SICP - Solution: Exercise 1.35

October 28, 2018

Exercise 1.35 #

Show that the golden ratio $\varphi$ (1.2.2) is a fixed point of the transformation ${x\mapsto1+1/x}$, and use this fact to compute $\varphi$ by means of the fixed-point procedure.

Solution #

The fixed point for ${x\mapsto1+1/x}$ is defined as:

$$x=1+\frac1x$$

Which can be rewritten as:

$$x^2=x+1$$ $$x^2-x-1=0$$

This is a second-order polynomial whose solution is:

$$x=\frac{1+\sqrt5}2=\varphi$$

In order to compute $\varphi$ by means of the fixed-point, you can just insert the function with a lambda in 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))

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

Which evaluates to:

1.6180327868852458