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Extrapolating Solution Paths of Polynomial Homotopies Towards Singularities with PHCpack and Phcpy

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Mathematical Software – ICMS 2024 (ICMS 2024)

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Abstract

PHCpack is a software package for polynomial homotopy continuation, which provides a robust path tracker [Telen, Van Barel, Verschelde, SISC 2020]. This tracker computes the radius of convergence of Newton’s method, estimates the distance to the nearest path, and then applies Padé approximants to predict the next point on the path. A priori step size control is less sensitive to finely tuned tolerances than a posteriori step size control, and is therefore robust. The Python interface phcpy is extended with a new step-by-step tracker and is applied to experiment with extrapolation methods to accurately locate the singular points at the end of solution paths.

Supported by the National Science Foundation under grant DMS 1854513.

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Notes

  1. 1.

    Version 1.2.2 of alr, GNAT 12.2.1 and gprbuild 22.0.1.

References

  1. Bliss, N., Verschelde, J.: The method of Gauss-Newton to compute power series solutions of polynomial homotopies. Linear Algebra Appl. 542, 569–588 (2018)

    Article  MathSciNet  Google Scholar 

  2. Brezinski, C., Redivo Zaglia, M.: Extrapolation Methods. Studies in Computational Mathematics, vol. 2. North-Holland, Amsterdam (1991)

    Google Scholar 

  3. Cuyt, A., Wuytack, L.: Nonlinear Methods in Numerical Analysis. North-Holland, Elsevier Science Publishers, Amsterdam (1987)

    Google Scholar 

  4. Eröcal, B., Stein, W.: The Sage project: unifying free mathematical software to create a viable alternative to magma, maple, mathematica and MATLAB. In: Fukuda, K., Hoeven, J., Joswig, M., Takayama, N. (eds.) ICMS 2010. LNCS, vol. 6327, pp. 12–27. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15582-6_4

    Chapter  Google Scholar 

  5. Fabry, E.: Sur les points singuliers d’une fonction donnée par son développement en série et l’impossibilité du prolongement analytique dans des cas très généraux. Annales scientifiques de l’École Normale Supérieure 13, 367–399 (1896)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  7. Henrici, P.: An algorithm for analytic continuation. SIAM J. Numer. Anal. 3(1), 67–78 (1966)

    Article  MathSciNet  Google Scholar 

  8. 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 

  9. Hunter, J.D.: Matplotlib: a 2D graphics environment. Comput. Sci. Eng. 9(3), 90–95 (2007)

    Article  Google Scholar 

  10. Joldes, M., Muller, J.-M., Popescu, V., Tucker, W.: CAMPARY: Cuda multiple precision arithmetic library and applications. In: Greuel, G.-M., Koch, T., Paule, P., Sommese, A. (eds.) ICMS 2016. LNCS, vol. 9725, pp. 232–240. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-42432-3_29

    Chapter  Google Scholar 

  11. Joyner, D., Čertík, O., Meurer, A., Granger, B.E.: Open source computer algebra systems: SymPy. ACM Commun. Comput. Algebra 45(4), 225–234 (2011)

    Google Scholar 

  12. Kluyver, T., et al.: Jupyter notebooks–a publishing format for reproducible computational workflows. In: Loizides, F., Schmidt, B. (eds.) Positioning and Power in Academic Publishing: Players, Agents, and Agendas, pp. 87–90. IOS Press, Amsterdam (2016)

    Google Scholar 

  13. Kowalewski, C.: Acceleration de la convergence pour certaines suites a convergence logarithmique. In: de Bruin, M.G., van Rossum, H. (eds.) Padé Approximation and its Applications Amsterdam 1980. LNM, vol. 888, pp. 263–272. Springer, Heidelberg (1981). https://doi.org/10.1007/BFb0095592

    Chapter  Google Scholar 

  14. Kuvychko, I.: Complexplorer. https://github.com/kuvychko/complexplorer

  15. Mizutani, T., Takeda, A.: DEMiCs: a software package for computing the mixed volume via dynamic enumeration of all mixed cells. In: Stillman, M., Takayama, N., Verschelde, J. (eds.) Software for Algebraic Geometry. The 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  Google Scholar 

  16. Ojika, T.: Modified deflation algorithm for the solution of singular problems. I. A system of nonlinear algebraic equations. J. Math. Anal. Appl. 123, 199–221 (1987)

    Google Scholar 

  17. Otto, J., Forbes, A., Verschelde, J.: Solving polynomial systems with phcpy. In: Proceedings of the 18th Python in Science Conference, pp. 563–582 (2019)

    Google Scholar 

  18. Sidi, A.: Practical Extrapolation Methods. Theory and Applications, Cambridge Monographs on Applied and Computational Mathematics, vol. 10. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  19. Telen, S., Van Barel, M., Verschelde, J.: A robust numerical path tracking algorithm for polynomial homotopy continuation. SIAM J. Sci. Comput. 42(6), A3610–A3637 (2020)

    Article  MathSciNet  Google Scholar 

  20. Trefethen, L.N.: Approximation Theory and Approximation Practice. Extented edn. SIAM (2020)

    Google Scholar 

  21. Trefethen, L.N.: Numerical analytic continuation. Jpn. J. Ind. Appl. Math. 40(3), 1587–1636 (2023)

    Article  MathSciNet  Google Scholar 

  22. Verschelde, J.: Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw. 25(2), 251–276 (1999). https://github.com/janverschelde/PHCpack

  23. Verschelde, J.: Modernizing PHCpack through phcpy. In: de Buyl, P., Varoquaux, N. (eds.) Proceedings of the 6th European Conference on Python in Science (EuroSciPy 2013), pp. 71–76 (2014)

    Google Scholar 

  24. Verschelde, J.: Exporting Ada software Julia and Python. Ada User J. 43(1), 75–77 (2022)

    Google Scholar 

  25. Verschelde, J., Viswanathan, K.: Extrapolating on Taylor series solutions of homotopies with nearby poles. arXiv:2404.17681v1 [math.NA]. Accessed 26 Apr 2024

  26. Verschelde, J., Viswanathan, K.: Locating the closest singularity in a polynomial homotopy. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds.) CASC 2022. LNCS, vol. 13366, pp. 333–352. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-14788-3_19

    Chapter  Google Scholar 

  27. Wegert, E.: Visual Complex Functions: An Introduction with Phase Portraits. Birkhäuser (2012)

    Google Scholar 

  28. Weniger, E.: Nonlinear sequence transformations for the acceleration of convergence and the summation of series. Comput. Phys. Rep. 10, 189–371 (1989)

    Article  Google Scholar 

  29. Wynn, P.: On a procrustean technique for the numerical transformation of slowly convergent sequences and series. Proc. Camb. Phil. Soc. 52, 663–672 (1956)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    Jan Verschelde & Kylash Viswanathan

Authors
  1. Jan Verschelde
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  2. Kylash Viswanathan
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Correspondence to Jan Verschelde .

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Editors and Affiliations

  1. Imperial College London, London, UK

    Kevin Buzzard

  2. University of Buenos Aires, Buenos Aires, Argentina

    Alicia Dickenstein

  3. TU Braunschweig, Braunschweig, Germany

    Bettina Eick

  4. Georgia Institute of Technology, Atlanta, GA, USA

    Anton Leykin

  5. Durham University, Durham, UK

    Yue Ren

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Verschelde, J., Viswanathan, K. (2024). Extrapolating Solution Paths of Polynomial Homotopies Towards Singularities with PHCpack and Phcpy. In: Buzzard, K., Dickenstein, A., Eick, B., Leykin, A., Ren, Y. (eds) Mathematical Software – ICMS 2024. ICMS 2024. Lecture Notes in Computer Science, vol 14749. Springer, Cham. https://doi.org/10.1007/978-3-031-64529-7_37

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  • DOI: https://doi.org/10.1007/978-3-031-64529-7_37

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