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 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.
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(define (root3 x) ((nth-root 3) x)) (define (root4 x) ((nth-root 4) x)) (check-equal? (round (root3 27)) 3.0) (check-equal? (round (root3 64)) 4.0) (check-equal? (round (root4 81)) 3.0) (check-equal? (round (root4 10000)) 10.0)