sqrt.scm

(define (sqrt x)
  (define tolerance 0.00001)
  (define (good-enough? y)
    (< (abs (- (square y) x))
       tolerance))
  (define (improve y)
    (average (/ x y) y))
  (define (try y)
    (if (good-enough? y)
        y
        (try (improve y))))
  (try 1))




(define (fixed-point improve i)
  (define tolerance 0.00001)
  (define (close-enough? old new)
    (< (abs (- old new)) tolerance))
  (define (iter old new)
    (if (close-enough? old new)
        new
        (iter new (improve new))))
  (iter i (improve i)))

(define (average-damp f)
  (lambda (x) (average (f x) x)))

(define (sqrt x)
  (fixed-point
   (average-damp (lambda(y) (/ x y)))
   1))



;; Newton's method
(define (sqrt x)
  (newton (lambda(y) (- x (square y)))
          1))

(define (newton f i)
  (define df (deriv f))
  (fixed-point
   (lambda(x) (- x (/ (f x) (df x))))
   i))

(define (deriv f)
  (define dx 0.000001)
  (lambda(x) (/ (- (f (+ x dx))
                   (f x))
                dx)))