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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',2);
x=-2:.01:4; // defining the range of x.
deff('[y]=f(x)','y=x^3-5*x+1'); //defining the function.
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)
x0=.5;
//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=1/5*(x^3+1)');
deff('[y]=gp(x)','y=1/5*(3*x^2)');
generaliteration2(x0,g,gp)
// from the above iterations performed we can infer that-
x1=0.225;
x2=0.202278;
aitken(x0,x1,x2,g) // calling the aitken method for one iteration
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