1.3.4. Procedures as Returned Values
Exercise 1.45

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 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.

There are no comments yet.

Authentication required

You must log in to post a comment.