1 (define tolerance 0.00001)
2 (define (fixed-point f first-guess)
3   (define (close-enough? v1 v2)
4     (< (abs (- v1 v2)) tolerance))
5   (define (try guess)
6     (let ((next (f guess)))
7       (if (close-enough? guess next)
8 	  next
9 	  (try next))))
10   (try first-guess))
11 
12 (define (average-damp f)
13   (lambda (x) (average x (f x))))
14 (define (sqrt x)
15   (fixed-point (average-damp (lambda (y) (/ x y)))
16 	       1.0))
17 (define (cube-root x)
18   (fixed-point (average-damp (lambda (y) (/ x (square y))))
19 	       1.0))
20 (define (deriv g)
21   (lambda (x)
22     (/ (- (g (+ x dx)) (g x))
23        dx)))
24 (define dx 0.00001)
25 (define (newton-transform g)
26   (lambda (x)
27     (- x (/ (g x) ((deriv g) x)))))
28 (define (newtons-method g guess)
29   (fixed-point (newton-transform g) guess))
30 (define (sqrt x)
31   (newtons-method (lambda (y) (- (square y) x))
32 		  1.0))
33 (define (fixed-point-of-transform g transform guess)
34   (fixed-point (transform g) guess))
35 (define (sqrt x)
36   (fixed-point-of-transform (lambda (y) (/ x y))
37 			    average-damp
38 			    1.0))
39 (define (sqrt x)
40   (fixed-point-of-transform (lambda (y) (- (square y) x))
41 			    newton-transform
42 			    1.0))
43 (define (cubic a b c)
44   (define (cube x) (* x x x))
45   (define (square x) (* x x))
46   (lambda (x)
47     (+ (cube x)
48        (* a (square x))
49        (* b x)
50        c)))
51 
52 (test-case (newtons-method (cubic 5 2 3) 1.0) -4.710623963172891)
53