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+#lang bracket
+% Compute sqrt(a) using Newton iteration.
+a:=9;
+% The function f has sqrt(a) as root.
+f(x):=x^2-a;
+% The deriviative f'(x) is needed.
+df(x):=2*x;
+% Given a number x close to sqrt(a) compute a number even closer.
+next(x):=x-f(x)/df(x);
+% Iterate and save temporary values.
+x0:=1.0;
+x1:=next(x0);
+x2:=next(x1);
+x3:=next(x2);
+x4:=next(x3);
+x5:=next(x4);
+x6:=next(x5);
+x7:=next(x6);
+% The list of the first approximations.
+{x0,x1,x2,x3,x4,x5,x6,x7};
+
+% The function NestList(f,expr,n) returns the first n elements of {x,f(x), f(f(x)), ...}
+NestList(f,expr,n):=if(n=0, List(expr), Cons(expr, NestList(f,f(expr),n-1)));
+% Use it!
+NestList(next,x0,7);
+% NestList also works symbolically.
+DeclareVars(g,x);
+NestList(g,x,5)
+