Computing nth roots

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 y → x/y³ ) the fixed-point search does converge. Do some experiments to determine how many average damps are required to compute n th 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 n th roots using fixed-point , average-damp , and the repeated procedure of exercise 1.43 . Assume that any arithmetic operations you need are available as primitives.

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