Exercise 1.36. Modify fixed-point so that it prints the sequence of approximations it generates, using the newline and display primitives shown in exercise 1.22. Then find a solution to xx = 1000 by finding a fixed point of x log(1000)/log(x). (Use Scheme’s primitive log procedure, which computes natural logarithms.) Compare the number of steps this takes with and without average damping. (Note that you cannot start fixed-point with a guess of 1, as this would cause division by log(1) = 0.)
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| (define tolerance 0.00001) | |
| (define (fixed-point f first-guess) | |
| (define (close-enough? v1 v2) | |
| (newline) | |
| (display "old guess ") | |
| (display v1) | |
| (newline) | |
| (display "new guess ") | |
| (display v2) | |
| (newline) | |
| (< (abs (- v1 v2)) tolerance)) | |
| (define (try guess) | |
| (let ((next (f guess))) | |
| (if (close-enough? guess next) | |
| next | |
| (try next)))) | |
| (try first-guess)) | |
| (define f (lambda (x) (/ (log 1000) (log x)))) | |
| (fixed-point f 4.0) | |
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