// The equation cos(x)-x*%e^x==0 has  real roots.
                                   // the graph of this function can be observed here.
xset('window',25);                                   
x=0:.01:2;                                             // defining the range of x.
deff('[y]=f(x)','y=cos(x)-x*%e^x');                  //defining the cunction.
y=feval(x,f);
 
a=gca(); 
 
a.y_location = "origin";
 
a.x_location = "origin"; 
plot(x,y)                                                  // instruction to plot the graph 
title(' y = cos(x)-x*%e^x')

// from the above plot we can infre that the function has root between   
// the interval (0,1)

x0=0;

         //solution using linear iteration method for the first two iterations and aitken's process two times for the third iteration.
         
         deff('[y]=g(x)','y=x+1/2*(cos(x)-x*%e^x)');
         deff('[y]=gp(x)','y=1+1/2*(-sin(x)-x*%e^x-%e^x)');
         
         
generaliteration2(x0,g,gp)


// from the above iterations performed we can infer that-
x1=0.50000000;
x2=0.5266110;



aitken(x0,x1,x2,g)          // calling the aitken method for one iteration