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

Vasile Berinde; Abdul Rahim Khan; Mădălina Păcurar

doi:10.33993/jnaat441-1059

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: 423

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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