination (using rifsimp) with numerical homotopy continuation (using phc).
{"title":"Symbolic-numeric completion of differential systems by homotopy continuation","authors":"G. Reid, J. Verschelde, A. Wittkopf, Wenyuan Wu","doi":"10.1145/1073884.1073922","DOIUrl":"https://doi.org/10.1145/1073884.1073922","url":null,"abstract":"Two ideas are combined to construct a hybrid symbolic-numeric differential-elimination method for identifying and including missing constraints arising in differential systems. First we exploit the fact that a system once differentiated becomes linear in its highest derivatives. Then we apply diagonal homotopies to incrementally process new constraints, one at a time. The method is illustrated on several examples, combining symbolic differential elimination (using rifsimp) with numerical homotopy continuation (using phc).","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"21 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":"123503704","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}
Sufficient conditions are given for validity of the discrete Newton-Leibniz formula when the indefinite sum is obtained either by Gosper's algorithm or by Accurate Summation algorithm. It is shown that sometimes a polynomial can be factored from the summand in such a way that the safe summation range is increased.
{"title":"Gosper's algorithm, accurate summation, and the discrete Newton-Leibniz formula","authors":"S. Abramov, M. Petkovssek","doi":"10.1145/1073884.1073888","DOIUrl":"https://doi.org/10.1145/1073884.1073888","url":null,"abstract":"Sufficient conditions are given for validity of the discrete Newton-Leibniz formula when the indefinite sum is obtained either by Gosper's algorithm or by Accurate Summation algorithm. It is shown that sometimes a polynomial can be factored from the summand in such a way that the safe summation range is increased.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"154 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":"124620254","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 the computation of selfintersections as a major problem in Computer Aided Geometric Design (CAD) and Geometric Modeling, and particularly for patches of parametrized bicubic surfaces. Then we expose two complementary contributions on that subject with Computer Algebra tools: First, a specific sparse bivariate resultant adapted to the corresponding elimination problem, second a semi-numeric polynomial solver able to deal with large system of equations with floating point coefficients. Examples and timings are provided.
{"title":"Selfintersections of a bézier bicubic surface","authors":"A. Galligo, J. Pavone","doi":"10.1145/1073884.1073906","DOIUrl":"https://doi.org/10.1145/1073884.1073906","url":null,"abstract":"We present the computation of selfintersections as a major problem in Computer Aided Geometric Design (CAD) and Geometric Modeling, and particularly for patches of parametrized bicubic surfaces. Then we expose two complementary contributions on that subject with Computer Algebra tools: First, a specific sparse bivariate resultant adapted to the corresponding elimination problem, second a semi-numeric polynomial solver able to deal with large system of equations with floating point coefficients. Examples and timings are provided.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"570 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":"123403505","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}
James C. Beaumont, R. Bradford, J. Davenport, Nalina Phisanbut
Given an elementary function with algebraic branch cuts, we show how to decide which sheet of the associated Riemann surface we are on at any given point. We do this by establishing a correspondence between the Cylindrical Algebraic Decomposition (CAD) of the complex plane defined by the branch cuts and a finite subset of sheets of the Riemann surface. The key advantage is that we no longer have to deal with the difficult 'constant problem'.
{"title":"Adherence is better than adjacency: computing the Riemann index using CAD","authors":"James C. Beaumont, R. Bradford, J. Davenport, Nalina Phisanbut","doi":"10.1145/1073884.1073892","DOIUrl":"https://doi.org/10.1145/1073884.1073892","url":null,"abstract":"Given an elementary function with algebraic branch cuts, we show how to decide which sheet of the associated Riemann surface we are on at any given point. We do this by establishing a correspondence between the Cylindrical Algebraic Decomposition (CAD) of the complex plane defined by the branch cuts and a finite subset of sheets of the Riemann surface. The key advantage is that we no longer have to deal with the difficult 'constant problem'.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"108 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":"115659387","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 a new and general approach for analyzing the stability of a large class of biological networks, modeled as autonomous systems of differential equations, using real solving and solution classification. The proposed approach, based on the classical technique of linearization from the qualitative theory of ordinary differential equations yet with exact symbolic computation, is applied to analyzing the local stability of the Cdc2-cyclin B/Wee1 system and the Mos/MEK/p42 MAPK cascade, two well-known models for cell and protein signaling that have been studied extensively in the literature. We provide rigorous proofs and generalizations for some of the previous results established experimentally and report our new findings.
{"title":"Stability analysis of biological systems with real solution classification","authors":"Dongming Wang, Bican Xia","doi":"10.1145/1073884.1073933","DOIUrl":"https://doi.org/10.1145/1073884.1073933","url":null,"abstract":"This paper presents a new and general approach for analyzing the stability of a large class of biological networks, modeled as autonomous systems of differential equations, using real solving and solution classification. The proposed approach, based on the classical technique of linearization from the qualitative theory of ordinary differential equations yet with exact symbolic computation, is applied to analyzing the local stability of the Cdc2-cyclin B/Wee1 system and the Mos/MEK/p42 MAPK cascade, two well-known models for cell and protein signaling that have been studied extensively in the literature. We provide rigorous proofs and generalizations for some of the previous results established experimentally and report our new findings.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"104 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":"128038750","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 method is developed for constructing a series of exact analytical solutions of the nonlinear Schrödinger equation model (NLSE) with varying dispersion, nonlinearity, and gain or absorption. With the help of symbolic computation, a broad class of analytical solutions of NLSE are obtained. From our results, many previous known results of NLSE obtained by some authors can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. Further, the formation, interaction and stability of solitons have been investigated.
{"title":"Exact analytical solutions to the nonlinear Schrödinger equation model","authors":"Biao Li, Yong Chen, Qi Wang","doi":"10.1145/1073884.1073916","DOIUrl":"https://doi.org/10.1145/1073884.1073916","url":null,"abstract":"A method is developed for constructing a series of exact analytical solutions of the nonlinear Schrödinger equation model (NLSE) with varying dispersion, nonlinearity, and gain or absorption. With the help of symbolic computation, a broad class of analytical solutions of NLSE are obtained. From our results, many previous known results of NLSE obtained by some authors can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. Further, the formation, interaction and stability of solitons have been investigated.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"35 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":"132588210","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 μ-bases of rational curves/surfaces are newly developed tools which play an important role in connecting parametric forms and implicit forms of the rational curves/surfaces. They provide efficient algorithms to implicitize rational curves/surfaces as well as algorithms to compute singular points of rational curves and to reparametrize rational ruled surfaces. In this paper, we present an efficient algorithm to compute the μbasis of a rational curve/surface by using polynomial matrix factorization followed by a technique similar to Gaussian elimination. The algorithm is shown superior than previous algorithms to compute the μ-basis of a rational curve, and it is the only known algorithm that can rigorously compute the μ-basis of a general rational surface. We present some examples to illustrate the algorithm.
{"title":"Computing μ-bases of rational curves and surfaces using polynomial matrix factorization","authors":"J. Deng, Falai Chen, L. Shen","doi":"10.1145/1073884.1073904","DOIUrl":"https://doi.org/10.1145/1073884.1073904","url":null,"abstract":"The μ-bases of rational curves/surfaces are newly developed tools which play an important role in connecting parametric forms and implicit forms of the rational curves/surfaces. They provide efficient algorithms to implicitize rational curves/surfaces as well as algorithms to compute singular points of rational curves and to reparametrize rational ruled surfaces. In this paper, we present an efficient algorithm to compute the μbasis of a rational curve/surface by using polynomial matrix factorization followed by a technique similar to Gaussian elimination. The algorithm is shown superior than previous algorithms to compute the μ-basis of a rational curve, and it is the only known algorithm that can rigorously compute the μ-basis of a general rational surface. We present some examples to illustrate the algorithm.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"60 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":"115737262","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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