Optimal Polyhedral Homotopies on Supercomputers for Algebraic Sets

代数集是多项式系统的解。利用多项式系统的稀疏结构，多面体同伦对于计算一般系统的所有孤立解是最优的，因为每个解路径收敛于一个解。本提案的目标是开发新的同伦算法，这将是计算正维解集的数值表示的一般最优算法。为了求解大型多项式系统，新的同伦算法将适合在超级计算机上运行。多项式系统经常出现在科学和工程的许多问题中。求解多项式系统的算法是在一个开源软件包PHCpack中实现的，可以在网上免费获得。PHCpack的用户友好界面（在计算机代数系统中，如Maple（商业），SAGE（开源），以及在科学软件系统中，如MATLAB（商业），Octave（开源））将继续使许多在研究中解决多项式系统的科学家和工程师受益。首席研究员教他的学生通过软件转移数学技术，着眼于高性能计算。

Algebraic sets are solutions of polynomial systems. Exploiting the sparse structure of a polynomial system, polyhedral homotopies are optimal for computing all isolated solutions of general systems in the sense that every solution path converges to a solution. The goal of this proposal is to develop new homotopy algorithms that will be generically optimal for computing numerical representations of positive dimensional solution sets. To solve large polynomial systems, the new homotopy algorithms will be made suitable to run on supercomputers.Polynomial systems arise frequently in many problems in science and engineering. The algorithms to solve polynomial systems are implemented in an open source software package PHCpack, available for free on the web.User-friendly interfaces to PHCpack (in computer algebra systems such as Maple (commercial), SAGE (open source), and in scientific software systems such as MATLAB (commercial), Octave (open source)) will continue to benefit many scientists and engineers who solve polynomial systems in their research.The principal investigator teaches his students to transfer mathematical technology through software with an eye towards high performance computing.