Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation最新文献_第5页

On Exact Polya and Putinar's Representations 关于Polya和Putinar的精确表述

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation Pub Date : 2018-02-28 DOI: 10.1145/3208976.3208986

Victor Magron, M. S. E. Din

引用次数: 19

Symmetric Indefinite Triangular Factorization Revealing the Rank Profile Matrix 揭示秩轮廓矩阵的对称不定三角分解

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation Pub Date : 2018-02-26 DOI: 10.1145/3208976.3209019

J. Dumas, Clément Pernet

{"title":"Symmetric Indefinite Triangular Factorization Revealing the Rank Profile Matrix","authors":"J. Dumas, Clément Pernet","doi":"10.1145/3208976.3209019","DOIUrl":"https://doi.org/10.1145/3208976.3209019","url":null,"abstract":"We present a novel recursive algorithm for reducing a symmetric matrix to a triangular factorization which reveals the rank profile matrix. That is, the algorithm computes a factorization P TA P = L D L T where P is a permutation matrix, L is lower triangular with a unit diagonal and D is symmetric block diagonal with 1 x 1 and 2 x 2 antidiagonal blocks. This algorithm requires O(n2rømega-2) arithmetic operations, with n the dimension of the matrix, r its rank and ømega an admissible exponent for matrix multiplication. Furthermore, experimental results demonstrate that our algorithm has very good performance: its computational speed matches that of its numerical counterpart and is twice as fast as the unsymmetric exact Gaussian factorization. By adapting the pivoting strategy developed in the unsymmetric case, we show how to recover the rank profile matrix from the permutation matrix and the support of the block-diagonal matrix. We also note that there is an obstruction in characteristic 2 for revealing the rank profile matrix, which requires to relax the shape of the block diagonal by allowing the 2-dimensional blocks to have a non-zero bottom-right coefficient. This relaxed decomposition can then be transformed into a standard PLDLTP T decomposition at a negligible cost.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128742997","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: 3

On the Maximal Number of Real Embeddings of Spatial Minimally Rigid Graphs 空间最小刚性图的最大实数嵌入

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation Pub Date : 2018-02-16 DOI: 10.1145/3208976.3208994

Evangelos Bartzos, I. Emiris, Jan Legerský, Elias P. Tsigaridas

{"title":"On the Maximal Number of Real Embeddings of Spatial Minimally Rigid Graphs","authors":"Evangelos Bartzos, I. Emiris, Jan Legerský, Elias P. Tsigaridas","doi":"10.1145/3208976.3208994","DOIUrl":"https://doi.org/10.1145/3208976.3208994","url":null,"abstract":"The number of embeddings of minimally rigid graphs in RD is (by definition) finite, modulo rigid transformations, for every generic choice of edge lengths. Even though various approaches have been proposed to compute it, the gap between upper and lower bounds is still enormous. Specific values and its asymptotic behavior are major and fascinating open problems in rigidity theory. Our work considers the maximal number of real embeddings of minimally rigid graphs in R3. We modify a commonly used parametric semi-algebraic formulation that exploits the Cayley-Menger determinant to minimize the a priori number of complex embeddings, where the parameters correspond to edge lengths. To cope with the huge dimension of the parameter space and find specializations of the parameters that maximize the number of real embeddings, we introduce a method based on coupler curves that makes the sampling feasible for spatial minimally rigid graphs. Our methodology results in the first full classification of the number of real embeddings of graphs with 7 vertices in R3, which was the smallest open case. Building on this and certain 8-vertex graphs, we improve the previously known general lower bound on the maximum number of real embeddings in R3.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128077843","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: 4

Frobenius Additive Fast Fourier Transform Frobenius加性快速傅里叶变换

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation Pub Date : 2018-02-12 DOI: 10.1145/3208976.3208998

Wen-Ding Li, Ming-Shing Chen, Po-Chun Kuo, Chen-Mou Cheng, Bo-Yin Yang

引用次数: 9

Exact Algorithms for Semidefinite Programs with Degenerate Feasible Set 具有退化可行集的半定规划的精确算法

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation Pub Date : 2018-02-08 DOI: 10.1145/3208976.3209022

D. Henrion, Simone Naldi, M. S. E. Din

{"title":"Exact Algorithms for Semidefinite Programs with Degenerate Feasible Set","authors":"D. Henrion, Simone Naldi, M. S. E. Din","doi":"10.1145/3208976.3209022","DOIUrl":"https://doi.org/10.1145/3208976.3209022","url":null,"abstract":"Let A0, ..., An be m x m symmetric matrices with entries in Q, and let A(x) be the linear pencil A0+x1 A1 + ··· + xn An, where x=(x1,...,xn) are unknowns. The linear matrix inequality (LMI) A(x) ≥ 0 defines the subset of Rn, called spectrahedron, containing all points x such that A(x) has non-negative eigenvalues. The minimization of linear functions over spectrahedra is called semidefinite programming (SDP). Such problems appear frequently in control theory and real algebra, especially in the context of nonnegativity certificates for multivariate polynomials based on sums of squares. Numerical software for solving SDP are mostly based on the interior point method, assuming some non-degeneracy properties such as the existence of interior points in the admissible set. In this paper, we design an exact algorithm based on symbolic homotopy for solving semidefinite programs without assumptions on the feasible set, and we analyze its complexity. Because of the exactness of the output, it cannot compete with numerical routines in practice but we prove that solving such problems can be done in polynomial time if either n or m is fixed.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114381443","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: 13

Error Correction in Fast Matrix Multiplication and Inverse 快速矩阵乘法与逆的误差校正

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation Pub Date : 2018-02-07 DOI: 10.1145/3208976.3209001

Daniel S. Roche

引用次数: 3

Additive Decompositions in Primitive Extensions 基元扩展中的加性分解

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation Pub Date : 2018-02-07 DOI: 10.1145/3208976.3208987

Shaoshi Chen, Hao Du, Ziming Li

引用次数: 6

On the Chordality of Polynomial Sets in Triangular Decomposition in Top-Down Style 关于自顶向下三角分解中多项式集的弦性

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation Pub Date : 2018-02-06 DOI: 10.1145/3208976.3208997

Chenqi Mou, Yang Bai

引用次数: 9

Certification of Minimal Approximant Bases 最小近似基的认证

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation Pub Date : 2018-02-06 DOI: 10.1145/3208976.3208991

Pascal Giorgi, Vincent Neiger

{"title":"Certification of Minimal Approximant Bases","authors":"Pascal Giorgi, Vincent Neiger","doi":"10.1145/3208976.3208991","DOIUrl":"https://doi.org/10.1145/3208976.3208991","url":null,"abstract":"For a given computational problem, a certificate is a piece of data that one (the prover) attaches to the output with the aim of allowing efficient verification (by the verifier) that this output is correct. Here, we consider the minimal approximant basis problem, for which the fastest known algorithms output a polynomial matrix of dimensions m x m and average degree D/m using O~(mømega D/m) field operations. We propose a certificate which, for typical instances of the problem, is computed by the prover using O(mømega D/m) additional field operations and allows verification of the approximant basis by a Monte Carlo algorithm with cost bound O(mømega + m D). Besides theoretical interest, our motivation also comes from the fact that approximant bases arise in most of the fastest known algorithms for linear algebra over the univariate polynomials; thus, this work may help in designing certificates for other polynomial matrix computations. Furthermore, cryptographic challenges such as breaking records for discrete logarithm computations or for integer factorization rely in particular on computing minimal approximant bases for large instances: certificates can then be used to provide reliable computation on outsourced and error-prone clusters.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"128 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115253917","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

Computing Popov and Hermite Forms of Rectangular Polynomial Matrices 计算矩形多项式矩阵的Popov和Hermite形式

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation Pub Date : 2018-02-06 DOI: 10.1145/3208976.3208988

Vincent Neiger, J. Rosenkilde, Grigory Solomatov

引用次数: 10