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We saw in section 1.3.3 that attempting to compute square roots by naively finding a
fixed point of y → x/y does not converge, and that this can be fixed by average damping. The same method works for finding cube
roots as fixed points of the average-damped y → x/y2.
Unfortunately, the process does not work for fourth roots — a single
average damp is not enough to make a fixed-point search for y →
x/y3 converge. On the other hand, if we average damp twice (i.e., use the average damp of the average damp of x/y3) the
fixed-point search does converge. Do some experiments to determine
how many average damps are required to compute nth roots as a
fixed-point search based upon repeated average damping of y →
x/yn-1. Use this to implement a simple procedure for computing nth roots using
average-damp, and the
repeated procedure of exercise 1.43.
Assume that any arithmetic operations you need are available as primitives.