, Volume 77, Issue 4, pp 387–411 | Cite as

PHoMpara – Parallel Implementation of the Polyhedral Homotopy Continuation Method for Polynomial Systems

  • T. GunjiEmail author
  • S. Kim
  • K. Fujisawa
  • M. Kojima


The polyhedral homotopy continuation method is known to be a successful method for finding all isolated solutions of a system of polynomial equations. PHoM, an implementation of the method in C++, finds all isolated solutions of a polynomial system by constructing a family of modified polyhedral homotopy functions, tracing the solution curves of the homotopy equations, and verifying the obtained solutions. A software package PHoMpara parallelizes PHoM to solve a polynomial system of large size. Many characteristics of the polyhedral homotopy continuation method make parallel implementation efficient and provide excellent scalability. Numerical results include some large polynomial systems that had not been solved.

AMS Subject Classifications

65H10 65H20 


Polynomials parallel computation homotopy continuation methods equations polyhedral homotopy numerical experiments software package 


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  1. Akl, S. G. 1984Optimal parallel algorithms for computing convex hulls and for sortingComputing33111zbMATHMathSciNetCrossRefGoogle Scholar
  2. Allgower, E., Georg, K.: Numerical continuation methods. Springer 1990.Google Scholar
  3. Allison, D. C. S., Chakaborty, A., Watson, L. T. 1989Granularity issues for solving polynomial systems via globally convergent algorithms on a hypercubeJ. Supercomputing3520CrossRefGoogle Scholar
  4. Bernshtein, D. N. 1975The number of roots of a system of equationsFunct. Anal. Appl.9183185zbMATHCrossRefGoogle Scholar
  5. Boege, W., Gebauer, R., Kredel, H. 1986Some examples for solving systems of algebraic equations by calculating Groebner basesJ. Symb. Comput.28398MathSciNetCrossRefzbMATHGoogle Scholar
  6. Björck, G., Fröberg, R. 1991A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic n-rootsJ. Symb. Comput.12329336zbMATHCrossRefGoogle Scholar
  7. Dai, Y., Kim S., Kojima, M.: Computing all nonsingular solutions of cyclic-n polynomial using polyhedral homotopy continuation methods. J. Comput. Appl. Math. 151, 1–2:83–97 (2003).Google Scholar
  8. Garcia, C. B., Zangwill, W. I. 1979Determining all solutions to certain systems of nonlinear equationsMath. Oper. Res.4114MathSciNetzbMATHCrossRefGoogle Scholar
  9. Gao, T., Li, T. Y., Verschelde, J., Wu, M. 2000Balancing the lifting values to improve the numerical stability of polyhedral homotopy continuation methodsAppl. Math. Comput.114233247MathSciNetCrossRefzbMATHGoogle Scholar
  10. Gunji, T., Kim, S., Kojima, M., Takeda, A., Fujisawa, K., Mizutani, T. 2004PHoM – a polyhedral homotopy continuation method for polynomial systemsComputing735577MathSciNetCrossRefGoogle Scholar
  11. Huber, B., Sturmfels, B. 1995A Polyhedral method for solving sparse polynomial systemsMath. Comput.6415411555MathSciNetCrossRefzbMATHGoogle Scholar
  12. Huber, B., Verschelde, J. 1998Polyhedral end games for polynomial continuationNumer. Algorithms1891108MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kim, S., Kojima, M. 2004Numerical stability of path tracing in polynomial homotopy continuation methodsComputing73329348MathSciNetCrossRefzbMATHGoogle Scholar
  14. Li, T. Y. 1987Solving polynomial systemsThe mathematical intelligencer93339zbMATHMathSciNetCrossRefGoogle Scholar
  15. Li, T. Y. 1999Solving polynomial systems by polyhedral homotopiesTaiwan J. Math.3251279zbMATHGoogle Scholar
  16. Li, T. Y., Li, X. 2001Finding mixed cells in the mixed volume computationFoundation of Computational Mathematics1161181MathSciNetCrossRefzbMATHGoogle Scholar
  17. Morgan, A.: Solving polynomial systems using continuation for engineering and scientific problems. Prentice-Hall, 1987.Google Scholar
  18. Morgan, A. P., Sommese, A. J. 1989Coefficient-parameter polynomial continuationAppl. Math. Comput.29123160MathSciNetCrossRefzbMATHGoogle Scholar
  19. Morgan, A. P., Watson, L. T. 1989A globally convergent parallel algorithm for zeros of polynomial systemsNonlinear Analysis1313391350MathSciNetCrossRefzbMATHGoogle Scholar
  20. Morgan, A. P., Sommese, A. J., Wampler, C. W. 1992Computing singular solutions to polynomial systemsAdv. Appl. Math.13305327MathSciNetCrossRefzbMATHGoogle Scholar
  21. MPI: Scholar
  22. Nemhauser,G. L.,Wolsey, L. A.: Integer and combinatorial optimization.Wiley-Interscience, 1988Google Scholar
  23. Noonberg, V. W. 1989A neural network modeled by an adaptive Lotka-Volterra systemSIAM J. Appl. Math.4917791792MathSciNetCrossRefGoogle Scholar
  24. Pelz, W., Watson, L. T. 1989Message length effects for solving polynomial systems on a hypercubeParallel Computing10161176MathSciNetCrossRefzbMATHGoogle Scholar
  25. Sato, M., Nakada, H., Sekiguchi, S., Matsuoka, S., Nagashima, U., Takagi, H.: Ninf: A network based information library for a global world-wide computing infrastructure. HPCN'97 (LNCS-1225), 1997, pp. 491–502.Google Scholar
  26. Su, H. -J., McCarthy, J. M.: Kinematic synthesis of RPS serial chains. In: Proc. ASME Design Engineering Technical Conferences (CDROM), Chicago, IL, September 2–6, 2003.Google Scholar
  27. Sturmfels, B.: Solving systems of polynomial equations. CBMS Regional Conf. Series in Mathematics, No. 97. American Mathematical Society, 2002.Google Scholar
  28. Su, H. J., McCarthy, J. M., Sosonkina, M., Watson, L. T.: POLSYS-GLP: A parallel general linear product homotopy code for solving polynomial systems of equations. ACM Trans. Math. Softw. (to appear).Google Scholar
  29. Takeda, A., Kojima, M., Fujisawa, K. 2002Enumeration of all solutions of a combinatorial linear inequality system arising from the polyhedral homotopy continuation methodJ. Oper. Soc. Japan456482MathSciNetzbMATHGoogle Scholar
  30. Traverso, C.: The PoSSo test suite examples. Available at Scholar
  31. Verschelde, J.: The database of polynomial systems is in his web site: ``http://www.math.uic. edu/~jan/''.Google Scholar
  32. Verschelde, J., Verlinden, P., Cools, R. 1994Homotopies exploiting Newton polytopes for solving sparse polynomial systemsSIAM J. Numer. Anal.31915930MathSciNetCrossRefzbMATHGoogle Scholar
  33. Verschelde, J.: Homotopy continuation methods for solving polynomial systems. Ph.D. thesis, Department of Computer Science, Katholieke Universiteit Leuven, 1996.Google Scholar
  34. Verschelde, J., Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw. 25, 251–276 (1999).Google Scholar

Copyright information

© Springer-Verlag Wien 2006

Authors and Affiliations

  1. 1.Department of Mathematical and Computing SciencesTokyo Institute of TechnologyTokyoJapan
  2. 2.Department of MathematicsEwha Women's UniversitySeoulKorea
  3. 3.Department of Mathematical SciencesTokyo Denki UniversitySaitamaJapan

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