Analytic and empirical study of the rate of convergence of some iterative methods

Authors

  • Vasile Berinde North University of Baia Mare, Romania
  • Abdul Rahim Khan King Fahd University of Petroleum and Minerals, Saudi Arabia
  • Mădălina Păcurar Babeş-Bolyai University, Romania

DOI:

https://doi.org/10.33993/jnaat441-1059

Keywords:

metric space, contractive mapping, fixed point, k-step fixed point, iterative method, rate of convergence
Abstract views: 375

Abstract

We study analytically and empirically the rate of convergence of two \(k\)-step fixed point iterative methods in the family of methods
\[
x_{n+1}=T(x_{i_0+n-k+1},x_{i_1+n-k+1},\dots, x_{{i_{k-1}+n-k+1}}),\,n\geq k-1,
\]

where \(T\colon X^k\rightarrow X\) is a mapping satisfying some Presic type contraction conditions and \((i_0,i_1,\dots,i_{k-1})\) is a permutation of  \((0,1,\dots,k-1)\).   We also consider the Picard iteration associated to the fixed point problem  \(x=T(x,\dots,x)\) and compare analytically and empirically the rate and speed of convergence of the three iterative methods. Our approach opens a new perspective on the study of the rate of convergence / speed of convergence of fixed point iterative methods and also illustrates the essential difference between them by means of some concrete numerical experiments.

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Published

2015-12-18

How to Cite

Berinde, V., Khan, A. R., & Păcurar, M. (2015). Analytic and empirical study of the rate of convergence of some iterative methods. J. Numer. Anal. Approx. Theory, 44(1), 25–37. https://doi.org/10.33993/jnaat441-1059

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