Parallel Algorithms for Polynomials

多项式的并行算法

• 批准号: RGPIN-2014-04238
• 负责人: Monagan, Michael
• 金额: $ 2.33万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=610915
• 关键词: Parallel; Algorithms; Polynomials

This research proposal is about designing and implementing algorithms for computing with mathematical formulas. It focusses on computing with large polynomials in more than one variable and with large formulas involving algebraic numbers like sqrt(2) and algebraic functions and sqrt(1+y^2). Such formulas arise in many applications in Science, Engineering and Mathematics. This research is part of a field known as Computer Algebra. It is also known as Symbolic Computation. Researchers in this field design algorithms and software systems, called Computer Algebra Systems, for doing algebra and calculus on the computer. Computer Algebra Systems like Maple and Mathematica are used by Scientists, Engineers and Mathematicians in both industry and academia for their work. The researcher on this proposal has been and continues to be involved with the design and development of Maple which is a Canadian product. A focus of the proposal is the development and implementation of parallel algorithms. This is because today's laptops, desktops and servers are all multi-core computers and the only way now to improve performance of software is to exploit their multi-core processing capability. We propose to use Cilk for implementing parallel algorithms. Cilk was developed at MIT and has now been adopted by Intel. We also propose to begin experimenting with using Graphics Processing Units (GPUs) for implementing algorithms. A second focus of the research is how to interpolate, that is, how to reconstruct polynomials and rational functions in more than one variable from their values. Often formulas involving many variables are sparse and have structure. We propose to try to automate the detection of this structure so we can interpolate them rapidly. Some of the software we develop in this research will be written in Maple and some will be written in C and/or Cilk. We do plan to make it available to the Maple user community so that Scientists, Engineers and Mathematicians can use it. For the students working on these projects, it is very encouraging for them to see other people using their software. Students who work on the research projects in this proposal will learn how to design and implement algorithms for solving various problems in computational mathematics like factoring polynomials and fast algorithms for solving problems involving matrices. They will learn how to design and implement parallel algorithms using Cilk. They will be learning mathematics (e.g. about algebraic numbers) and computer science (e.g. how to analyze the efficiency of algorithms.)

该研究建议是关于使用数学公式设计和实施用于计算的算法。 它的重点是在多个变量中使用大多项式计算，并且具有代数的大公式 SQRT（2）和代数函数和SQRT（1+Y^2）之类的数字。 这种公式在科学，工程和数学中的许多应用中产生。 这项研究是称为计算机代数的领域的一部分。它也称为符号计算。 该领域设计算法和软件系统的研究人员称为计算机代数系统， 用于在计算机上执行代数和微积分。诸如Maple和Mathematica之类的计算机代数系统由 工业和学术界的科学家，工程师和数学家的工作。研究人员 提案已经并且继续参与了加拿大产品的枫树的设计和开发。 该提案的重点是平行算法的开发和实施。 这是因为今天的笔记本电脑，台式机和服务器都是多核计算机， 现在提高软件性能的唯一方法是利用其多核 处理能力。我们建议使用CILK实现并行算法。 CILK是在麻省理工学院开发的，现已被英特尔采用。我们还建议开始 尝试使用图形处理单元（GPU）实现算法。 该研究的第二个重点是如何插值，即如何重建 多项式和合理函数来自多个变量的值。 通常涉及许多变量的公式是稀疏且具有结构的。 我们建议尝试自动化该结构的检测，以便我们可以迅速插入它们。 我们在这项研究中开发的一些软件将用枫木编写，其中一些将用C和/或CILK编写。 我们确实计划将其提供给枫木用户社区，以便科学家，工程师和数学家可以使用它。 对于从事这些项目的学生来说，看到其他人使用他们的软件是非常令人鼓舞的。 在此提案中从事研究项目的学生将学习如何设计和实施算法 用于解决计算数学中的各种问题，例如分解多项式和快速算法来解决 涉及矩阵的问题。他们将学习如何使用CILK设计和实施并行算法。 他们将学习数学（例如代数数字）和计算机科学 （例如，如何分析算法的效率。）