Exercise 1.45. 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 / y 2 . 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 / y 3 converge. On the other hand, if we average damp twice (i.e., use the average damp of the average damp of y x / y 3 ) 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 / y n -1 . Use this to implement a simple procedure for computing n th roots using fixed-point , averag...
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