Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Relative error propagation in the recursive solution of linear recurrence relations

  • 196 Accesses

  • 22 Citations

Abstract

This paper is concerned with the numerical solution of the general initial value problem for linear recurrence relations. An error analysis of direct recursion is given, based on relative rather than absolute error, and a theory of relative stability developed.Miller's algorithm for second order homogeneous relations is extended to more general cases, and the propagation of errors analysed in a similar manner. The practical significance of the theoretical results is indicated by applying them to particular classes of problem.

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

References

  1. [1]

    British Association for the Advancement of Science: Bessel functions, Part II. Mathematical Tables, vol. X. Cambridge: Cambridge University Press 1952.

  2. [2]

    Fort, T.: Finite differences. Oxford: Oxford University Press 1948.

  3. [3]

    Gautschi, W.: Recursive computation of certain integrals. J. Assoc. Comp. Mach.8, 21–40 (1960).

  4. [4]

    Metropolis, N.: Algorithms in unnormalized arithmetic. I. Recurrence relations. Numer. Math.7, 104–112 (1965).

  5. [5]

    Mitchell, A. R., andJ. W. Craggs: Stability of difference relations in the solution of ordinary differential equations. M.T.A.C.7, 127–129 (1953).

  6. [6]

    Oliver, J.: The numerical solution of the initial value problem for linear recurrence relations. Ph. D. Thesis, University of Cambridge 1965.

  7. [7]

    — A general algorithm for the numerical solution of linear recurrence relations. To appear.

  8. [8]

    Olver, F. W. J.: Error analysis ofMiller's recurrence algorithm. Math. Comp.18, 65–74 (1964).

  9. [9]

    Watson, G. N.: A treatise on the theory of Bessel functions, p. 328, 2nd Ed. Cambridge: Cambridge University Press 1958.

  10. [10]

    Wilkinson, J. H.: Rounding errors in algebraic processes. London: Her Majesty's Stationery Office 1963.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Oliver, J. Relative error propagation in the recursive solution of linear recurrence relations. Numer. Math. 9, 323–340 (1967). https://doi.org/10.1007/BF02162423

Download citation

Keywords

  • Relative Error
  • Theoretical Result
  • Mathematical Method
  • Absolute Error
  • Error Analysis