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Analytic and empirical study of the rate of convergence of some iterative methods. (English) Zbl 1363.47110

Summary: We study analytically and empirically the rate of convergence of two \(k\)-step fixed point iterative methods in the family of methods \[ \qquad x_{n+1}=T(x_{i_0+n-k+1},x_{i_1+n-k+1},\dots,x_{i_{k-1}+n-k+1}),\quad n\geq k-1, {(1)} \] where \(T:X^k\to X\) is a mapping satisfying some Prešić 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 three iterative methods. Our approach opens a new perspective on the study of the rate of convergence resp. the speed of convergence of fixed point iterative methods and also illustrates the essential difference between them by means of some concrete numerical experiments.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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