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+                                   // The equation x^3-5*x+1==0 has  real roots.

+                                   // the graph of this function can be observed here.

+xset('window',6);                                   

+x=-2:.01:4;                                             // defining the range of x.

+deff('[y]=f(x)','y=x^3-5*x+1');                  //defining the function.

+deff('[y]=fp(x)','y=3*x^2-5');

+y=feval(x,f);

+ 

+a=gca(); 

+ 

+a.y_location = "origin";

+ 

+a.x_location = "origin"; 

+plot(x,y)                                                  // instruction to plot the graph 

+title(' y = x^3-5*x+1')

+

+// from the above plot we can infre that the function has roots between   

+// the intervals (0,1),(2,3).

+// since we have been asked for the smallest positive root of the equation,

+// we are intrested on the interval (0,1)

+// a=0;b=1,

+

+// since in the example 2.7 we have been asked to perform 4 itterations ,

+// the approximate  root after 4 iterations can be observed.

+

+

+newton4(0.5,f,fp)  

+

+

+// hence the approximate root after 4 iterations is 0.201640 witin the permissible error of 10^-15,
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