Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation最新文献_第5页

GBLA: Gröbner Basis Linear Algebra Package GBLA: Gröbner基础线性代数包

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-02-19 DOI: 10.1145/2930889.2930914

Brice Boyer, C. Eder, J. Faugère, Sylvain Lachartre, Fayssal Martani

引用次数: 11

Computing with Quasiseparable Matrices 拟可分矩阵的计算

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-02-03 DOI: 10.1145/2930889.2930915

Clément Pernet

{"title":"Computing with Quasiseparable Matrices","authors":"Clément Pernet","doi":"10.1145/2930889.2930915","DOIUrl":"https://doi.org/10.1145/2930889.2930915","url":null,"abstract":"The class of quasiseparable matrices is defined by a pair of bounds, called the quasiseparable orders, on the ranks of the sub-matrices entirely located in their strictly lower and upper triangular parts. These arise naturally in applications, as e.g. the inverse of band matrices, and are widely used for they admit structured representations allowing to compute with them in time linear in the dimension. We show, in this paper, the connection between the notion of quasiseparability and the rank profile matrix invariant, presented in [Dumas & al. ISSAC'15]. This allows us to propose an algorithm computing the quasiseparable orders (rL,rU) in time O{n2sω-2} where s=max(rL,rU) and ω the exponent of matrix multiplication. We then present two new structured representations, a binary tree of PLUQ decompositions, and the Bruhat, using respectively O{ns log n/s and O{ns} field elements instead of O{ns2} for the classical generator and O{ns log n} for the hierarchically semiseparable representations. We present algorithms computing these representations in time O{n2sω-2}. These representations allow a matrix-vector product in time linear in the size of their representation. Lastly we show how to multiply two such structured matrices in time O{n2sω-2}.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125199312","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}

引用次数: 6

Algorithms for Simultaneous Padé Approximations 同时逼近的算法

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-02-02 DOI: 10.1145/2930889.2930933

J. R. N. Nielsen, A. Storjohann

引用次数: 13

Linear Time Interactive Certificates for the Minimal Polynomial and the Determinant of a Sparse Matrix 最小多项式和稀疏矩阵行列式的线性时间交互证书

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-02-02 DOI: 10.1145/2930889.2930908

J. Dumas, E. Kaltofen, Emmanuel Thomé, G. Villard

引用次数: 15

Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts 任意移位波波夫形式最小插值基的快速计算

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-02-01 DOI: 10.1145/2930889.2930928

C. Jeannerod, Vincent Neiger, É. Schost, G. Villard

{"title":"Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts","authors":"C. Jeannerod, Vincent Neiger, É. Schost, G. Villard","doi":"10.1145/2930889.2930928","DOIUrl":"https://doi.org/10.1145/2930889.2930928","url":null,"abstract":"We compute minimal bases of solutions for a general interpolation problem, which encompasses Hermite-Pade approximation and constrained multivariate interpolation, and has applications in coding theory and security. This problem asks to find univariate polynomial relations between m vectors of size σ; these relations should have small degree with respect to an input degree shift. For an arbitrary shift, we propose an algorithm for the computation of an interpolation basis in shifted Popov normal form with a cost of O~(mω-1 σ) field operations, where ω is the exponent of matrix multiplication and the notation O~(·) indicates that logarithmic terms are omitted. Earlier works, in the case of Hermite-Pade approximation and in the general interpolation case, compute non-normalized bases. Since for arbitrary shifts such bases may have size Θ(m2 σ), the cost bound O~(mω-1 σ) was feasible only with restrictive assumptions on the shift that ensure small output sizes. The question of handling arbitrary shifts with the same complexity bound was left open. To obtain the target cost for any shift, we strengthen the properties of the output bases, and of those obtained during the course of the algorithm: all the bases are computed in shifted Popov form, whose size is always O(m σ). Then, we design a divide-and-conquer scheme. We recursively reduce the initial interpolation problem to sub-problems with more convenient shifts by first computing information on the degrees of the intermediate bases.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"123 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124190003","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}

引用次数: 23

Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations 用模多项式方程组快速计算多项式矩阵的移位波波夫形式

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-02-01 DOI: 10.1145/2930889.2930936

Vincent Neiger

{"title":"Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations","authors":"Vincent Neiger","doi":"10.1145/2930889.2930936","DOIUrl":"https://doi.org/10.1145/2930889.2930936","url":null,"abstract":"We give a Las Vegas algorithm which computes the shifted Popov form of an m x m nonsingular polynomial matrix of degree d in expected ~O(mω d) field operations, where ω is the exponent of matrix multiplication and ~O(·) indicates that logarithmic factors are omitted. This is the first algorithm in ~O(mω d) for shifted row reduction with arbitrary shifts. Using partial linearization, we reduce the problem to the case d ≤ ⌈ σ/m ⌉ where σ is the generic determinant bound, with σ / m bounded from above by both the average row degree and the average column degree of the matrix. The cost above becomes ~O(mω ⌈ σ/m ⌉), improving upon the cost of the fastest previously known algorithm for row reduction, which is deterministic. Our algorithm first builds a system of modular equations whose solution set is the row space of the input matrix, and then finds the basis in shifted Popov form of this set. We give a deterministic algorithm for this second step supporting arbitrary moduli in ~O(mω-1 σ) field operations, where m is the number of unknowns and σ is the sum of the degrees of the moduli. This extends previous results with the same cost bound in the specific cases of order basis computation and M-Pade approximation, in which the moduli are products of known linear factors.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130235261","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}

引用次数: 22

Fast Computation of the Nth Term of an Algebraic Series over a Finite Prime Field 有限素数域上代数级数第n项的快速计算

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-02-01 DOI: 10.1145/2930889.2930904

A. Bostan, G. Christol, P. Dumas

引用次数: 7

Solving Rank-Constrained Semidefinite Programs in Exact Arithmetic 用精确算法求解秩约束半定规划

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-02-01 DOI: 10.1145/2930889.2930925

Simone Naldi

{"title":"Solving Rank-Constrained Semidefinite Programs in Exact Arithmetic","authors":"Simone Naldi","doi":"10.1145/2930889.2930925","DOIUrl":"https://doi.org/10.1145/2930889.2930925","url":null,"abstract":"We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active, this is a non-convex optimization problem, otherwise it is a semidefinite program. Both find numerous applications especially in systems control theory and combinatorial optimization, but even in more general contexts such as polynomial optimization or real algebra. While numerical algorithms exist for solving this problem, such as interior-point or Newton-like algorithms, in this paper we propose an approach based on symbolic computation. We design an exact algorithm for solving rank-constrained semidefinite programs, whose complexity is essentially quadratic on natural degree bounds associated to the given optimization problem: for subfamilies of the problem where the size of the feasible matrix is fixed, the complexity is polynomial in the number of variables. The algorithm works under assumptions on the input data: we prove that these assumptions are generically satisfied. We also implement it in Maple and discuss practical experiments.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"255 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122153978","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}

引用次数: 11

Reduction-Based Creative Telescoping for Algebraic Functions 基于约简的代数函数创造性伸缩

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-02-01 DOI: 10.1145/2930889.2930901

Shaoshi Chen, Manuel Kauers, C. Koutschan

引用次数: 32

Validating the Completeness of the Real Solution Set of a System of Polynomial Equations 验证多项式方程组实解集的完备性

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-02-01 DOI: 10.1145/2930889.2930910

Daniel A. Brake, J. Hauenstein, Alan C. Liddell

引用次数: 15