Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation最新文献

Complexity Analysis of Root Clustering for a Complex Polynomial 复多项式根聚类的复杂度分析

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-07-20 DOI: 10.1145/2930889.2930939

R. Becker, Michael Sagraloff, Vikram Sharma, Juan Xu, C. Yap

引用次数: 40

Numerical Sparsity Determination and Early Termination 数值稀疏度的确定和早期终止

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-07-20 DOI: 10.1145/2930889.2930924

Z. Hao, E. Kaltofen, L. Zhi

{"title":"Numerical Sparsity Determination and Early Termination","authors":"Z. Hao, E. Kaltofen, L. Zhi","doi":"10.1145/2930889.2930924","DOIUrl":"https://doi.org/10.1145/2930889.2930924","url":null,"abstract":"Ankur Moitra in his paper at STOC 2015 has given an in-depth analysis of how oversampling improves the conditioning of the arising Prony systems for sparse interpolation and signal recovery from numeric data. Moitra assumes that oversampling is done for a number of samples beyond the actual sparsity of the polynomial/signal. We give an algorithm that can be used to compute the sparsity and estimate the minimal number of samples needed in numerical sparse interpolation. The early termination strategy of polynomial interpolation has been incorporated in the algorithm: by oversampling at a small number of extra sample points we can diagnose that the sparsity has not been reached. Our algorithm still has to make a guess, the number ζ of oversamples, and we show by example that if ζ is guessed too small, premature termination can occur, but our criterion is numerically more accurate than that by Kaltofen, Lee and Yang (Proc. SNC 2011, ACM [12]), but not as efficiently computable. For heuristic justification one has available the multivariate early termination theorem by Kaltofen and Lee (JSC vol. 36(3--4) 2003 [11]) for exact arithmetic, and the numeric Schwartz-Zippel Lemma by Kaltofen, Yang and Zhi (Proc. SNC 2007, ACM [13]). A main contribution here is a modified proof of the Theorem by Kaltofen and Lee that permits starting the sequence at the point (1,...,1), for scalar fields of characteristic ≠ 2 (in characteristic 2 counter-examples are given).","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"110 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127982147","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: 7

Baby-Step Giant-Step Algorithms for the Symmetric Group 对称群的Baby-Step - Giant-Step算法

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-07-20 DOI: 10.1145/2930889.2930930

E. Bach, Bryce Sandlund

引用次数: 0

Comprehensive Gröbner Systems in Rings of Differential Operators, Holonomic D-modules and B-functions 微分算子、完整d模和b函数环上的综合Gröbner系统

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-07-20 DOI: 10.1145/2930889.2930918

Katsusuke Nabeshima, Katsuyoshi Ohara, S. Tajima

引用次数: 6

Computer Assisted Proof for Apwenian Sequences Apwenian数列的计算机辅助证明

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-07-20 DOI: 10.1145/2930889.2930891

Hao Fu, Guo-Niu Han

引用次数: 10

Analysis of the Brun Gcd Algorithm 布朗Gcd算法的分析

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-07-20 DOI: 10.1145/2930889.2930899

V. Berthé, Loïck Lhote, B. Vallée

引用次数: 2

Fast Polynomial Multiplication over F260 快速多项式乘法在F260

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-07-20 DOI: 10.1145/2930889.2930920

David Harvey, J. Hoeven, Grégoire Lecerf

引用次数: 17

Guessing Linear Recurrence Relations of Sequence Tuplesand P-recursive Sequences with Linear Algebra 用线性代数猜测序列元组和p递归序列的线性递归关系

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-07-20 DOI: 10.1145/2930889.2930926

Jérémy Berthomieu, J. Faugère

{"title":"Guessing Linear Recurrence Relations of Sequence Tuplesand P-recursive Sequences with Linear Algebra","authors":"Jérémy Berthomieu, J. Faugère","doi":"10.1145/2930889.2930926","DOIUrl":"https://doi.org/10.1145/2930889.2930926","url":null,"abstract":"Given several n-dimensional sequences, we first present an algorithm for computing the Grobner basis of their module of linear recurrence relations. A P-recursive sequence (ui)i ∈ Nn satisfies linear recurrence relations with polynomial coefficients in i, as defined by Stanley in 1980. Calling directly the aforementioned algorithm on the tuple of sequences ((ij, ui)i ∈ Nn)j for retrieving the relations yields redundant relations. Since the module of relations of a P-recursive sequence also has an extra structure of a 0-dimensional right ideal of an Ore algebra, we design a more efficient algorithm that takes advantage of this extra structure for computing the relations. Finally, we show how to incorporate Grobner bases computations in an Ore algebra K t1,...,tn,x1,...,xn, with commutators xk,xl-xl,xk=tk,tl-tl,tk= tk,xl-xl,tk=0 for k ≠ l and tk,xk-xk,tk=xk, into the algorithm designed for P-recursive sequences. This allows us to compute faster the elements of the Grobner basis of which are in the ideal spanned by the first relations, such as in 2D/3D-space walks examples.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121785135","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: 10

On the Bit Complexity of Solving Bilinear Polynomial Systems 求解双线性多项式系统的位复杂度

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-07-20 DOI: 10.1145/2930889.2930919

I. Emiris, Angelos Mantzaflaris, Elias P. Tsigaridas

引用次数: 8

A Superfast Randomized Algorithm to Decompose Binary Forms 一种分解二进制形式的超快速随机算法

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation Pub Date : 2016-07-20 DOI: 10.1145/2930889.2930896

M. Bender, J. Faugère, Ludovic Perret, Elias P. Tsigaridas

{"title":"A Superfast Randomized Algorithm to Decompose Binary Forms","authors":"M. Bender, J. Faugère, Ludovic Perret, Elias P. Tsigaridas","doi":"10.1145/2930889.2930896","DOIUrl":"https://doi.org/10.1145/2930889.2930896","url":null,"abstract":"Symmetric Tensor Decomposition is a major problem that arises in areas such as signal processing, statistics, data analysis and computational neuroscience. It is equivalent to write a homogeneous polynomial in $n$ variables of degree $D$ as a sum of $D$-th powers of linear forms, using the minimal number of summands. This minimal number is called the rank of the polynomial/tensor. We consider the decomposition of binary forms, that corresponds to the decomposition of symmetric tensors of dimension $2$ and order $D$. This problem has its roots in Invariant Theory, where the decompositions are known as canonical forms. As part of that theory, different algorithms were proposed for the binary forms. In recent years, those algorithms were extended for the general symmetric tensor decomposition problem. We present a new randomized algorithm that enhances the previous approaches with results from structured linear algebra and techniques from linear recurrent sequences. It achieves a softly linear arithmetic complexity bound. To the best of our knowledge, the previously known algorithms have quadratic complexity bounds. We compute a symbolic minimal decomposition in O(M(D) log(D)) arithmetic operations, where M(D) is the complexity of multiplying two polynomials of degree D. We approximate the terms of the decomposition with an error of 2-ε, in O(D log2(D) (log2(D) + log(ε))) arithmetic operations. To bound the size of the representation of the coefficients involved in the decomposition, we bound the algebraic degree of the problem by min(rank, D-rank+1). When the input polynomial has integer coefficients, our algorithm performs, up to poly-logarithmic factors, OB(D l + D4 + D3 τ) bit operations, where τ is the maximum bitsize of the coefficients and 2-l is the relative error of the terms in the decomposition.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124832927","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