Example - Iteration Using Streams

Back in Unit 1, we wrote a procedure for approximating the square root of a given number—let's call it x. The idea was to generate a sequence of better and better guesses for the square root of x by applying over and over again the procedure that improves guesses:

(define (sqrt-improve guess x)
(average guess (/ x guess)))


We can create an infinite stream of guesses, starting with an initial guess of 1:

(define (sqrt-stream x)
(define guesses
(cons-stream 1.0 (stream-map (lambda (guess)
(sqrt-improve guess x))
guesses)))
guesses)


The first few elements of (sqrt-stream 2) would be:

1
1.5
1.4166666666666665
1.4142156862745097
1.4142135623746899


Each successive element of the stream gets closer and closer to the square root of 2.

Similarly, we used the following formula to approximate pi:

Now, let's calculate pi with an infinite stream:

(define (pi-summands n)
(cons-stream (/ 1.0 n)
(stream-map - (pi-summands (+ n 2)))))

(define pi-stream
(scale-stream (partial-sums (pi-summands 1)) 4))


The first few elements look like this:

4.
2.666666666666667
3.466666666666667
2.8952380952380956
3.3396825396825403
2.9760461760461765
3.2837384837384844
3.017071817071818


As you can tell, the numbers are converging on pi—after looking at the first eight elements, we know pi is somewhere between 3.28 and 3.02.