SI2-SSE: Solving Polynomial Systems with PHCpack and phcpy

SI2-SSE：使用 PHCpack 和 phcpy 求解多项式系统

基本信息

批准号：
1440534
负责人：
Jan Verschelde
金额：
$ 46.44万
依托单位：
University of Illinois at Chicago
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-10-01 至 2019-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1440534&HistoricalAwards=false
关键词：
SI2 SSE Solving Polynomial Systems

项目摘要

Solving polynomial systems is a fundamental problem in mathematics with applications to various fields of science and engineering. The free and open source software PHCpack applies symbolic-numeric and polyhedral methods to solve polynomial systems. As a new interface to PHCpack written in the Python scripting language, phcpy improves the functionality of PHCpack. The implementation on parallel computers to compensate for the cost overhead of multi-precision arithmetic will enable scientists and engineers to solve larger systems faster and more accurately. A web server developed with phcpy will give anyone with an internet connection access to the developed software.The solvers in PHCpack apply homotopy continuation methods, blending symbolic-numeric with polyhedral algorithms. Numerical approximations to solutions are computed with Newton's method. Solutions are approximated symbolically by Puiseux series, which originate at initial forms defined by the Newton polytopes of the polynomials in the system. The design of phcpy gives a flexible interactive scripting interface, without sacrificing efficiency as compiled code in PHCpack is executed. The package phcpy will provide the tools for a scalable compute server to serve requests submitted to the web server. Multithreaded implementations on multicore processors accelerated by graphics processing units compensate for the cost overhead of double double and quad double arithmetic.

求解多项式系统是数学中的一个基本问题，可应用于科学和工程的各个领域。免费开源软件 PHCpack 应用符号数值和多面体方法来求解多项式系统。 phcpy 作为用 Python 脚本语言编写的 PHCpack 的新接口，改进了 PHCpack 的功能。在并行计算机上实施以补偿多精度算术的成本开销将使科学家和工程师能够更快、更准确地解决更大的系统。使用 phcpy 开发的 Web 服务器将允许任何有互联网连接的人访问所开发的软件。PHCpack 中的求解器应用同伦延拓方法，将符号数字与多面体算法混合在一起。使用牛顿法计算解的数值近似值。解通过 Puiseux 级数象征性地近似，该级数源自由系统中多项式的牛顿多胞形定义的初始形式。 phcpy 的设计提供了灵活的交互式脚本接口，而不会牺牲 PHCpack 中编译代码的执行效率。 phcpy 包将为可扩展的计算服务器提供工具，以服务提交到 Web 服务器的请求。由图形处理单元加速的多核处理器上的多线程实现补偿了双双精度和四双精度算术的成本开销。