Parallel Homotopy Algorithms to Solve Polynomial Systems

  • Anton Leykin
  • Jan Verschelde
  • Yan Zhuang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4151)


Homotopy continuation methods to compute numerical approximations to all isolated solutions of a polynomial system are known as “embarrassingly parallel”, i.e.: because of their low communication overhead, these methods scale very well for a large number of processors. Because so many important problems remain unsolved mainly due to their intrinsic computational complexity, it would be embarrassing not to develop parallel implementations of polynomial homotopy continuation methods. This paper concerns the development of “parallel PHCpack”, a project which started a couple of years ago in collaboration with Yusong Wang, and which currently continues with Anton Leykin (parallel irreducible decomposition) and Yan Zhuang (parallel polyhedral homotopies). We report on our efforts to make PHCpack ready to solve large polynomial systems which arise in applications.

2000 Mathematics Subject Classification. Primary 65H10. Secondary 14Q99, 68W30.

keywords and phrases

Continuation methods high performance continuation jumpstarting homotopies linear-product systems parallel computation path following polynomial systems polyhedral homotopies simplex system 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allison, D.C.S., Chakraborty, A., Watson, L.T.: Granularity issues for solving polynomial systems via globally convergent algorithms on a hypercube. J. of Supercomputing 3, 5–20 (1989)CrossRefGoogle Scholar
  2. 2.
    Bernshteǐn, D.N.: The number of roots of a system of equations. Functional Anal. Appl., 9(3), 183–185 (1975); Translated from Funktsional. Anal. i Prilozhen., 9(3), 1–4 (1975)Google Scholar
  3. 3.
    Boege, W., Gebauer, R., Kredel, H.: Some examples for solving systems of algebraic equations by calculating Groebner bases. J. Symbolic Computation 2, 83–98 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chakraborty, A., Allison, D.C.S., Ribbens, C.J., Watson, L.T.: Note on unit tangent vector computation for homotopy curve tracking on a hypercube. Parallel Computing 17(12), 1385–1395 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chakraborty, A., Allison, D.C.S., Ribbens, C.J., Watson, L.T.: The parallel complexity of embedding algorithms for the solution of systems of nonlinear equations. IEEE Transactions on Parallel and Distributed Systems 4(4), 458–465 (1993)CrossRefGoogle Scholar
  6. 6.
    Gao, T., Li, T.Y., Wu, M.: Algorithm 846: MixedVol: A software package for mixed volume computation. ACM Trans. Math. Softw. 31(4), 555–560 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gunji, T., Kim, S., Fujisawa, K., Kojima, M.: PHoMpara – parallel implementation of the Polyhedral Homotopy continuation Method for polynomial systems. Research report b-419, Tokyo Institute of Technology (2005) available via: zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gunji, T., Kim, S., Kojima, M., Takeda, A., Fujisawa, K., Mizutani, T.: PHoM – a polyhedral homotopy continuation method for polynomial systems. Computing 73(4), 55–77 (2004)MathSciNetGoogle Scholar
  9. 9.
    Harimoto, S., Watson, L.T.: The granularity of homotopy algorithms for polynomial systems of equations. In: Rodrigue, G. (ed.) Parallel processing for scientific computing, pp. 115–120. SIAM, Philadelphia (1989)Google Scholar
  10. 10.
    Huber, B., Sottile, F., Sturmfels, B.: Numerical Schubert calculus. J. Symbolic Computation 26(6), 767–788 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Huber, B., Sturmfels, B.: A polyhedral method for solving sparse polynomial systems. Math. Comp. 64(212), 1541–1555 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Huber, B., Verschelde, J.: Pieri homotopies for problems in enumerative geometry applied to pole placement in linear systems control. SIAM J. Control Optim. 38(4), 1265–1287 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Katsura, S.: Users posing problems to PoSSO. In: Gonzelez-Vega, L., Recio, T. (eds.) The PoSSO Newsletter, vol. 2 (July 1994)Google Scholar
  14. 14.
    Leykin, A., Verschelde, J.: Decomposing solution sets of polynomial systems: a new parallel monodromy breakup algorithm. The International Journal of Computational Science and Engineering (accepted for publication)Google Scholar
  15. 15.
    Leykin, A., Verschelde, J.: Interfacing with the numerical homotopy algorithms in phcpack. In: Iglesias, A., Takayama, N. (eds.) ICMS 2006. LNCS, vol. 4151, pp. 354–360. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Leykin, A., Verschelde, J.: Factoring solution sets of polynomial systems in parallel. In: Skeie, T., Yang, C.-S. (eds.) Proceedings of the 2005 International Conference on Parallel Processing Workshops, June 14-17, 2005. High Performance Scientific and Engineering Computing, pp. 173–180. IEEE Computer Society Press, Los Alamitos (2005)Google Scholar
  17. 17.
    Li, T.Y.: Numerical solution of polynomial systems by homotopy continuation methods. In: Cucker, F. (ed.) Handbook of Numerical Analysis. Foundations of Computational Mathematics, vol. XI, pp. 209–304. North-Holland, Amsterdam (2003)Google Scholar
  18. 18.
    Li, T.Y., Wang, X., Wu, M.: Numerical schubert calculus by the pieri homotopy algorithm. SIAM J. Numer. Anal. 40(2), 578–600 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Morgan, A., Sommese, A.: A homotopy for solving general polynomial systems that respects m-homogeneous structures. Appl. Math. Comput. 24(2), 101–113 (1987)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Morgan, A.P., Watson, L.T.: A globally convergent parallel algorithm for zeros of polynomial systems. Nonlinear Analysis 13(11), 1339–1350 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Pelz, W., Watson, L.T.: Message length effects for solving polynomial systems on a hypercube. Parallel Computing 10(2), 161–176 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Samet, H.: The quadtree and related hierarchical data structures. ACM Computing Surveys 16(2) (1984)Google Scholar
  23. 23.
    Sommese, A.J., Verschelde, J., Wampler, C.W.: Numerical decomposition of the solution sets of polynomial systems into irreducible components. SIAM J. Numer. Anal. 38(6), 2022–2046 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Sommese, A.J., Verschelde, J., Wampler, C.W.: Using monodromy to decompose solution sets of polynomial systems into irreducible components. In: Ciliberto, C., Hirzebruch, F., Miranda, R., Teicher, M. (eds.) Application of Algebraic Geometry to Coding Theory, Physics and Computation. Proceedings of a NATO Conference, Eilat, Israel, February 25 - March 1, 2001, pp. 297–315. Kluwer Academic Publishers, Dordrecht (2001)Google Scholar
  25. 25.
    Sommese, A.J., Verschelde, J., Wampler, C.W.: Symmetric functions applied to decomposing solution sets of polynomial systems. SIAM J. Numer. Anal. 40(6), 2026–2046 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Sommese, A.J., Verschelde, J., Wampler, C.W.: Introduction to numerical algebraic geometry. In: Dickenstein, A., Emiris, I.Z. (eds.) Solving Polynomial Equations. Foundations, Algorithms and Applications. Algorithms and Computation in Mathematics, vol. 14, pp. 301–337. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  27. 27.
    Sommese, A.J., Wampler, C.W.: The Numerical solution of systems of polynomials arising in engineering and science. World Scientific, Singapore (2005)zbMATHCrossRefGoogle Scholar
  28. 28.
    Su, H.-J.: Computer-Aided Constrained Robot Design Using Mechanism Synthesis Theory. PhD thesis, University of California, Irvine (2004)Google Scholar
  29. 29.
    Su, H.-J., McCarthy, J.M.: Kinematic synthesis of RPS serial chains. In: the Proceedings of the ASME Design Engineering Technical Conferences (CDROM), Chicago, IL, September 2-6 (2003)Google Scholar
  30. 30.
    Su, H.-J., McCarthy, J.M., Sosonkina, M., Watson, L.T.: Algorithm 8xx: POLSYS_GLP: A parallel general linear product homotopy code for solving polynomial systems of equations. ACM Trans. Math. Softw. (to appear)Google Scholar
  31. 31.
    Su, H.-J., McCarthy, J.M., Watson, L.T.: Generalized linear product homotopy algorithms and the computation of reachable surfaces. ASME Journal of Information and Computer Sciences in Engineering 4(3), 226–234 (2004)CrossRefGoogle Scholar
  32. 32.
    Su, H.-J., Wampler, C.W., McCarthy, J.M.: Geometric design of cylindric PRS serial chains. ASME Journal of Mechanical Design 126(2), 269–277 (2004)CrossRefGoogle Scholar
  33. 33.
    Verschelde, J.: Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw. 25(2), 251–276 (1999), Software available at: zbMATHCrossRefGoogle Scholar
  34. 34.
    Verschelde, J., Cools, R.: Symbolic homotopy construction. Applicable Algebra in Engineering, Communication and Computing 4(3), 169–183 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Verschelde, J., Verlinden, P., Cools, R.: Homotopies exploiting Newton polytopes for solving sparse polynomial systems. SIAM J. Numer. Anal. 31(3), 915–930 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Verschelde, J., Wang, Y.: Computing feedback laws for linear systems with a parallel Pieri homotopy. In: Yang, Y. (ed.) Proceedings of the 2004 International Conference on Parallel Processing Workshops, Montreal, Quebec, Canada, August 15-18, 2004. High Performance Scientific and Engineering Computing, pp. 222–229. IEEE Computer Society Press, Los Alamitos (2004)Google Scholar
  37. 37.
    Verschelde, J., Zhuang, Y.: Parallel implementation of the polyhedral homotopy method. In: Proceedings of The 8th Workshop on High Performance Scientific and Engineering Computing (HPSEC 2006), Columbus, Ohio, USA, August 18 (2006) (to appear)Google Scholar
  38. 38.
    Wampler, C.W., Morgan, A.P., Sommese, A.J.: Numerical continuation methods for solving polynomial systems arising in kinematics. ASME J. of Mechanical Design 112(1), 59–68 (1990)CrossRefGoogle Scholar
  39. 39.
    Watson, L.T., Billups, S.C., Morgan, A.P.: Algorithm 652: HOMPACK: a suite of codes for globally convergent homotopy algorithms. ACM Trans. Math. Softw. 13(3), 281–310 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Wise, S.M., Sommese, A.J., Watson, L.T.: Algorithm 801: POLSYS_PLP: a partitioned linear product homotopy code for solving polynomial systems of equations. ACM Trans. Math. Softw. 26(1), 176–200 (2000)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Anton Leykin
    • 1
  • Jan Verschelde
    • 1
  • Yan Zhuang
    • 1
  1. 1.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA

Personalised recommendations