Skip to main content
SpringerLink
Log in

A New View of the Computational Complexity of IVP for ODE

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

The cost of solving an initial value problem for ordinary differential equations to accuracy 2 is polynomial in ln ε. Adaptive step-size control is never theoretically more costly than fixed step-size control, and can be an unbounded factor less costly. These results contradict the standard theory, but are based on more realistic assumptions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. U. Ascher, R.M.M. Mattheij and R.D. Russell, The Numerical Solution of Two-Point Boundary-Value Problems (Prentice-Hall, 1988, Englewood Clifs, NJ, reprinted as a SIAM Classic, Vol. 13).

    Google Scholar 

  2. J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods (Wiley, New York, 1987).

    Google Scholar 

  3. R.M. Corless, An elementary solution of a minimax problem arising in algorithms for automatic mesh selection, SIGSAM Bulletin: Commun. Comput. Algebra 34(4) (2001) 7–15.

    Google Scholar 

  4. W.H. Enright, Analysis of error control strategies for continuous Runge-Kutta methods, SIAM J. Numer. Anal. 26(3) (1989) 588–599.

    Google Scholar 

  5. W.H. Enright, A new error control for initial value solvers, Appl. Math. Comput. 31 (1989) 288–301.

    Google Scholar 

  6. A. Griewank, ODE Solving via Automatic Differentiation and Rational Prediction, Pitman Research Notes in Mathematics, Vol. 344 (Addison-Wesley/Longman, 1995).

  7. E. Hairer, S.P. N¸rsett and G. Wanner, Solving Ordinary Differential Equations I, Computational Mathematics, Vol. 8 (Springer, Berlin, 1987).

    Google Scholar 

  8. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Computational Mathematics, Vol. 14 (Springer, Berlin, 1991).

    Google Scholar 

  9. G.H. Hardy, J. Littlewood and G. Pólya, Inequalities (Cambridge Univ. Press, Cambridge, 1952).

    Google Scholar 

  10. N.S. Nedialkov and K.R. Jackson, An interval Hermite-Obreschkoff method for computing rigorous bounds on the solution of an inital value problem for an ordinary differential equation, Reliable Computing (1999) 289-310.

  11. G. Söderlind, Automatic control and adaptive time-stepping, Numer. Algorithms 31 (2002) 281–310.

    Google Scholar 

  12. A.G. Werschulz, The Computational Complexity of Differential and Integral Equations (Oxford Science, Oxford, 1991).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ontario Research Centre for Computer Algebra, Deptartment Applied Mathematics, University of Western Ontario, London, Canada, N6A 5B7

    Robert M. Corless

Authors
  1. Robert M. Corless
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Corless, R.M. A New View of the Computational Complexity of IVP for ODE. Numerical Algorithms 31, 115–124 (2002). https://doi.org/10.1023/A:1021108323034

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021108323034

Search

Navigation