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

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.

Keywords

• metric space
• contractive mapping
• fixed point
• k-step fixed point
• iterative method
• rate of convergence