BIT Numerical Mathematics

, Volume 22, Issue 1, pp 79–93 | Cite as

Stable computation of solutions of unstable linear initial value recursions

  • R. M. M. Mattheij
Part II Numerical Mathematics

Abstract

An algorithm is given for approximating dominated solutions of linear recursions, when some initial conditions are given. The stability of this algorithm is investigated and expressions for the truncation and rounding errors are derived. A number of practical questions concerning the algorithm is considered, and several numerical examples sustain the theory.

Keywords

Computational Mathematic Rounding Error Practical Question Stable Computation Linear Recursion 

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References

  1. 1.
    M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions, Dover, New York, (1968).Google Scholar
  2. 2.
    P. Alfeld and J. D. Lambert,Correction in the dominant space: A numerical technique for a certain class of stiff initial value problems. Math. Comp. 31 (1977), 922–938.Google Scholar
  3. 3.
    J. R. Cash,Stable recursions with applications to the numerical solution of stiff systems, Academic Press, New York, (1979).Google Scholar
  4. 4.
    G. Dahlquist,A numerical method for some ordinary differential equations with large Lipschitz constants. In:Information Processing 68, 183–186. North Holland, Amsterdam (1968).Google Scholar
  5. 5.
    W. Gautschi,Computational aspects of three term recurrence relations, SIAM Review 9 (1967), 24–82.Google Scholar
  6. 6.
    G. Golub and J. H. Wilkinson,Ill-conditioned eigen-systems and the computation of the Jordan canonical form, SIAM Review 18 (1976), 578–619.Google Scholar
  7. 7.
    F. R. de Hoog and R. Weiss,An approximation theory for boundary value problems on infinite intervals. Computing 24 (1980), 227–239.Google Scholar
  8. 8.
    M. Lentini and H. B. Keller,Boundary value problems on semi-infinite intervals and their numerical solution, SIAM J. Numer. Anal. 17 (1980), 577–604.Google Scholar
  9. 9.
    R. M. M. Mattheij,Accurate estimates of solutions of second order recursions, Lin. alg. and its Applic. 12 (1975), 29–54.Google Scholar
  10. 10.
    R. M. M. Mattheij,Estimating and determining solutions of matrix vector recursions, Thesis, Utrecht (1977).Google Scholar
  11. 11.
    R. M. M. Mattheij,On approximating smooth solutions of linear singularly perturbed ODE. In:Proceedings of the Conference on the Numerical Analysis of Singular Perturbation Problems, ed. P. W. Hemker and J. J. H. Miller, 457–465, Ac. Press, London (1979).Google Scholar
  12. 12.
    R. M. M. Mattheij,Characterizations of dominant and dominated solutions of linear recursions, Numer. Math. 35 (1980), 441–442.Google Scholar
  13. 13.
    R. M. M. Mattheij,Stable computation of solutions of unstable linear initial value recursions, Report 8108, Mathematical Institute, Nijmegen, (1981).Google Scholar
  14. 14.
    R. M. M. Mattheij and A. van der Sluis,Error estimates for Miller's algorithm, Numer. Math. 26 (1976), 61–78.Google Scholar
  15. 15.
    F. W. J. Olver,Numerical solution of second order linear difference equations, J. Res. NBS 71B (1967), 111–129.Google Scholar
  16. 16.
    A. van der Sluis,Estimating the solutions of slowly varying recursions, SIAM J. Math. Anal. 7 (1976), 662–695.Google Scholar
  17. 17.
    J. H. Wilkinson,The Algebraic Eigenvalue Problem, Clarendon Press, Oxford (1965).Google Scholar

Copyright information

© BIT Foundations 1982

Authors and Affiliations

  • R. M. M. Mattheij
    • 1
  1. 1.Mathematisch Instituut Katholieke UniversiteitNijmegenThe Netherlands

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