Advertisement

BIT Numerical Mathematics

, Volume 15, Issue 4, pp 362–366 | Cite as

Contraction mappings in interval analysis

  • Ole Caprani
  • Kaj Madsen
Article

Abstract

Under certain conditions, the contraction mapping fixed point theorem guarantees the convergence of the iterationxi+1=f(x i ) toward a fixed point of the functionf:RnRn. When an interval extensionF off is used in a similar iteration scheme to obtain a sequence of interval vectors these conditions need not provide convergence to a degenerate interval vector representing the fixed point, even if the width of the initial interval vector is chosen arbitrarily small. We give a sufficient condition on the extensionF in order that the convergence is guaranteed. The centered form of Moore satisfies this condition.

Keywords

Computational Mathematic Point Theorem Fixed Point Theorem Contraction Mapping Interval Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. M. Ortega, W. C. Reinboldt,Iterative solution of nonlinear equations in several variables, Academic Press, 1970.Google Scholar
  2. 2.
    R. E. Moore,Interval Analysis, Prentice-Hall, 1966.Google Scholar
  3. 3.
    E. R. Hansen,Topics in Interval Analysis, Oxford University Press, 1969.Google Scholar
  4. 4.
    E. R. Hansen,A generalized interval arithmetic, inInterval Mathematics, ed. K. Nickel, Springer Verlag, 1975.Google Scholar

Copyright information

© BIT Foundations 1975

Authors and Affiliations

  • Ole Caprani
    • 1
    • 2
  • Kaj Madsen
    • 1
    • 2
  1. 1.Datalogisk InstitutKøbenhavn nDenmark
  2. 2.Numerisk Institut Danmarks Tekniske HøjskoleLyngbyDenmark

Personalised recommendations