Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On the R-order of some recurrences with applications to inclusion-methods II

Über die R-Ordnung einiger Rekursionsfolgen bei Einschließungsverfahren II

  • 12 Accesses

  • 5 Citations


In this note we are considering a class of recurrences which is a generalization of those in [3]. Estimates for theirR-order are derived and the results are then applied to a family of higher-order interval methods for the inclusion of the inverse of a matrix. An efficient algorithm is determined.


Wir betrachten hier eine Klasse von Rekursionsfolgen, die eine Verallgemeinerung von den in [3] behandelten ist. Dazu werden Abschätzungen für dieR-Ordnung hergeleitet und dann auf eine Klasse von Iterationsverfahren zur Einschließung der Inversen einer Matrix angewendet.

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


  1. [1]

    Albrecht, J.: Bemerkungen zum Iterationsverfahren von Schulz zur Matrixinversion. Z. Angew. Math. Mech.41, 262–263 (1961).

  2. [2]

    Alefeld, G., Herzberger, J.: Introduction to Interval Computations. New York: Academic Press 1983.

  3. [3]

    Herzberger, J.: On theR-order of some recurrences with applications to inclusion-methods. Computing36, 175–180 (1986).

  4. [4]

    Herzberger, J.: Über ein intervallmäßiges Newton-Verfahren. Z. Angew. Math. Mech. 66 (to appear).

  5. [5]

    Ortega, J. W., Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic Press 1970.

  6. [6]

    Schmidt, J. W.: On theR-order of coupled squences. Computing26, 333–342 (1981).

  7. [7]

    Traub, J. F.: Iterative methods for the solution of equations. Englewood Cliffs, N.J.: Prentice-Hall 1964.

Added in proof

  1. [8]

    Burmeister, W., Schmidt, J. W.: Characterization of the bestR-orders of coupled sequences arising in iterative processes. Numerical Methods and Applications '84, Sofia, 191–202 (1985).

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Herzberger, J. On the R-order of some recurrences with applications to inclusion-methods II. Computing 37, 255–259 (1986). https://doi.org/10.1007/BF02252516

Download citation

AMS Subject Classifications

  • 65 G 10
  • 65 H 05
  • 65 F 30

Key words

  • Order of convergence
  • interval methods