A software package for the numerical integration of odes by means of high-order taylor methods

Experimental Mathematics

Volumn 14, Issue 1, 2005, Pages 99-117

  Jorba, Àngel ^a   Zou, Maorong ^b  

^a UNIVERSITY OF BARCELONA   (Spain)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

Automatic differentiation; Software; Taylor method

Indexed keywords

EID: 22544468094     PISSN: 10586458     EISSN: 1944950X     Source Type: Journal    
DOI: 10.1080/10586458.2005.10128904     Document Type: Article

References (38)

1. Alfred V. Aho, Ravi Sethi, and Jeffrey D. Ullman. Compilers, Principles, Techniques, and Tools. Reading, MA: Addison-Wesley Publishing Company, 1986
2. D. H. Bailey, Y. Hida, K. Jeyabalan, X. S. Li, and B. Thompson. “High-Precision Software Directory.” Available from World Wide Web (http://www.nersc.gov/∼dhbailey/mpdist/mpdist.html), 2005.
3. D. Barton. “On Taylor Series and Stiff Equations.” ACM Trans. Math. Software 6: 3 (1980), 280–294.
4. D. Barton, I. M. Willers, and R. V. M. Zahar. “The Automatic Solution of Ordinary Differential Equations by the Method of Taylor Series.” Computer J. 14: 3 (1970), 243–248.
5. L. M. Beda, L. N. Korolev, N. V. Sukkikh, and T. S. Frolova. “Programs for Automatic Differentiation for the Machine BESM.” Technical Report, Institute for Precise Mechanics and Computation Techniques, Academy of Science, Moscow, USSR, 1959. (In Russian).
6. M. Berz, C. Bischof, G. F. Corliss, and A. Griewank, editors. Computational Differentiation: Techniques, Applications, and Tools. Philadelphia, Penn.: SIAM, 1996.
7. C. H. Bischof, A. Carle, G. F. Corliss, and A. Griewank. “ADIFOR: Automatic Differentiation in a Source Translation Environment.” In Proceedings of the International Symposium on Symbolic and Algebraic Computation, edited by Paul S. Wang, pp. 294–302. New York: ACM Press, 1992.
8. K. Briggs. “The Doubledouble Homepage.” Available from World Wide Web (http://members.lycos.co.uk/keithmbriggs/doubledouble.html), 2002.
9. R. Broucke. “Solution of the N-Body Problem with Recurrent Power Series.” Celestial Mech. 4: 1 (1971), 110–115.
10. Y. F. Chang and G. F. Corliss. “ATOMFT: Solving ODEs and DAEs Using Taylor Series.” Computers and Mathematics with Applications 28 (1994), 209–233.
11. G. F. Corliss. “Guaranteed Error Bounds for Ordinary Differential Equations.” In Theory of Numerics in Ordinary and Partial Differential Equations, edited by M. Ainsworth, J. Levesley, W. A. Light, and M. Marletta, pp. 1–75. Oxford: Oxford University Press, 1995.
12. G. F. Corliss and Y. F. Chang. “Solving Ordinary Differential Equations Using Taylor Series.” ACM Trans. Math. Software 8: 2 (1982), 114–144.
13. G. F. Corliss, A. Griewank, P. Henneberger, G. Kirlinger, F. A. Potra, and H. J. Stetter. “High-Order Stiff ODE Solvers via Automatic Differentiation and Rational Prediction.” In Numerical Analysis and Its Applications (Rousse, 1996), pp. 114–125. Berlin: Springer, 1997.
14. A. Delshams and R. Ramírez- Ros. “Singular Separatrix Splitting and the Melnikov Method: An Experimental Study.” Exp. Math. 8: 1 (1999), 29–48.
15. A. Delshams and T. M. Seara. “Splitting of Separatrices in Hamiltonian Systems with One and a Half Degrees of Freedom.” Math. Phys. Electron. J. 3 (1997), Paper 4, 40 pp.
16. A. Gibbons. “A Program for the Automatic Integration of Differential Equations Using the Method of Taylor Series.” Comp. J. 3 (1960), 108–111.
17. GNU Multiple Precision Library. “GMP Arithmetic without Limitations.” Available from World Wide Web (http://www.swox.com/gmp/), 2005.
18. A. Griewank. Evaluating Derivatives. Philadelphia, Penn.: SIAM, 2000.
19. A. Griewank and G. F. Corliss, editors. Automatic Differentiation of Algorithms: Theory, Implementation, and Application. Philadelphia, Penn.: SIAM, 1991.
20. E. Hairer, S. P. Norsett, and G. Wanner. Solving Ordinary Differential Equations I. Nonstiff Problems, Springer Series in Computational Mathematics, 8, Second revised edition. Berlin: Springer-Verlag, 2000.
21. J. Hoefkens. Rigorous Numerical Analysis with High Order Taylor Methods. PhD. diss., Michigan State University, 2001.
22. D. H. Irvine and M. A. Savageau. “Efficient Solution of Nonlinear Ordinary Differential Equations Expressed in S-System Canonical Form.” SIAM J. Numer. Anal. 27: 3 (1990), 704–735.
23. F. Jalbert and R. V. M. Zahar. “A Highly Precise Taylor Series Method for Stiff ODEs.” In Proceedings of the Fourteenth Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1984), Congressus Numerantium 46 (1985), 347–358.
24. G. Kirlinger and G. F. Corliss. “On Implicit Taylor Series Methods for Stiff ODEs.” In Computer Arithmetic and Enclosure Methods, edited by L. Atanassova and J. Herzberger, pp. 371–379. Amsterdam: North-Holland, 1992.
25. R. Martínez and C. Simó. “Simultaneous Binary Collisions in the Planar Four-Body Problem.” Nonlinearity 12: 4 (1999), 903–930.
26. K. R. Meyer and G. R. Hall. Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. New York: Springer, 1992.
27. R. E. Moore. Interval Analysis. Englewood Cliffs, N.J.: Prentice-Hall, 1966.
28. N. S. Nedialkov, K. R. Jackson, and G. F. Corliss. “Validated Solutions of Initial Value Problems for Ordinary Differential Equations.” Appl. Math. Comput. 105: 1 (1999), 21–68.
29. L. B. Rall. Automatic Differentiation: Techniques and Applications, Lecture Notes in Computer Science, 120. Berlin: Springer Verlag, 1981.
30. M. A. Savageau and E. O. Voit. “Recasting Nonlinear Differential Equations as S-Systems: A Canonical Nonlinear Form.” Math. Biosci. 87: 1 (1987), 83–115.
31. C. Simó. “Global Dynamics and Fast Indicators.” In Global Analysis of Dynamical Systems, edited by H. W. Broer, B. Krauskopf, and G. Vegter, pp. 373–389. Bristol: IOP Publishing, 2001.
32. C. Simó. “Dynamical Properties of the Figure Eight Solution of the Three-Body Problem.” In Celestial Mechanics (Evanston, IL, 1999), edited by A. Chenciner, R. Cushman, C. Robinson, and Z. Xia, pp. 209–228, Contemporary Mathematics, 292. Providence, RI: Amer. Math. Soc., 2002.
33. C. Simó and C. Valls. “A Formal Approximation of the Splitting of Separatrices in the Classical Arnold’s Example of Diffusion with Two Equal Parameters.” Nonlinearity 14: 4 (2001), 1707–1760.
34. J. F. Steffensen. “On the Restricted Problem of Three Bodies.” Danske Vid. Selsk. Mat.-Fys. Medd. 30: 18 (1956), 17.
35. J. F. Steffensen. “On the Problem of Three Bodies in the Plane.” Mat.-Fys. Medd. Danske Vid. Selsk. 31: 3 (1957), 18.
36. V. Szebehely. Theory of Orbits. New York: Academic Press, 1967.
37. R. E. Wengert. “A Simple Automatic Derivative Evaluation Program.” Comm. ACM 7: 8 (1964), 463–464.
38. M. Zou and A. Jorba. “Taylor 1.4.” Available from World Wide Web in the United States (http://www.ma.utexas.edu/∼mzou/taylor/) and in Europe (http: //www.maia.ub.es/∼angel/taylor/), 2001