Solution of a non-linear algebraic equation using 
the Wegstein Method
Author's Data: Housam BINOUS
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Solution of a non-linear algebraic equation using FindRoot

FindRoot[Cos[x+0.25]== 2 Exp[-x+1]-1,{x,5.5}]

Solution of a non-linear algebraic equation using the fixed point method

f[x_]:=x-(Cos[x+0.25]- 2 Exp[-x+1]+1)

i=0;x[0]=1.5;

While[i<20,{i+=1,x[i]=f[x[i-1]],Print[i,"  ",x[i]]}]

tbl1=Table[{i,x[i]},{i,0,20}];

Solution of a non-linear algebraic equation using the wegstein method

i=0;x[0]=1.5;q=-0.5;

While[i<20,{i+=1,x[i]=q x[i-1]+(1-q) f[x[i-1]],Print[i,"  ",x[i]]}]

tbl2=Table[{i,x[i]},{i,0,20}];

Convergence is faster using the wegstein method (blue dots)

ListPlot[{tbl1,tbl2},PlotStyle->{RGBColor[1,0,0],RGBColor[0,0,1]}]