{"title":"Skew-polynomial-sparse matrix multiplication","authors":"Qiao-Long Huang , Ke Ye , Xiao-Shan Gao","doi":"10.1016/j.jsc.2023.102240","DOIUrl":null,"url":null,"abstract":"<div><p>Based on the observation that <span><math><msup><mrow><mi>Q</mi></mrow><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>×</mo><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span> is isomorphic to a quotient skew polynomial ring, we propose a new deterministic algorithm for <span><math><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>×</mo><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> matrix multiplication over <span><math><mi>Q</mi></math></span>, where <em>p</em> is a prime number. The algorithm has complexity <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>ω</mi><mo>−</mo><mn>2</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>, where <span><math><mi>T</mi><mo>≤</mo><mi>p</mi><mo>−</mo><mn>1</mn></math></span> is a parameter determined by the skew-polynomial-sparsity of input matrices and <em>ω</em><span> is the asymptotic exponent of matrix multiplication<span>. 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 </span></span><span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>∼</mo></mrow></msup><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>ω</mi><mo>−</mo><mn>2</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mo></mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>ν</mi></mrow></mfrac><mo>)</mo></math></span>, where <span><math><mi>t</mi><mo>≤</mo><mi>p</mi><mo>−</mo><mn>1</mn></math></span> is the skew-polynomial-sparsity of the product and <em>ν</em><span> 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.</span></p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"121 ","pages":"Article 102240"},"PeriodicalIF":0.6000,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Symbolic Computation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0747717123000548","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
Based on the observation that is isomorphic to a quotient skew polynomial ring, we propose a new deterministic algorithm for matrix multiplication over , where p is a prime number. The algorithm has complexity , where 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 , where 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.
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.