PHCpack: a general-purpose solver for polynomial systems
by homotopy continuation

Jan Verschelde

Abstract:

Polynomial systems occur in a wide variety of application domains. Homotopy continuation methods have proven to be reliable and powerful methods to compute numerically approximations to all isolated complex solutions. During the last decade considerable progress has been accomplished on exploiting structure in a polynomial system, in particular its sparsity. In this paper the structure and design of the software package PHC is described. The main program operates in several modes, is menu-driven and file-oriented. This package features a great variety of root-counting methods among its tools. The outline of one black-box solver is sketched and a report is given on its performance on a large database of test problems. The software has been developed on three different machine architectures. Its portability is ensured by the gnu-ada compiler.


Categories and Subject Descriptors: D.3.2 [Programming Languages]: Language Classification - Ada; G.1.5 [Numerical Analysis]: Roots of Nonlinear Equations - Systems of equations, Polynomials, methods for; G.2.1 [Discrete Mathematics]: Combinatorics - Counting problems; G.4 [Mathematics of Computing]: Mathematical Software.

General Terms: Algorithms, Theory.

Additional Key Words and Phrases: homotopy continuation, polynomial systems, start system, root count, Bézout's theorem, polyhedral homotopy, mixed volume.

ACM Transactions on Mathematical Software 25(2): 251-276, 1999.