Exercise 1.35

This is the $35^{th}$ exercise from Sicp. Here, we calculate the golden ratio ($\varphi$) by finding the fixed point of a function.

The Question

Exercsie 1.35: Show that the golden ratio $\varphi$ (Section 1.2.2) is a fixed point of the transformation $x \mapsto 1 + 1/x$ , and use this fact to compute $\varphi$ by means of the fixed-point procedure.

The Answer

This is simple. We already know that $\varphi$ is equal to $\frac{1 + \sqrt{5}}{2}$. (Refer Exercise 1.13) A quick division gives us 1.6180 as the representation in float.

All we now need to do is use the fixed-point function and pass it a lambda of the function $1 + 1/x$

;; fixed-point function
(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))

We shall use 1 as the initial guess. The following is the lambda

(lambda (x)(+ 1 (/ 1 x)))

And testing:

(fixed-point (lambda (x)(+ 1 (/ 1 x))) 1.0)

;Value: 1.6180327868852458

There you go! That wasn’t too complicated was it ?