SI2-SSE: Solving Polynomial Systems with PHCpack and phcpy

求解多项式系统是数学中的一个基本问题，在科学和工程的各个领域都有应用。免费开源软件PHCpack应用符号-数值和多面体方法求解多项式系统。phcpy是用Python脚本语言编写的PHCpack的新接口，它改进了PHCpack的功能。在并行计算机上的实现弥补了多精度算法的成本开销，将使科学家和工程师能够更快、更准确地解决更大的系统。用phcpy开发的web服务器可以让任何有互联网连接的人访问开发的软件。PHCpack中的求解器采用同伦延拓方法，混合了符号-数值算法和多面体算法。用牛顿法计算了解的数值近似。解由Puiseux级数象征性地逼近，它起源于由系统中多项式的牛顿多面体定义的初始形式。phcpy的设计提供了一个灵活的交互式脚本界面，在执行PHCpack中编译的代码时不会牺牲效率。phcpy包将为可伸缩的计算服务器提供工具，以处理提交给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和四双算法的开销。

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.