Solution of a System of Nonlinear Equations using the Fixed Point 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 few iterations.
Solution is compared to direct computation using FindRoot.

Off[General::"spell"]

F[x1_, x2_, 
        x3_] := {-1/81 Cos[x1] + 1/9 x2^2 + 1/3 Sin[x3] - 
          x1, 
1/3 Sin[x1] + 1/3 Cos[x1] - 
          x2, 
-1/9 Cos[x1] + 1/3 x2 + 1/6 Sin[x3] - 
          x3}

Grad[fon_, var_List] := D[fon, #] & /@ var

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

x1 = -0.9; x2 = -0.9; x3 = -1;

For[i = 1, 
      i < 20, {x1 = G[x1, x2, x3][[1]] // Chop; 
        x2 = G[x1, x2, x3][[2]] // Chop; 
        x3 = G[x1, x2, x3][[3]] // Chop, i++}]

{x1, x2, x3} // Chop

x1 =. ; x2 =. ; x3 =.

FindRoot[{-1/81 Cos[x1] + 1/9 x2^2 + 1/3 Sin[x3] == x1, 
          1/3 Sin[x1] + 1/3 Cos[x1] ==  
            x2, 
-1/9 Cos[x1] + 1/3 x2 + 
              1/6 Sin[x3]  ==  x3}, {x1, 0.5}, {x2, 0.5}, {x3, 1}] // 
      Chop