SI2-SSE: Solving Polynomial Systems with PHCpack and phcpy

求解多项式方程组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。免费和开源软件PHCpack应用符号-数值和多面体方法来求解多项式系统。phcpy作为一个用Python脚本语言编写的PHCPack的新接口，改进了PHCPack的功能。在并行计算机上实现多精度算法以补偿成本开销，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中的求解器应用同伦延拓方法，将符号-数值与多面体算法混合在一起。用牛顿法计算解的数值近似。解决方案是近似的象征性的Puixix系列，它起源于初始形式所定义的牛顿多面体的多项式在系统中。phcpy的设计提供了一个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包将为可伸缩的计算服务器提供工具，以处理提交到Web服务器的请求。在多核处理器上的多线程实现由图形处理单元加速，补偿了双倍和四倍运算的成本开销。

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.