Solving Polynomial Systems by the Polyhedral Homotopy

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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.