The Granularity of Parallel Homotopy Algorithms for Polynomial Systems of Equations

Polynomial systems consist of n polynomial functions in n variables, with real or complex coefficients. Finding zeros of such systems is challenging because there may be a large number of solutions, and Newton-type methods can rarely be guaranteed to find the complete set of solutions. There are homotopy algorithms for polynomial systems of equations that are globally convergent from an arbitrary starting point with probability one, are guaranteed to find all the solutions, and are robust, accurate, and reasonably efficient. There is inherent parallelism at several levels in these algorithms. Several parallel homotopy algorithms with different granularities are studied on several different parallel machines, using actual industrial problems from chemical engineering and solid modeling.