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Convergence of Iterations

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Computer Aided Systems Theory – EUROCAST 2005 (EUROCAST 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3643))

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Abstract

Convergence is a central problem in both computer science and in population biology.

Will a program terminate? Will a population go to an equilibrium?

In general these questions are quite difficult – even unsolvable.

In this paper we will concentrate on very simple iterations of the form

x t + 1 = f(x t )

where each x t is simply a real number and f(x) is a reasonable real function with a single fixed point. For such a system, we say that an iteration is “globally stable” if it approaches the fixed point for all starting points. We will show that there is a simple method which assures global stability. Our method uses bounding of f(x) by a self-inverse function. We call this bounding “enveloping” and we show that enveloping implies global stability. For a number of standard population models, we show that local stability implies enveloping by a self-inverse linear fractional function and hence global stability. We close with some remarks on extensions and limitations of our method.

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Author information

Authors and Affiliations

  1. Computer Science Dept., Oregon State University, Corvallis, OR, 97331, USA

    Paul Cull

Authors
  1. Paul Cull
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Editor information

Editors and Affiliations

  1. Instituto Universitario de Ciencias y Tecnologícas Cibernéticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira, 35017, Las Palmas de Gran Canaria,, Las Palmas, Spain

    Roberto Moreno Díaz

  2. Systems Theory, Johannes Kepler University Linz, Altenbergerstrasse 69, A-4040, Linz, Austria

    Franz Pichler

  3. Universidad de Las Palmas de Gran Canaria, Instituto Universitario de Ciencias y Tecnologícas Cibernéticas, Campus de Tafira, 35017, Las Palmas de Gran Canaria, Las Palmas, Spain

    Alexis Quesada Arencibia

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© 2005 Springer-Verlag Berlin Heidelberg

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Cull, P. (2005). Convergence of Iterations. In: Moreno Díaz, R., Pichler, F., Quesada Arencibia, A. (eds) Computer Aided Systems Theory – EUROCAST 2005. EUROCAST 2005. Lecture Notes in Computer Science, vol 3643. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11556985_59

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  • DOI: https://doi.org/10.1007/11556985_59

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-29002-5

  • Online ISBN: 978-3-540-31829-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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