Acceleration of nonhyperbolic sequences of Mann (Q1854172)

Two functions \( g \) and \( h \) having a common fixed point \( x^\ast \) may be composed by setting ( \( {\mathcal D} \) denotes differentiation) \[ H(x) = \{ h(x) - g(x) {\mathcal D} h(x) \} / \{ 1 - {\mathcal D} h(x) \} \] In this composition, \( h \) may be made to depend upon \( g \) by taking \( h(x) = x^\ast \) when \( x = x^\ast \) and \[ h(x) = \{ g(x) - x {\mathcal D} g(x) \} / \{ 1 - {\mathcal D} g(x) \} \] otherwise. In this way an iterative scheme of the form \( x(n+1) = g\{ x(n) \} \) ( \( n \geq 0 \)) to obtain \( x^\ast \) may be accelerated to one of the form \( x(n+1) = H\{ x(n) \} \) also for obtaining \( x^\ast \). This artifice is considered with reference to functions of the form \( g(x) = x + \{ x(1-x)^{p} \} \{ f(x) - x \} \), where \( f(x) \) is a polynomial and \( x^\ast \) is a repeated root of \( f(x) - x = 0 \).