HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method

• Published in: Computing Vol. 83; no. 2-3; pp. 109 - 133
• Main Authors: Lee, T. L., Li, T. Y., Tsai, C. H.
• Format: Journal Article
• Language: English
• Published: Vienna Springer Vienna 01.11.2008; Springer; Springer Nature B.V
• Subjects: Algebra; Algebraic geometry; Algorithmics. Computability. Computer arithmetics; Algorithms; Applied mathematics; Applied sciences; Artificial Intelligence; Computer Appl. in Administrative Data Processing; Computer Communication Networks; Computer Science; Computer science; control theory; systems; Euclidean space; Exact sciences and technology; Information Systems Applications (incl.Internet); Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Numerical analysis; Numerical analysis. Scientific computation; Polynomials; Sciences and techniques of general use; Software; Software Engineering; Software packages; Studies; Theoretical computing; Polyhedral homotopy; 90B99; Software package; 65H10; Numerical experiments; Polynomial systems; Homotopy continuation methods; 65H15; Polynomial; Software package; 65H10; FORTRAN 90; Large system; Polynomial method; Continuation method; Algorithm; Mixed method; Implementation; Numerical analysis; Scientific computation; Homotopy; Software; Algebraic geometry
• ISSN: 0010-485X; 1436-5057
• DOI: 10.1007/s00607-008-0015-6

HOM4PS-2.0 is a software package in FORTRAN 90 which implements the polyhedral homotopy continuation method for solving polynomial systems. It updates its original version HOM4PS in three key aspects: (1) new method for finding mixed cells, (2) combining the polyhedral and linear homotopies in one step, (3) new way of dealing with curve jumping. Numerical results show that this revision leads to a spectacular speed-up, ranging up to 1950s, over its original version on all benchmark systems, especially for large ones. It surpasses the existing packages in finding isolated zeros, such as PHCpack (Verschelde in ACM Trans Math Softw 25:251–276, 1999), PHoM (Gunji et al. in Computing 73:57–77, 2004), and Bertini (Bates et al. in Software for numerical algebraic geometry. Available at http://www.nd.edu/~sommese/bertini ), in speed by big margins.