Fixed Point Computation using 
Aitken's method
Author's Data: Housam BINOUS
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Reference :
A. Quarteroni, R. Sacco and F. Saleri, Méthodes numériques pour le calcul scientifique, Programmes en Matlab, Springer, Paris, 2000

Fixed point is obtained in fewer iterations using Aitken's method when compared to direct computation.

f[x_] := Cos[x] + Exp[-x] + x

NestList[f, 1. , 10]

FindRoot[f[x] ==  x, {x, -1}]

ϕ[
        x_] := (x f[f[x]] - f[x]^2)/(f[f[x]] - 2  f[x] + 
            x)

NestList[ϕ, 1. , 10]

FindRoot[ϕ[x] ==  x, {x, -1}]

Fixed point is obtained only with Aitken's method. Direct method does not converge.

f[x_] := x^2 - Exp[x]

NestList[f, -1. , 10]

FindRoot[f[x] ==  x, {x, -1}]

ϕ[
        x_] := (x f[f[x]] - f[x]^2)/(f[f[x]] - 2  f[x] + 
            x)

NestList[ϕ, -1. , 10]

FindRoot[ϕ[x] ==  x, {x, -1}]