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+                                   // The equation 3*x^3+4*x^2+4*x+1==0 has three real roots.
+                                   // the graph of this function can be observed here.
+xset('window',22);                                   
+x=-1.5:.001:1.5;                                             // defining the range of x.
+deff('[y]=f(x)','y=3*x^3+4*x^2+4*x+1');                  //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 =3*x^3+4*x^2+4*x+1')
+
+// from the above plot we can infre that the function has root between   
+// the interval (-1,0),
+
+x0=-.5;                 // initial approximation 
+
+
+// let the iterative function g(x) be x+A*(3*x^3+4*x^2+4*x+1) =g(x);
+
+// gp(x)=(1+A*(9*x^2+8*x+4) )
+// we need to choose a value for A , which makes abs(gp(x0))<1
+
+// hence abs(gp(x0))=abs(1+9*A/4)
+
+A=-1:.1:1;
+
+abs(1+9*A/4)            // tryin to check the values of abs(gp(x0)) for different values of A.
+
+
+// from the above values of 'A' and the values of 'abs(gp(x0))',
+// we can infer that for the vales of 'A 'in the range (-.8,0) g(x ) will be giving a converging solution ,
+
+// hence deliberatele we choose a to be -0.5,
+
+A=-0.5;
+
+deff('[y]=g(x)','y= x-0.5*(3*x^3+4*x^2+4*x+1)');
+deff('[y]=gp(x)','y= 1-0.5*(9*x^2+8*x+4)');      // hence defining g(x) and gp(x),
+generaliteration(x0,g,gp)
+
+
+
+