iBet uBet web content aggregator. Adding the entire web to your favor.
iBet uBet web content aggregator. Adding the entire web to your favor.



Link to original content: https://doi.org/10.1007/978-3-319-99639-4_25
A Blackbox Polynomial System Solver on Parallel Shared Memory Computers | SpringerLink
Skip to main content
Springer Nature Link
Log in

A Blackbox Polynomial System Solver on Parallel Shared Memory Computers

  • Conference paper
  • First Online:
Computer Algebra in Scientific Computing (CASC 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11077))

Included in the following conference series:

Abstract

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.

This material is based upon work supported by the National Science Foundation under Grant No. 1440534.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  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). https://doi.org/10.1007/978-3-319-02297-0_2

    Chapter  Google Scholar 

  2. Backelin, J.: Square multiples n give infinitely many cyclic n-roots. Reports, Matematiska Institutionen 8, Stockholms universitet (1989)

    Google Scholar 

  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). https://doi.org/10.1007/978-0-387-78133-4_1

    Chapter  MATH  Google Scholar 

  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. 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). https://doi.org/10.1007/978-3-662-44199-2_30

    Chapter  Google Scholar 

  6. Chen, T., Lee, T.L., Li, T.Y.: Mixed volume computation in parallel. Taiwan. J. Math. 18(1), 93–114 (2014)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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. 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. 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)

    Article  MathSciNet  Google Scholar 

  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. Malajovich, G.: Computing mixed volume and all mixed cells in quermassintegral time. Found. Comput. Math. 17, 1293–1334 (2016)

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1007/978-0-387-78133-4_5

    Chapter  MATH  Google Scholar 

  15. Mizutani, T., Takeda, A., Kojima, M.: Dynamic enumeration of all mixed cells. Discret. Comput. Geom. 37(3), 351–367 (2007)

    Article  MathSciNet  Google Scholar 

  16. Sabeti, R.: Numerical-symbolic exact irreducible decomposition of cyclic-12. LMS J. Comput. Math. 14, 155–172 (2011)

    Article  MathSciNet  Google Scholar 

  17. Sandén, B.I.: Design of Multithreaded Software. The Entity-Life Modeling Approach. IEEE Computer Society (2011)

    Google Scholar 

  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). https://doi.org/10.1007/978-3-662-05148-1_6

    Chapter  Google Scholar 

  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). https://doi.org/10.1007/3-540-27357-3_8

    Chapter  Google Scholar 

  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: http://www.phcpack.org

  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 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street (m/c 249), Chicago, IL, 60607-7045, USA

    Jan Verschelde

Authors
  1. Jan Verschelde
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan Verschelde .

Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Verschelde, J. (2018). A Blackbox Polynomial System Solver on Parallel Shared Memory Computers. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2018. Lecture Notes in Computer Science(), vol 11077. Springer, Cham. https://doi.org/10.1007/978-3-319-99639-4_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-99639-4_25

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-99638-7

  • Online ISBN: 978-3-319-99639-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation