Finding all isolated solutions to polynomial systems using HOMPACK

Although the theory of polynomial continuation has been established for over a decade (following the work of Garcia, Zangwill, and Drexler), it is difficult to solve polynomial systems using continuation in practice. Divergent paths (solutions at infinity), singular solutions, and extreme scaling of coefficients can create catastrophic numerical problems. Further, the large number of paths that typically arise can be discouraging. In this paper we summarize polynomial-solving homotopy continuation and report on the performance of three standard path-tracking algorithms (as implemented in HOMPACK) in solving three physical problems of varying degrees of difficulty. Our purpose is to provide useful information on solving polynomial systems, including specific guidelines for homotopy construction and parameter settings. The m-homogeneous strategy for constructing polynomial homotopies is outlined, along with more traditional approaches. Computational comparisons are included to illustrate and contrast the major HOMPACK options. The conclusions summarize our numerical experience and discuss areas for future research.