Efficient path tracking methods

Numerical Algorithms

Volumn 58, Issue 4, 2011, Pages 451-459

  Bates, Daniel J ^a   Hauenstein, Jonathan D ^b   Sommese, Andrew J ^c  

^a Veterinary Dentistry and Oral Surgery   (United States)

^b TEXAS A AND M UNIVERSITY   (United States)

^c UNIVERSITY OF NOTRE DAME   (United States)

Author keywords

Adaptive precision; Euler's method; Homotopy continuation; Numerical algebraic geometry; Ordinary differential equations; Path tracking; Polynomial systems; Precision; Runge Kutta methods

Indexed keywords

EID: 85027918961     PISSN: 10171398     EISSN: 15729265     Source Type: Journal    
DOI: 10.1007/s11075-011-9463-8     Document Type: Article

References (17)

1. Allgower, E. L., Georg, K.: Numerical continuation methods, An introduction. Springer Series in Computational Mathematics, vol. 13. Springer, Berlin (1990).
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).
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).
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).
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).
7. Fehlberg, E.: Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweiten-Kontrolle. Computing (Arch. Elektron. Rechnen), 4, 93-106 (1969).
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).
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).
13. Prince, P. J., Dormand, J. R.: High order embedded Runge-Kutta formulae. J. Comput. Appl. Math. 7(1), 67-75 1981.
14. Shampine, L. F.: Numerical Solution of Ordinary Differential Equations. Chapman & Hall, New York (1994).
15. Sommese, A. J., Wampler, C. W.: The Numerical Solution of Systems of Polynomials Arising in Engineering and Science. World Scientific, Singapore (2005).
16. Verner, J. H.: Explicit Runge-Kutta methods with estimates of the local truncation error. SIAM J. Numer. Anal. 15(4), 772-790 (1978).
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).