Solving Polynomial Systems by the Polyhedral Homotopy

多项式系统经常出现在科学和工程的各个领域，如公式构造、几何相交、逆运动学、机器人、视觉和化学反应方程的平衡态计算。 多年来，同伦延拓方法在数值求解多项式方程组时是有效、可靠和强大的。 最近，一个主要的计算突破称为多面体同伦方法出现，使该方法更加有效和强大。 基于这种新方法，PI开发的HOM 4PS-2.0软件包在求解大量多项式方程组时表现出了惊人的性能，并在效率和存储需求方面大大领先于所有其他现有代码。该项目的实质是进一步开发求解器HOM 4PS-2.0及其并行版本HOM 4PS-2.0para的各个方面，在进行进一步研究的基础上，大大扩大其应用范围，特别是对超大型系统的应用。 该项目的最终目标是更完整的高质量黑盒软件，这将为一般科学界提供一个可靠的来源，用于在各种先进的架构上求解多项式系统。

Polynomial systems arise very frequently in various fields of science and engineering, such as formula construction, geometric intersection, inverse kinematics, robotics, vision, and the computation of equilibrium states of chemical reaction equations. Algorithms for solving such systems numerically are therefore highly in demand.Over the years, practical evidence has been given that homotopy continuation methods are efficient, reliable, and powerful in solving polynomial systems numerically. Recently, a major computational breakthrough called the polyhedral homotopy method emerged, making the method considerably more efficient and powerful. Based on this new method, the software package HOM4PS-2.0 developed by the PI has produced marvelous performance on solving a large collection of polynomial systems and has led all the other existing codes in efficiency and storage requirement by a huge margin.The essence of the proposed project is the further development in all aspects of the solver HOM4PS-2.0 as well as its parallelized version HOM4PS-2.0para, based on the conduction of further research to greatly enlarge the scope of the applications, especially applications to very large systems. The ultimate goal of the project is more-complete high-quality black-box software which will provide the general scientific community a reliable source for solving polynomial systems on a wide variety of advanced architectures.