Skew-polynomial-sparse matrix multiplication

IF 0.6 4区 数学 Q4 COMPUTER SCIENCE, THEORY & METHODS
Qiao-Long Huang , Ke Ye , Xiao-Shan Gao
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引用次数: 0

Abstract

Based on the observation that Q(p1)×(p1) is isomorphic to a quotient skew polynomial ring, we propose a new deterministic algorithm for (p1)×(p1) matrix multiplication over Q, where p is a prime number. The algorithm has complexity O(Tω2p2), where Tp1 is a parameter determined by the skew-polynomial-sparsity of input matrices and ω is the asymptotic exponent of matrix multiplication. Here a matrix is skew-polynomial-sparse if its corresponding skew polynomial is sparse. Moreover, by introducing randomness, we also propose a probabilistic algorithm with complexity O(tω2p2+p2log1ν), where tp1 is the skew-polynomial-sparsity of the product and ν is the probability parameter. The main feature of the algorithms is the acceleration for matrix multiplication if the input matrices or their products are skew-polynomial-sparse.

斜多项式-稀疏矩阵乘法
在观察到Q(p−1)×(p−2)同构于商斜多项式环的基础上,我们提出了一种新的Q上(p−3)×(p−1)矩阵乘法的确定算法,其中p是素数。该算法的复杂度为O(Tω−2p2),其中T≤p−1是由输入矩阵的斜多项式稀疏性决定的参数,ω是矩阵乘法的渐近指数。这里,如果矩阵对应的偏斜多项式是稀疏的,则矩阵是偏斜多项式稀疏的。此外,通过引入随机性,我们还提出了一种复杂度为O~(tω−2p2+p2log)的概率算法⁡1Γ),其中t≤p−1是乘积的偏斜多项式稀疏性,Γ是概率参数。如果输入矩阵或其乘积是斜多项式稀疏的,则该算法的主要特征是矩阵乘法的加速。
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来源期刊
Journal of Symbolic Computation
Journal of Symbolic Computation 工程技术-计算机:理论方法
CiteScore
2.10
自引率
14.30%
发文量
75
审稿时长
142 days
期刊介绍: An international journal, the Journal of Symbolic Computation, founded by Bruno Buchberger in 1985, is directed to mathematicians and computer scientists who have a particular interest in symbolic computation. The journal provides a forum for research in the algorithmic treatment of all types of symbolic objects: objects in formal languages (terms, formulas, programs); algebraic objects (elements in basic number domains, polynomials, residue classes, etc.); and geometrical objects. It is the explicit goal of the journal to promote the integration of symbolic computation by establishing one common avenue of communication for researchers working in the different subareas. It is also important that the algorithmic achievements of these areas should be made available to the human problem-solver in integrated software systems for symbolic computation. To help this integration, the journal publishes invited tutorial surveys as well as Applications Letters and System Descriptions.
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