# Exercise 1.37 of SICP

Exercise 1.37:

a. An infinite continued fraction is an expression of the form

$$f = \cfrac{N_1}{D_1 + \cfrac{N_2}{D_2 + \cfrac{N_3}{D_3 +\cdots}}}$$

As an example, one can show that the infinite continued fraction expansion with the Ni and the Di all equal to 1 produces $$\frac{1}{\phi}$$, where $$\phi$$ is the golden ratio (described in section 1.2.2). One way to approximate an infinite continued fraction is to truncate the expansion after a given number of terms. Such a truncation – a so-called k-term finite continued fraction – has the form

$$f = \cfrac{N_1}{D_1 + \cfrac{N_2}{{\ddots,} + \cfrac{N_k}{D_k}}}$$

Suppose that n and d are procedures of one argument (the term index i) that return the Ni and Di of the terms of the continued fraction. Define a procedure cont-frac such that evaluating (cont-frac n d k) computes the value of the k-term finite continued fraction. Check your procedure by approximating $$\frac{1}{\phi}$$ using

(cont-frac (lambda (i) 1.0)
(lambda (i) 1.0)
k)

for successive values of k. How large must you make k in order to get an approximation that is accurate to 4 decimal places?

(define (cont-frac n d k)
(define (cf i)
(if (= i (+ k 1))
0
(/ (n i)
(+ (d i) (cf (+ i 1))))))
(if (not (> k 0))
0
(cf 1)))
> (cont-frac (lambda (i) 1.0)
(lambda (i) 1.0)
11)

.6180555555555556

It takes 11 iterations to get to 4 decimal places within the reciprocal of golden ratio.

$$\frac{1}{\phi} = 0.618033988$$

It’s amusing that the LaTeX looks almost identical to the above definition in scheme:

$$\cfrac{N_1}{D_1 + \cfrac{N_2}{D_2 + \cfrac{N_3}{D_3 +\cdots}}}$$

b. If your cont-frac procedure generates a recursive process, write one that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.

(define (cont-frac-iter n d k)
(define (iter k result)
(if (= k 0)
result
(iter (- k 1) (/ (n k) (+ (d k) result)))))
(iter k 0))
> (cont-frac-iter (lambda (i) 1.0)
(lambda (i) 1.0)
11)

.6180555555555556