We present a deterministic algorithm to interpolate an m-sparse n-variate polynomial which uses poly(n, m, log H, log d) bit operations. Our algorithm works over the integers. Here H is a bound on the magnitude of the coefficient values of the given polynomial. The degree of given polynomial is bounded by d and m is upper bound on number of monomials. This running time is polynomial in the output size. Our algorithm only requires modular black box access to the given polynomial, as introduced in [12]. As an easy consequence, we obtain an algorithm to interpolate polynomials represented by arithmetic circuits.
我们提出了一种确定性算法来插值一个m稀疏的n变量多项式，该多项式使用多(n, m, log H, log d)位运算。我们的算法适用于整数。这里H是给定多项式的系数值的大小的一个界。给定多项式的阶以d为界，m为单项式个数的上界。这个运行时间是输出大小的多项式。我们的算法只需要对给定多项式进行模块化黑盒访问，如[12]所述。作为一个简单的结果，我们得到了一个用算术电路表示多项式的插值算法。
{"title":"A new deterministic algorithm for sparse multivariate polynomial interpolation","authors":"M. Bläser, Gorav Jindal","doi":"10.1145/2608628.2608648","DOIUrl":"https://doi.org/10.1145/2608628.2608648","url":null,"abstract":"We present a deterministic algorithm to interpolate an m-sparse n-variate polynomial which uses poly(n, m, log H, log d) bit operations. Our algorithm works over the integers. Here H is a bound on the magnitude of the coefficient values of the given polynomial. The degree of given polynomial is bounded by d and m is upper bound on number of monomials. This running time is polynomial in the output size. Our algorithm only requires modular black box access to the given polynomial, as introduced in [12]. As an easy consequence, we obtain an algorithm to interpolate polynomials represented by arithmetic circuits.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"C-21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126784429","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 usual algorithm to solve polynomial systems using Gröbner bases consists of two steps: first computing the DRL Gröbner basis using the F5 algorithm then computing the LEX Gröbner basis using a change of ordering algorithm. When the Bézout bound is reached, the bottleneck of the total solving process is the change of ordering step. For 20 years, thanks to the FGLM algorithm the complexity of change of ordering is known to be cubic in the number of solutions of the system to solve. We show that, in the generic case or up to a generic linear change of variables, the multiplicative structure of the quotient ring can be computed with no arithmetic operation. Moreover, given this multiplicative structure we propose a change of ordering algorithm for Shape Position ideals whose complexity is polynomial in the number of solutions with exponent ω where 2 ≤ ω < 2.3727 is the exponent in the complexity of multiplying two dense matrices. As a consequence, we propose a new Las Vegas algorithm for solving polynomial systems with a finite number of solutions by using Gröbner basis for which the change of ordering step has a sub-cubic (i.e. with exponent ω) complexity and whose total complexity is dominated by the complexity of the F5 algorithm. In practice we obtain significant speedups for various polynomial systems by a factor up to 1500 for specific cases and we are now able to tackle some instances that were intractable.
{"title":"Sub-cubic change of ordering for Gröbner basis: a probabilistic approach","authors":"J. Faugère, P. Gaudry, Louise Huot, G. Renault","doi":"10.1145/2608628.2608669","DOIUrl":"https://doi.org/10.1145/2608628.2608669","url":null,"abstract":"The usual algorithm to solve polynomial systems using Gröbner bases consists of two steps: first computing the DRL Gröbner basis using the F5 algorithm then computing the LEX Gröbner basis using a change of ordering algorithm. When the Bézout bound is reached, the bottleneck of the total solving process is the change of ordering step. For 20 years, thanks to the FGLM algorithm the complexity of change of ordering is known to be cubic in the number of solutions of the system to solve.\u0000 We show that, in the generic case or up to a generic linear change of variables, the multiplicative structure of the quotient ring can be computed with no arithmetic operation. Moreover, given this multiplicative structure we propose a change of ordering algorithm for Shape Position ideals whose complexity is polynomial in the number of solutions with exponent ω where 2 ≤ ω < 2.3727 is the exponent in the complexity of multiplying two dense matrices. As a consequence, we propose a new Las Vegas algorithm for solving polynomial systems with a finite number of solutions by using Gröbner basis for which the change of ordering step has a sub-cubic (i.e. with exponent ω) complexity and whose total complexity is dominated by the complexity of the F5 algorithm.\u0000 In practice we obtain significant speedups for various polynomial systems by a factor up to 1500 for specific cases and we are now able to tackle some instances that were intractable.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134521260","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 Newton homotopy is a homotopy that involves changing only the constant terms. They arise naturally, for example, when performing monodromy loops, moving end effectors of robots, and simply when trying to compute a solution to a square system of equations. Previous certified path tracking techniques have focused on using an a priori certified tracking scheme which means that the stepsize is constructed so that the result automatically satisfies some conditions. These schemes use pessimistic stepsizes that can be much smaller than those used by heuristic tracking methods. This article designs an a posteriori certification scheme that uses the result of a heuristic tracking scheme as input to produce a certificate that the path was indeed tracked correctly, e.g., no path jumpings occurred. By using an a posteriori approach, each step can be certified independently and thus certification of the path can be performed in parallel. Examples are presented demonstrating the efficiency of this a posteriori certification approach.
{"title":"An a posteriori certification algorithm for Newton homotopies","authors":"J. Hauenstein, Ian Haywood, Alan C. Liddell","doi":"10.1145/2608628.2608651","DOIUrl":"https://doi.org/10.1145/2608628.2608651","url":null,"abstract":"A Newton homotopy is a homotopy that involves changing only the constant terms. They arise naturally, for example, when performing monodromy loops, moving end effectors of robots, and simply when trying to compute a solution to a square system of equations. Previous certified path tracking techniques have focused on using an a priori certified tracking scheme which means that the stepsize is constructed so that the result automatically satisfies some conditions. These schemes use pessimistic stepsizes that can be much smaller than those used by heuristic tracking methods. This article designs an a posteriori certification scheme that uses the result of a heuristic tracking scheme as input to produce a certificate that the path was indeed tracked correctly, e.g., no path jumpings occurred. By using an a posteriori approach, each step can be certified independently and thus certification of the path can be performed in parallel. Examples are presented demonstrating the efficiency of this a posteriori certification approach.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"45 6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115527835","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 reports on a high-performance implementation of the truncated Fourier transform (TFT). A general Cooley-Tukey like algorithm for the TFT is developed that allows the implementation to automatically adapt to the memory hierarchy. Then the algorithm introduces a small relaxation for larger transform sizes which trades off slightly higher arithmetic cost for improved data flow which allows full vectorization and parallelization. The implementation is automatically derived and tuned using the SPIRAL system for code generation and adaptation. The resulting arbitrary-size TFT library smooths out the staircase performance associated with power-of-two modular FFT implementations while retaining the performance associated with state-of-the-art FFT libraries. This provides significant performance improvement over approaches that pad to the next power of two even when using high-performance FFT libraries.
{"title":"High performance implementation of the TFT","authors":"Lingchuan Meng, Jeremy R. Johnson","doi":"10.1145/2608628.2608661","DOIUrl":"https://doi.org/10.1145/2608628.2608661","url":null,"abstract":"This paper reports on a high-performance implementation of the truncated Fourier transform (TFT). A general Cooley-Tukey like algorithm for the TFT is developed that allows the implementation to automatically adapt to the memory hierarchy. Then the algorithm introduces a small relaxation for larger transform sizes which trades off slightly higher arithmetic cost for improved data flow which allows full vectorization and parallelization. The implementation is automatically derived and tuned using the SPIRAL system for code generation and adaptation. The resulting arbitrary-size TFT library smooths out the staircase performance associated with power-of-two modular FFT implementations while retaining the performance associated with state-of-the-art FFT libraries. This provides significant performance improvement over approaches that pad to the next power of two even when using high-performance FFT libraries.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"82 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128730182","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 tutorial, we will give an overview of typical algorithms of quantifier elimination over the reals and illustrate their actual applications in industry. Some recent research results on computational efficiency improvement of quantifier elimination algorithms, in particular for solving practical industrial problems, will be also mentioned. Moreover, we will briefly explain valuable techniques and tips to effectively utilize quantifier elimination in practice.
{"title":"Effective quantifier elimination for industrial applications","authors":"H. Anai","doi":"10.1145/2608628.2627494","DOIUrl":"https://doi.org/10.1145/2608628.2627494","url":null,"abstract":"In this tutorial, we will give an overview of typical algorithms of quantifier elimination over the reals and illustrate their actual applications in industry. Some recent research results on computational efficiency improvement of quantifier elimination algorithms, in particular for solving practical industrial problems, will be also mentioned. Moreover, we will briefly explain valuable techniques and tips to effectively utilize quantifier elimination in practice.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128239250","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 new algorithm is given for computing parametric local cohomology classes associated with semi-quasihomogeneous singularities. The essential point of the proposed algorithm involves Poincaré polynomials and weighted degrees. The proposed algorithm gives a suitable decomposition of the parameter space depending on the structure of the parametric local cohomology classes. As an application, an algorithm for computing parametric standard bases of zero-dimensional ideals, is given. These algorithms work for non-parametric cases, too.
{"title":"On efficient algorithms for computing parametric local cohomology classes associated with semi-quasihomogeneous singularities and standard bases","authors":"Katsusuke Nabeshima, S. Tajima","doi":"10.1145/2608628.2608639","DOIUrl":"https://doi.org/10.1145/2608628.2608639","url":null,"abstract":"A new algorithm is given for computing parametric local cohomology classes associated with semi-quasihomogeneous singularities. The essential point of the proposed algorithm involves Poincaré polynomials and weighted degrees. The proposed algorithm gives a suitable decomposition of the parameter space depending on the structure of the parametric local cohomology classes. As an application, an algorithm for computing parametric standard bases of zero-dimensional ideals, is given. These algorithms work for non-parametric cases, too.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127920033","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 previous work, we have introduced several fast algorithms for relaxed power series multiplication (also known under the name on-line multiplication) up to a given order n. The fastest currently known algorithm works over an effective base field K with sufficiently many 2p-th roots of unity and has algebraic time complexity O(n log ne2[EQUATION]). In this paper, we will generalize this algorithm to the cases when K is replaced by an effective ring of positive characteristic or by an effective ring of characteristic zero, which is also torsion-free as a Z-module and comes with an additional algorithm for partial division by integers. In particular, we may take K to be any effective field. We will also present an asymptotically faster algorithm for relaxed multiplication of p-adic numbers.
{"title":"Faster relaxed multiplication","authors":"J. Hoeven","doi":"10.1145/2608628.2608657","DOIUrl":"https://doi.org/10.1145/2608628.2608657","url":null,"abstract":"In previous work, we have introduced several fast algorithms for relaxed power series multiplication (also known under the name on-line multiplication) up to a given order n. The fastest currently known algorithm works over an effective base field K with sufficiently many 2p-th roots of unity and has algebraic time complexity O(n log ne2[EQUATION]). In this paper, we will generalize this algorithm to the cases when K is replaced by an effective ring of positive characteristic or by an effective ring of characteristic zero, which is also torsion-free as a Z-module and comes with an additional algorithm for partial division by integers. In particular, we may take K to be any effective field. We will also present an asymptotically faster algorithm for relaxed multiplication of p-adic numbers.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131244673","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 procedure for solving autonomous algebraic ordinary differential equations (AODEs) of first order. This method covers the known case of rational solutions and depends crucially on the use of radical parametrizations for algebraic curves. We can prove that certain classes of AODEs permit a radical solution, which can be determined algorithmically. However, this approach is not limited to rational and radical solutions of AODEs.
{"title":"Radical solutions of first order autonomous algebraic ordinary differential equations","authors":"Georg Grasegger","doi":"10.1145/2608628.2608636","DOIUrl":"https://doi.org/10.1145/2608628.2608636","url":null,"abstract":"We present a procedure for solving autonomous algebraic ordinary differential equations (AODEs) of first order. This method covers the known case of rational solutions and depends crucially on the use of radical parametrizations for algebraic curves. We can prove that certain classes of AODEs permit a radical solution, which can be determined algorithmically. However, this approach is not limited to rational and radical solutions of AODEs.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132546600","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 R be an Ore extension of a skew-field. A basic computational problem is to decide effectively whether two given Ore polynomials f, g ∈ R (of the same degree) are similar, that is, if there exists an isomorphism of left R--modules between R/Rf and R/Rg. Since these modules are of finite length, we consider the more general problem of deciding when two given left R--modules of finite length are isomorphic. We show that if R is free of finite rank as a module over its center C, then this problem can be reduced to check the existence of an isomorphism of C--modules. This method works for a large class of left R--modules of finite length. Our result is proven in the realm of non-commutative principal ideal domains, and generalizes a result by Jacobson for some Ore extensions of a skew field by an automorphism. As a consequence, we propose an algorithm to check whether two given left R--modules of finite length are isomorphic by associating a matrix with coefficients in C to each of the modules, and checking if the corresponding rational canonical forms are equal. Our method is illustrated with examples of computations for Ore extensions of finite fields, and of the Hamilton quaternions.
{"title":"On isomorphisms of modules over non-commutative PID","authors":"J. Gómez-Torrecillas, F. J. Lobillo, G. Navarro","doi":"10.1145/2608628.2608665","DOIUrl":"https://doi.org/10.1145/2608628.2608665","url":null,"abstract":"Let R be an Ore extension of a skew-field. A basic computational problem is to decide effectively whether two given Ore polynomials f, g ∈ R (of the same degree) are similar, that is, if there exists an isomorphism of left R--modules between R/Rf and R/Rg. Since these modules are of finite length, we consider the more general problem of deciding when two given left R--modules of finite length are isomorphic. We show that if R is free of finite rank as a module over its center C, then this problem can be reduced to check the existence of an isomorphism of C--modules. This method works for a large class of left R--modules of finite length. Our result is proven in the realm of non-commutative principal ideal domains, and generalizes a result by Jacobson for some Ore extensions of a skew field by an automorphism. As a consequence, we propose an algorithm to check whether two given left R--modules of finite length are isomorphic by associating a matrix with coefficients in C to each of the modules, and checking if the corresponding rational canonical forms are equal. Our method is illustrated with examples of computations for Ore extensions of finite fields, and of the Hamilton quaternions.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116776716","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 provide bounds on the size of operators obtained by algorithms for executing D-finite closure properties. For operators of small order, we give bounds on the degree and on the height (bit-size). For higher order operators, we give degree bounds that are parameterized with respect to the order and reflect the phenomenon that higher order operators may have lower degrees (order-degree curves).
{"title":"Bounds for D-finite closure properties","authors":"Manuel Kauers","doi":"10.1145/2608628.2608634","DOIUrl":"https://doi.org/10.1145/2608628.2608634","url":null,"abstract":"We provide bounds on the size of operators obtained by algorithms for executing D-finite closure properties. For operators of small order, we give bounds on the degree and on the height (bit-size). For higher order operators, we give degree bounds that are parameterized with respect to the order and reflect the phenomenon that higher order operators may have lower degrees (order-degree curves).","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124033026","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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