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SICP exercise 1.40

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Problem

Define a procedure cubic that can be used together with the newtons-method procedure from the text in expressions of the form (newtons-method (cubic a b c) 1) to approximate zeros of the cubic <math>x^3 + ax^2 + bx + c</math>.

Solution

Here's cubic:

(define (cube x) (* x x x))
 
(define (square x) (* x x))
 
(define (cubic a b c)
  (lambda (x)
    (+ (cube x) (* a (square x)) (* b x) c)))

And here's newtons-method and friends from the text:

(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))
 
(define dx 0.00001)
 
(define (deriv g)
  (lambda (x)
    (/ (- (g (+ x dx)) (g x))
       dx)))
 
(define (newton-transform g)
  (lambda (x)
    (- x (/ (g x) ((deriv g) x)))))
 
(define (newtons-method g guess)
  (fixed-point (newton-transform g) guess))

Tests performed using Chicken Scheme 3.1 on a MacBook Pro running Mac OS X 10.5.

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

Output:

-0.99999999999978
(newtons-method (cubic 0 0 0) 1)

Output:

2.65319902917972e-05
(newtons-method (cubic 0 0 -1) 1)

Output:

1.0
(newtons-method (cubic 2 1 2) 1)

Output:

-1.99999999998065
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