It is a method of computing fixed points and iterated functions. This method is nearly similar to Newton-Raphson method.Directly mathematica program code is given here.
Clear["`*"];
f[x_] := 1 - x Exp[x];
Plot[f[x],{x,-5,5}];
x0 = Input["Enter x0"];
tol = Input["Enter tolerance"];
n = Input["Enter total iteration"];
a = x0;
g[x_] := Exp[-x];
Do[
If[Abs[g'[x0]]>1, {Print["The method is divergence"] Exit[]}];
x0 = g[a];
Print[i, PaddedForm[N[x0],{10,6}]];
If[Abs[a-x0]<tol, {Print["The solution is: ", N[x0]], Exit[]}];
a = x0;
,{i,1,n}];
Print["Maximum iteration failed"];
When you run it by mathematica, you will find the following graphics.
Graphically the root is near 0.5. So we will input
x0 = 1
tol = 0.001
n = 25
The result will be displayed then.