Solving Polynomial Systems by Polyhedral Homotopies

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The project is aiming at solving polynomial systems numerically by thehomotopy continuation method. Over the years, practical evidences have beengiven that the homotopy method is efficient, reliable and much powerful insolving polynomial systems. Recently, modeling the sparse structure of apolynomial system by its Newton polytopes leads to a major computationalbreakthrough called the polyhedral homotopy method. Based on it, a sourcecode HOM4PS, produced by the PI and his students, leads all the otherexisting codes for solving polynomial systems in efficiency and storagerequirement by a great margin. Nonetheless, there are numerous models oflarge polynomial systems in application still do not have a satisfactoryline of attack. And it has become apparent that the numerical techniques forfollowing homotopy paths are far from thoroughly developed for largesystems. The essence of the project is the advance development in allaspects of the solver HOM4PS based on the conduct of further research togreatly enlarge the scope of its applications, especially applications tolarge systems.The problem of solving polynomial systems arises very frequently in variousfields of science and engineering, such as formula construction, geometricintersection, inverse kinematics, robotics, power flow problems withPQ-specified bases, vision and the computation of equilibrium states ofchemical reaction equations, etc. This topic has been an important researchsubjectin Europe. In the last decade, a considerable research effortinvolving seven countries and twenty universities had been directed to thisproblem in two consecutive major projects, PoSSo and FRISCO, supported byEuropean Commission. As mentioned above, the code HOM4PS developed by the PIfor this problem is far more advanced and the ultimate goal of the currentproject is a more complete high-quality block-box software which willincorporate the best state of the art to provide the general scientificcommunity a reliable source for solving polynomial systems on a wide varietyof advanced architectures.