Ex-1.36

(define (fixed-point f first-guess)
  (define (close-enough? v1 v2)
    (< (abs (- v1 v2)) tolerance))
  (define (try guess)
    (display guess)
    (newline)
    (let ((next (f guess)))
      (if (close-enough? guess next)
          next
          (try next))))
  (try first-guess))


;solve x^x = 1000
;without average damping
(fixed-point
 (lambda (y) (/ (log 1000) (log y))) 2)
;Value: 4.555532270803653
;steps taken: 34

;with average damping
; y = log(1000)/log(y)
; add y on both side and get following
; y = (log(1000)/log(y) + y)/2

(fixed-point
 (lambda (y) (/ (+ y (/ (log 1000) (log y))) 2)) 2)
;Value: 4.555537551999825
;steps-taken: 9