1.40

Exercise 1.40: Define a procedure cubic that can be used together with the newtons-method procedure in expressions of the form

(newtons-method (cubic a b c) 1)

to approximate zeros of the cubic $x^3 + a x^2 + bx +c$ .

fixed-point

 
(define tolerance 0.00001)
​
(define (fixed-point f first-guess)
  (define (close-enough? v1 v2)
    (< (abs (- v1 v2)) 
       tolerance))
  (define (try guess)
    (let ((next (f guess)))
      (if (close-enough? guess next)
          next
          (try next))))
  (try first-guess))
​x
 
#<undef>

deriv

 
(define dx 0.00001)
​
(define (deriv g)
  (lambda (x)
    (/ (- (g (+ x dx)) (g x))
       dx)))
 
#<undef>

newton-transform

 
(define (newton-transform g)
  (lambda (x)
    (- x (/ (g x) 
            ((deriv g) x)))))
 
#<undef>

newtons-method

 
(define (newtons-method g guess)
  (fixed-point (newton-transform g) 
               guess))
 
#<undef>

Now let's define cubic

$f(x)=x^3 + a x^2 + bx +c$

 
(define (cubic a b c) (lambda (x) (+ (* x x x) (* a (* x x)) (* b x) c)))
((cubic 1 1 1) 1)
 
4

 
((cubic 1 1 1) -1)
 
0

Finally let's verify the result

 
(newtons-method (cubic 1 1 1) 1)
 
-0.9999999999997796