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 x^x = 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.)

; Answer
; -------------------------
; (i) Modified `fixed-point` procedure
(define tolerance 0.00001)
(define (average a b) (/ (+ a b) 2))
(define (fixed-point f first-guess)
    (define (close-enough? v1 v2)
        (< (abs (- v1 v2)) tolerance))
    (define (try guess)
        (newline)
        (display "Guess: ")
        (display guess)
        (let ((next (f guess)))
            (if (close-enough? guess next)
                next
                (try next))))
    (try first-guess))

; (ii) Solution to x^x = 1000 (without avg)
(fixed-point (lambda (x) (/ (log 1000) (log x))) 2.5) ;Value: 4.555539169119903

; (iii) Solution to x^x = 1000 (with avg)
(fixed-point (lambda (x) (average x
                                  (/ (log 1000) (log x))))
             2.5)

; Comparison
; -----------------
; Without average => 33 steps
; With average    =>  9 steps