Solution of a System of Nonlinear Equations using the Fixed Point Method and NestList​
​Authors: Housam Binous and Ahmed Bellagi

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 few iterations using NestList command.
​
Solution is compared to direct computation using FindRoot.
In[]:=
F[x1_,x2_,x3_]:={-1/81Cos[x1]+1/9x2^2+1/3Sin[x3]-x1,​​1/3Sin[x1]+1/3Cos[x1]-x2,​​-1/9Cos[x1]+1/3x2+1/6Sin[x3]-x3}
In[]:=
G[{x1_,x2_,x3_}]:={x1,x2,x3}-Inverse[Grad[F[x1a,x2a,x3a],{x1a,x2a,x3a}]].F[x1,x2,x3]/.x1ax1/.x2ax2/.x3ax3
In[]:=
x1=-0.9;x2=-0.9;x3=-1;
In[]:=
NestList[G,{x1,x2,x3},6]//Chop
Out[]=
{{-0.9,-0.9,-1},{-0.270855,0.240731,-0.103058},{-0.00139379,0.342573,-0.00112744},{-0.0000101855,0.33333,-1.35828×
-6
10
},{0,0.333333,0},{0,0.333333,0},{0,0.333333,0}}
In[]:=
x1=.;x2=.;x3=.
In[]:=
FindRoot[{-1/81Cos[x1]+1/9x2^2+1/3Sin[x3]x1,1/3Sin[x1]+1/3Cos[x1]x2,​​-1/9Cos[x1]+1/3x2+1/6Sin[x3]x3},{x1,0.5},{x2,0.5},{x3,1}]//Chop
Out[]=
{x10,x20.333333,x30}