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Efficient path tracking methods

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Abstract

Path tracking is the fundamental computational tool in homotopy continuation and is therefore key in most algorithms in the emerging field of numerical algebraic geometry. Though the basic notions of predictor-corrector methods have been known for years, there is still much to be considered, particularly in the specialized algebraic setting of solving polynomial systems. In this article, the effects of the choice of predictor method on the performance of a tracker is analyzed, and details for using Runge-Kutta methods in conjunction with adaptive precision are provided. These methods have been implemented in the Bertini software package, and several examples are described.

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References

  1. Allgower, E.L., Georg, K.: Numerical continuation methods, An introduction. Springer Series in Computational Mathematics, vol. 13. Springer, Berlin (1990)

    Google Scholar 

  2. Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Bertini: Software for Numerical Algebraic Geometry. Available at http://www.nd.edu/∼sommese/bertini

  3. Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Adaptive multiprecision path tracking. SIAM J. Numer. Anal., 46, 722–746 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Stepsize control for adaptive multiprecision path tracking. Contemp. Math. 496, 21–31 (2009)

    MathSciNet  Google Scholar 

  5. Cash, J.R., Karp, A.H.: A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides. ACM Trans. Math. Software, 16(3), 201–222 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Enright, W.H., Jackson, K.R., Nørsett, S.P., Thomsen, P.G.: Interpolants for Runge-Kutta formulas. ACM Trans. Math. Software, 12(3), 193–218 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fehlberg, E.: Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweiten-Kontrolle. Computing (Arch. Elektron. Rechnen), 4, 93–106 (1969)

    MathSciNet  MATH  Google Scholar 

  8. Fehlberg, E.: Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme. Computing (Arch. Elektron. Rechnen), 6, 65–71 (1970)

    MathSciNet  Google Scholar 

  9. Kincaid, D., Cheney, W.: Numerical Analysis: Mathematics of Scientific Computing, 3rd edn. Brooks/Cole Publishing Co., Pacific Grove, CA (2002)

  10. Li, T.Y.: Numerical solution of polynomial systems by homotopy continuation methods. In: Cucker, F. (ed.) Handbook of Numerical Analysis, Volume XI, Special Volume: Foundations of Computational Mathematics, North-Holland, pp. 209–304 (2003)

  11. Morgan, A.P.: Solving polynomial systems using continuation for engineering and scientific problems. Prentice Hall Inc., Englewood Cliffs, NJ (1987) Reprinted as Classics in Applied Mathematics (2009) 57, SIAM

  12. Morgan, A.P., Sommese, A.J.: Computing all solutions to polynomial systems using homotopy continuation. Appl. Math. Comput. 24(2), 115–138 (1987) Errata: Appl. Math. Comput., 51, 209 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Prince, P.J., Dormand, J.R.: High order embedded Runge-Kutta formulae. J. Comput. Appl. Math. 7(1), 67–75 1981

    Article  MathSciNet  MATH  Google Scholar 

  14. Shampine, L.F.: Numerical Solution of Ordinary Differential Equations. Chapman & Hall, New York (1994)

    MATH  Google Scholar 

  15. Sommese, A.J., Wampler, C.W.: The Numerical Solution of Systems of Polynomials Arising in Engineering and Science. World Scientific, Singapore (2005)

    Book  MATH  Google Scholar 

  16. Verner, J.H.: Explicit Runge-Kutta methods with estimates of the local truncation error. SIAM J. Numer. Anal. 15(4), 772–790 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wampler, C.W., Morgan, A., Sommese, A.J.: Complete solution of the nine-point path synthesis problem for four-bar linkages. ASME J. Mech. Des. 114(1), 153–159 (1992)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Colorado State University, Fort Collins, CO, 80523, USA

    Daniel J. Bates

  2. Department of Mathematics, Texas A&M University, College Station, Mailstop 3368, TX, 77843, USA

    Jonathan D. Hauenstein

  3. Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Andrew J. Sommese

Authors
  1. Daniel J. Bates
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  2. Jonathan D. Hauenstein
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  3. Andrew J. Sommese
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Corresponding author

Correspondence to Daniel J. Bates.

Additional information

D. J. Bates was supported by the Institute for Mathematics and its Applications (IMA), Colorado State University, and NSF DMS-0914674.

J. D. Hauenstein was supported by the Fields Institute, Texas A&M University, the Duncan Chair of the University of Notre Dame, and NSF grant DMS-0712910.

A. J. Sommese was supported by the Duncan Chair of the University of Notre Dame and NSF grant DMS-0712910.

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Bates, D.J., Hauenstein, J.D. & Sommese, A.J. Efficient path tracking methods. Numer Algor 58, 451–459 (2011). https://doi.org/10.1007/s11075-011-9463-8

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  • DOI: https://doi.org/10.1007/s11075-011-9463-8

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