BIT Numerical Mathematics

, Volume 45, Issue 3, pp 561–591 | Cite as

Solving Differential-Algebraic Equations by Taylor Series (I): Computing Taylor Coefficients

  • Nedialko S. NedialkovEmail author
  • John D. PryceEmail author


This paper is one of a series underpinning the authors’ DAETS code for solving DAE initial value problems by Taylor series expansion. First, building on the second author’s structural analysis of DAEs (BIT, 41 (2001), pp. 364–394), it describes and justifies the method used in DAETS to compute Taylor coefficients (TCs) using automatic differentiation. The DAE may be fully implicit, nonlinear, and contain derivatives of order higher than one. Algorithmic details are given.

Second, it proves that either the method succeeds in the sense of computing TCs of the local solution, or one of a number of detectable error conditions occurs.

Key words

differential-algebraic equations (DAEs) structural analysis Taylor series automatic differentiation 


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  1. 1.
    R. Barrio, Performance of the Taylor series method for ODEs/DAEs, Appl. Math. Comput., 163 (2005), pp. 525–545.Google Scholar
  2. 2.
    R. Barrio, F. Blesa, and M. Lara, VSVO formulation of the Taylor method for the numerical solution of ODEs, preprint, University of Zaragoza, submitted for publication, 2005.Google Scholar
  3. 3.
    D. Barton, I. M. Willers, and R. V. M. Zahar, The automatic solution of ordinary differential equations by the method of Taylor series, Comput. J., 14 (1970), pp. 243–248.Google Scholar
  4. 4.
    C. Bendsten and O. Stauning, TADIFF, a flexible C++ package for automatic differentiation using Taylor series, Technical Report 1997-x5-94, Department of Mathematical Modelling, Technical University of Denmark, DK-2800, Lyngby, Denmark, April 1997.Google Scholar
  5. 5.
    M. Berz, COSY INFINITY version 8 reference manual, Technical Report MSUCL–1088, National Superconducting Cyclotron Lab., Michigan State University, East Lansing, Mich., 1997.Google Scholar
  6. 6.
    S. L. Campbell and C. W. Gear, The index of general nonlinear DAEs, Numer. Math., 72 (1995), pp. 173–196.Google Scholar
  7. 7.
    Y. F. Chang and G. F. Corliss, ATOMFT: Solving ODEs and DAEs using Taylor series, Comput. Math. Appl., 28 (1994), pp. 209–233.Google Scholar
  8. 8.
    G. F. Corliss and W. Lodwick, Role of constraints in the validated solution of DAEs, Technical Report 430, Marquette University Department of Mathematics, Statistics, and Computer Science, Milwaukee, Wisc., March 1996.Google Scholar
  9. 9.
    J. J. B. de Swart, W. M. Lioen, and W. A. van der Veen, PSIDE—parallel software for implicit differential equations, December 1997. Scholar
  10. 10.
    A. Gofen, The Taylor Center for PCs: exploring, graphing and integrating ODEs with the ultimate accuracy, in Computational Science: ICCS 2002, P. Sloot et al., eds., Lect. Notes Comput. Sci., vol. 2329, Springer, Amsterdam, 2002.Google Scholar
  11. 11.
    G. H. Golub and C. F. V. Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, 3rd ed., 1996.Google Scholar
  12. 12.
    A. Griewank, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Frontiers in Applied Mathematics, SIAM, Philadelphia, PA, 2000.Google Scholar
  13. 13.
    A. Griewank, D. Juedes, and J. Utke, ADOL-C, a package for the automatic differentiation of algorithms written in C/C++, ACM Trans. Math. Softw., 22 (1996), pp. 131–167.Google Scholar
  14. 14.
    J. Hoefkens, Rigorous Numerical Analysis with High-Order Taylor Models, PhD thesis, Department of Mathematics and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, August 2001.Google Scholar
  15. 15.
    F. Iavernaro and F. Mazzia, Block-boundary value methods for the solution of ordinary differential equation, SIAM J. Sci. Comput., 21 (1999), pp. 323–339. GAMD web site is mazzia/ode/gamd.html.Google Scholar
  16. 16.
    K. R. Jackson and N. S. Nedialkov, Some recent advances in validated methods for IVPs for ODEs, Appl. Numer. Math., 42 (2002), pp. 269–284.Google Scholar
  17. 17.
    R. Jonker and A. Volgenant, A shortest augmenting path algorithm for dense and sparse linear assignment problems, Computing, 38 (1987), pp. 325–340. The assignment code is available at Scholar
  18. 18.
    A. Jorba and M. Zou, A software package for the numerical integration of ODE by means of high-order Taylor methods, Technical Report, Department of Mathematics, University of Texas at Austin, TX 78712-1082, USA, 2001.Google Scholar
  19. 19.
    C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, Frontiers in Applied Mathematics, vol. 16, SIAM, Philadelphia, 1995.Google Scholar
  20. 20.
    W. M. Lioen and J. J. B. de Swart, Test set for initial value problem solvers, Technical Report MAS-R9832, CWI, Amsterdam, The Netherlands, December 1998. Scholar
  21. 21.
    F. Mazzia and F. Iavernaro, Test set for initial value problem solvers, Technical Report 40, Department of Mathematics, University of Bari, Italy, 2003. testset/.Google Scholar
  22. 22.
    R. E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1966.Google Scholar
  23. 23.
    N. S. Nedialkov, Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation, PhD thesis, Department of Computer Science, University of Toronto, Toronto, Canada, M5S 3G4, February 1999.Google Scholar
  24. 24.
    N. S. Nedialkov, K. R. Jackson, and G. F. Corliss, Validated solutions of initial value problems for ordinary differential equations, Appl. Math. Comput., 105 (1999), pp. 21–68.Google Scholar
  25. 25.
    N. S. Nedialkov and J. D. Pryce, Solving differential-algebraic equations by Taylor series (II): Computing the System Jacobian, submitted to BIT, 2005.Google Scholar
  26. 26.
    C. C. Pantelides, The consistent initialization of differential-algebraic systems, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 213–231.Google Scholar
  27. 27.
    J. D. Pryce, Solving high-index DAEs by Taylor series, Numer. Algorithms, 19 (1998), pp. 195–211.Google Scholar
  28. 28.
    J. D. Pryce, A simple structural analysis method for DAEs, BIT, 41 (2001), pp. 364–394.Google Scholar
  29. 29.
    L. B. Rall, Automatic Differentiation: Techniques and Applications, Lect. Notes Comput. Sci., vol. 120, Springer, Berlin, 1981.Google Scholar
  30. 30.
    O. Stauning and C. Bendtsen, FADBAD++ web page, May 2003. FADBAD++ is availabe at Scholar
  31. 31.
    A. Wächter, An Interior Point Algorithm for Large-Scale Nonlinear Optimization with Applications in Process Engineering, PhD thesis, Carnegie Mellon University, Pittsburgh, PA, 2002.Google Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Department of Computing and SoftwareMcMaster UniversityHamiltonCanada
  2. 2.Computer Information Systems Engineering DepartmentCranfield UniversitySwindonUK

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