Skip to main content
SpringerLink
Log in

Über Verfahren von Steffensen und Householder zur Konvergenzverbesserung von Iterationen

  • Published:
Zeitschrift für angewandte Mathematik und Physik ZAMP Aims and scope Submit manuscript

Summary

Steffensen discovered that the iteration byy=ϕ(x) can be considerably improved if ϕ(x) is replaced by the function

$$\Phi (y) = \frac{{y\varphi (\varphi (y)) - \varphi (y)^2 }}{{y - 2 \varphi (y) + \varphi (\varphi (y))}},$$

This procedure has been generalized byHouseholder, who starts fromtwo iterationsy=ϕ1 (x), y=ϕ2 (x) with a common fix-point and builds up the new iterating function

$$\Phi ^ * (y) = \frac{{y \varphi _1 (\varphi _2 (y)) - \varphi _1 (y)\varphi _2 (y)}}{{y - \varphi _1 (y) - \varphi _2 (y) + \varphi _1 (\varphi _2 (y))}}.$$

We prove under very general conditions that any fix-point of ϕ(x), resp. any common fix-point of ϕ1(x) and ϕ2(x) becomes a point of attraction for Steffensen's or Householder's iterating function, and compare both procedures.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Vergleiche zum BeispielFr. A. Willers,Methoden der praktischen Analysis, 2. Aufl. (Walter de Gruyter, Berlin 1950), S. 256.

    Google Scholar 

  2. J. F. Steffensen, Skand. Aktuarietidskr.16, 64–72 (1933).

    Google Scholar 

  3. Fr. A. Willers,Methoden der praktischen Analysis, 2. Aufl. (Walter de Gruyter, Berlin 1950), S. 259.

    Google Scholar 

  4. Fr. A. Willers, Z. angew. Math. Mech.28, 125–126 (1948).

    Google Scholar 

  5. A. S. Householder,Principles of Numerical Analysis (McGraw-Hill, New York 1953), S. 126–128.

    Google Scholar 

  6. A. C. Aitken,Studies in Practical Mathematics. II.The Evaluation of the Latent Roots and Latent Vectors of a Matrix, Proc. roy. Soc., Edinburgh,57, 291–295 (1937).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematische Anstalt der Universität Basel, Basel

    Alexander Ostrowski

Authors
  1. Alexander Ostrowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Mauro Picone zum 70. Geburtstag

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ostrowski, A. Über Verfahren von Steffensen und Householder zur Konvergenzverbesserung von Iterationen. Journal of Applied Mathematics and Physics (ZAMP) 7, 218–229 (1956). https://doi.org/10.1007/BF02044467

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02044467

Search

Navigation