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Exercise 1.45

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We saw in section 1.3.3 that attempting to compute square roots by naively finding a fixed point of yx/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 yx/y2. Unfortunately, the process does not work for fourth roots — a single average damp is not enough to make a fixed-point search for yx/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 yx/yn-1. Use this to implement a simple procedure for computing nth 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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