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

Über die R-Ordnung einiger Rekursionsfolgen bei Einschließungsverfahren II

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Abstract

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.

Zusammenfassung

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.

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References

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

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

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

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

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

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

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

Added in proof

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    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).

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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

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AMS Subject Classifications

  • 65 G 10
  • 65 H 05
  • 65 F 30

Key words

  • Order of convergence
  • interval methods