A Blackbox Polynomial System Solver on Parallel Shared Memory Computers

  • Jan VerscheldeEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11077)


A numerical irreducible decomposition for a polynomial system provides representations for the irreducible factors of all positive dimensional solution sets of the system, separated from its isolated solutions. Homotopy continuation methods are applied to compute a numerical irreducible decomposition. Load balancing and pipelining are techniques in a parallel implementation on a computer with multicore processors. The application of the parallel algorithms is illustrated on solving the cyclic n-roots problems, in particular for \(n = 8, 9\), and 12.


Homotopy continuation Numerical irreducible decomposition Mathematical software Multitasking Pipelining Polyhedral homotopies Polynomial system Shared memory parallel computing 


  1. 1.
    Adrovic, D., Verschelde, J.: Polyhedral methods for space curves exploiting symmetry applied to the cyclic n-roots problem. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2013. LNCS, vol. 8136, pp. 10–29. Springer, Cham (2013). Scholar
  2. 2.
    Backelin, J.: Square multiples n give infinitely many cyclic n-roots. Reports, Matematiska Institutionen 8, Stockholms universitet (1989)Google Scholar
  3. 3.
    Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Software for numerical algebraic geometry: a paradigm and progress towards its implementation. In: Stillman, M.E., Takayama, N., Verschelde, J. (eds.) Software for Algebraic Geometry. IMA Volumes in Mathematics and its Applications, vol. 148, pp. 33–46. Springer, New York (2008). Scholar
  4. 4.
    Björck, G., Fröberg, R.: Methods to “divide out” certain solutions from systems of algebraic equations, applied to find all cyclic 8-roots. In: Gyllenberg, M., Persson, L.E. (eds.) Analysis, Algebra and Computers in Mathematical Research. LNM, vol. 564, pp. 57–70. Dekker, London (1994)Google Scholar
  5. 5.
    Chen, T., Lee, T.-L., Li, T.-Y.: Hom4PS-3: a parallel numerical solver for systems of polynomial equations based on polyhedral homotopy continuation methods. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 183–190. Springer, Heidelberg (2014). Scholar
  6. 6.
    Chen, T., Lee, T.L., Li, T.Y.: Mixed volume computation in parallel. Taiwan. J. Math. 18(1), 93–114 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Faugère, J.C.: Finding all the solutions of Cyclic 9 using Gröbner basis techniques. In: Computer Mathematics - Proceedings of the Fifth Asian Symposium (ASCM 2001). Lecture Notes Series on Computing, vol. 9, pp. 1–12. World Scientific (2001)Google Scholar
  8. 8.
    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)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hida, Y., Li, X.S., Bailey, D.H.: Algorithms for quad-double precision floating point arithmetic. In: 15th IEEE Symposium on Computer Arithmetic (Arith-15 2001), pp. 155–162. IEEE Computer Society (2001)Google Scholar
  10. 10.
    Leykin, A., Verschelde, J.: Decomposing solution sets of polynomial systems: a new parallel monodromy breakup algorithm. Int. J. Comput. Sci. Eng. 4(2), 94–101 (2009)Google Scholar
  11. 11.
    Leykin, A., Verschelde, J., Zhao, A.: Newton’s method with deflation for isolated singularities of polynomial systems. Theor. Comput. Sci. 359(1–3), 111–122 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Leykin, A., Verschelde, J., Zhao, A.: Evaluation of Jacobian matrices for Newton’s method with deflation to approximate isolated singular solutions of polynomial systems. In: Wang, D., Zhi, L. (eds.) Symbolic-Numeric Computation, Trends in Mathematics, pp. 269–278. Birkhauser (2007)Google Scholar
  13. 13.
    Malajovich, G.: Computing mixed volume and all mixed cells in quermassintegral time. Found. Comput. Math. 17, 1293–1334 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mizutani, T., Takeda, A.: DEMiCs: a software package for computing the mixed volume via dynamic enumeration of all mixed cells. In: Stillman, M.E., Takayama, N., Verschelde, J. (eds.) Software for Algebraic Geometry. IMA Volumes in Mathematics and Its Applications, vol. 148, pp. 59–79. Springer, New York (2008). Scholar
  15. 15.
    Mizutani, T., Takeda, A., Kojima, M.: Dynamic enumeration of all mixed cells. Discret. Comput. Geom. 37(3), 351–367 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Sabeti, R.: Numerical-symbolic exact irreducible decomposition of cyclic-12. LMS J. Comput. Math. 14, 155–172 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sandén, B.I.: Design of Multithreaded Software. The Entity-Life Modeling Approach. IEEE Computer Society (2011)Google Scholar
  18. 18.
    Sommese, A.J., Verschelde, J., Wampler, C.W.: Numerical irreducible decomposition using PHCpack. In: Joswig, M., Takayama, N. (eds.) Algebra, Geometry, and Software Systems, pp. 109–130. Springer, Heidelberg (2003). Scholar
  19. 19.
    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). Scholar
  20. 20.
    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:
  21. 21.
    Verschelde, J., Yoffe, G.: Polynomial homotopies on multicore workstations. In: Maza, M.M., Roch, J.-L. (eds.) Proceedings of the 4th International Workshop on Parallel Symbolic Computation (PASCO 2010), pp. 131–140. ACM (2010)Google Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA

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