Solving polynomial systems is both an important and in general a difficult problem which frequently occurs in many scientific and engineering models. Homotopy continuation methods have proven to be efficient and reliable to compute numerically approximations to all isolated solutions. See [15] for an introduction and [13] for a survey on recent research.
On average, these methods are optimal and suitable to distributed computing. Sequentially the solver computes one solution path at a time, where on average, each path requires only polynomial time. Because the paths can be tracked independently, communication cost is low and coarse-grained parallelism is applicable.
The performance of homotopy continuation slows down when confronted with deficient polynomial systems. Because applications are seldomly chosen at random, root counts often overestimate the actual number of solutions. This so-called deficiency leads to wasted computations, because too many solution paths need to be tracked. To obtain all isolated solutions, the complex roots must be computed as well.
The paper is organized in four parts. First we give a short description of the methods, mention related software and list the main features of the program. In the second part the four stages in the solver, the various tools and modes to invoke the program are explained. An overview of our database of polynomial systems illustrates the practical usefulness. The third part is concerned with programming aspects. Herein we comment on programming in Ada. We present PHC as a workbench and PHCPACK, as an interface to other software systems. A kind of installation guide is addressed in the fourth part, where we list the computers and compilers we used. To conclude, we list some future directions.