SICP Exercise 1.45 multi average damp

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/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/y3converge. 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 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 fixed-point, average-damp, and the repeated procedure of exercise 1.43. Assume that any arithmetic operations you need are available as primitives.

 SOLUTION

 Observations from the test below:
1. Fourth and Fifth roots need two average dampings. Three dampings also works but interestingly, it takes more iterations to converge to the same tolerance
2. I tried the 8th root of a number and found that even 3 average dampings were not enough but 4 dampings worked. 5 worked too but again, took more iterations than 4 dampings.

 The code and tests are here.

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