ACM Communications in Computer Algebra最新文献_第9页

Computing one billion roots using the tangent Graeffe method 使用正切Graeffe方法计算十亿根

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ACM Communications in Computer Algebra Pub Date : 2020-07-25 DOI: 10.1145/3457341.3457342

J. Hoeven, M. Monagan

{"title":"Computing one billion roots using the tangent Graeffe method","authors":"J. Hoeven, M. Monagan","doi":"10.1145/3457341.3457342","DOIUrl":"https://doi.org/10.1145/3457341.3457342","url":null,"abstract":"Let p be a prime of the form p = σ2k + 1 with σ small and let Fp denote the finite field with p elements. Let P(z) be a polynomial of degree d in Fp[z] with d distinct roots in Fp. For p =5 · 255 + 1 we can compute the roots of such polynomials of degree 109. We believe we are the first to factor such polynomials of size one billion. We used a multi-core computer with two 10 core Intel Xeon E5 2680 v2 CPUs and 128 gigabytes of RAM. The factorization takes just under 4,000 seconds on 10 cores and uses 121 gigabytes of RAM. We used the tangent Graeffe root finding algorithm from [27, 19] which is a factor of O(log d) faster than the Cantor-Zassenhaus algorithm. We implemented the tangent Graeffe algorithm in C using our own library of 64 bit integer FFT based in-place polynomial algorithms then parallelized the FFT and main steps using Cilk C. In this article we discuss the steps of the tangent Graeffe algorithm, the sub-algorithms that we used, how we parallelized them, and how we organized the memory so we could factor a polynomial of degree 109. We give both a theoretical and practical comparison of the tangent Graeffe algorithm with the Cantor-Zassenhaus algorithm for root finding. We improve the complexity of the tangent Graeffe algorithm by a factor of 2. We present a new in-place product tree multiplication algorithm that is fully parallelizable. We present some timings comparing our software with Magma's polynomial factorization command. Polynomial root finding over smooth finite fields is a key ingredient for algorithms for sparse polynomial interpolation that are based on geometric sequences. This application was also one of our main motivations for the present work.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"65 - 85"},"PeriodicalIF":0.1,"publicationDate":"2020-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3457341.3457342","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43840894","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: 2

Recent and upcoming events 最近和即将举行的活动

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ACM Communications in Computer Algebra Pub Date : 2020-06-01 DOI: 10.1145/3427218.3427227

Staff

引用次数: 0

Algebraic optimization degree 代数优化度

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ACM Communications in Computer Algebra Pub Date : 2020-06-01 DOI: 10.1145/3427218.3427222

Marc Härkönen, Benjamin Hollering, Fatemeh Tarashi Kashani, J. Rodriguez

引用次数: 2

A new Kenzo module for computing the Eilenberg-Moore spectral sequence 计算Eilenberg-Moore谱序列的新Kenzo模

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ACM Communications in Computer Algebra Pub Date : 2020-06-01 DOI: 10.1145/3427218.3427225

A. Romero, J. Rubio, F. Sergeraert, Markus Szymik

引用次数: 2

Sampling the feasible sets of SDPs and volume approximation 抽样可行集的sdp和体积逼近

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ACM Communications in Computer Algebra Pub Date : 2020-05-13 DOI: 10.1145/3457341.3457349

Tolis Chalkis, Vissarion Fisikopoulos, Panagiotis Repouskos, Elias P. Tsigaridas

{"title":"Sampling the feasible sets of SDPs and volume approximation","authors":"Tolis Chalkis, Vissarion Fisikopoulos, Panagiotis Repouskos, Elias P. Tsigaridas","doi":"10.1145/3457341.3457349","DOIUrl":"https://doi.org/10.1145/3457341.3457349","url":null,"abstract":"We present algorithmic, complexity, and implementation results on the problem of sampling points in the interior and the boundary of a spectrahedron, that is the feasible region of a semidefinite program. Our main tool is random walks. We define and analyze a set of primitive geometric operations that exploits the algebraic properties of spectrahedra and the polynomial eigenvalue problem and leads to the realization of a broad collection of efficient random walks. We demonstrate random walks that experimentally show faster mixing time than the ones used previously for sampling from spectrahedra in theory or applications, for example Hit and Run. Consecutively, the variety of random walks allows us to sample from general probability distributions, for example the family of log-concave distributions which arise frequently in numerous applications. We apply our tools to compute (i) the volume of a spectrahedron and (ii) the expectation of functions coming from robust optimal control. We provide a C++ open source implementation of our methods that scales efficiently up to dimension 200. We illustrate its efficiency on various data sets.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"114 - 118"},"PeriodicalIF":0.1,"publicationDate":"2020-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3457341.3457349","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43872475","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

International Symposium on Wen-Tsun Wu's Academic Thought and Mathematics Mechanization 温学术思想与数学机械化国际学术研讨会

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ACM Communications in Computer Algebra Pub Date : 2020-03-18 DOI: 10.1145/3388974.3388975

Academy of Mathematics and Systems Science

{"title":"International Symposium on Wen-Tsun Wu's Academic Thought and Mathematics Mechanization","authors":"Academy of Mathematics and Systems Science","doi":"10.1145/3388974.3388975","DOIUrl":"https://doi.org/10.1145/3388974.3388975","url":null,"abstract":"During May 12-17, 2019, shall we celebrate the late famous mathematician Wen-Tsun Wu's centenary birthday. Professor Wen-Tsun Wu is one of the most internationally influential mathematicians in China, who brought far-reaching influence on both mathematics and computer research by his great contributions to the field of topology and the opening of the new area of Mathematics Mechanization. His research includes algebraic topology, di.erential topology, mathematics mechanization, algebraic geometry, game theory as well as the history of mathematics. His work has been regarded as among the greatest classical influential results in topology. In late 1970s, he opened the area of mathematics mechanization by the so-called \"Wu method\" which automatically proves geometric theorems using computers and was taken as the pioneer work in automatic reasoning. The symposium focuses on his achievements in topology, mathematics mechanization as well as the history of mathematics, and casts light on their future prospects and potential applications. Specific topics for the symposium include but are not limited to: •Algebraic topology •Automated reasoning •Computational algebraic geometry and optimization •Computational geometry and topology •Computational number theory •Coding theory and cryptology •Di.erential and di.erence algebras •History of mathematics •Polynomial equation solving •Symbolic or symbolic-numeric computation •Applications of mathematical mechanization","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"53 1","pages":"155 - 181"},"PeriodicalIF":0.1,"publicationDate":"2020-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3388974.3388975","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48504133","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: 0

15 PhD positions in GRAPES 葡萄学院15个博士职位

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ACM Communications in Computer Algebra Pub Date : 2020-03-18 DOI: 10.1145/3388974.3388978

I. Emiris

{"title":"15 PhD positions in GRAPES","authors":"I. Emiris","doi":"10.1145/3388974.3388978","DOIUrl":"https://doi.org/10.1145/3388974.3388978","url":null,"abstract":"We are pleased to announce 15 open PhD Positions within the new European Network GRAPES: Learning, Processing and Optimising Shapes. GRAPES is a Marie Sk lodowska-Curie Innovative Training Network in the framework of H2020, started on December 1st, 2019, and lasting for 4 years. It includes 11 academic/industrial groups o.ering 15 PhD's, and 6 associated partners; it is coordinated by ATHENA Research & Innovation Center, Greece. GRAPES aims at advancing the state-of-the-art in Mathematics, Computer-Aided Design, and Machine Learning, in order to promote game changing approaches for generating, optimising, and learning 3D shapes, along with a multisectoral training for young researchers, supported by the active participation of SMEs. Recent advances in the above domains have solved numerous tasks concerning multimedia and 2D data. However, automation of 3D geometry processing and analysis lags severely behind, despite their importance in science, technology and everyday life, and some well-understood underlying mathematical principles. Concrete applications include simulation, manufacturing and 3D printing, visualisation and naval design, retrieval and mining, reconstruction and urban planning.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"53 1","pages":"189 - 189"},"PeriodicalIF":0.1,"publicationDate":"2020-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3388974.3388978","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44827423","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: 0

Maple Conference 2019 Maple Conference 2019

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ACM Communications in Computer Algebra Pub Date : 2020-03-18 DOI: 10.1145/3388974.3388976

J. Gerhard, I. Kotsireas