Exercise 1.07

Background: Section 1.1.7, Square Roots by Newton’s Method

(define (square x) (* x x))

(define (average x y)
  (/ (+ x y) 2))

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

(define (good-enough? guess x)
  (< (abs (- (square guess) x)) 0.001))

(define (sqrt-iter guess x)
  (if (good-enough? guess x)
      guess
      (sqrt-iter (improve guess x)
                 x)))

(define (sqrt x)
  (sqrt-iter 1.0 x))

Question:

The good-enough? test used in computing square roots will not be very effective for finding the square roots of very small numbers. Also, in real computers, arithmetic operations are almost always performed with limited precision. This makes our test inadequate for very large numbers. Explain these statements, with examples showing how the test fails for small and large numbers. An alternative strategy for implementing good-enough? is to watch how guess changes from one iteration to the next and to stop when the change is a very small fraction of the guess. Design a square-root procedure that uses this kind of end test. Does this work better for small and large numbers?

Answer:

For small numbers:

When we take the sqrt of - for example - 0.0001:

> (sqrt 0.0001)
0.03230844833048122

instead of the expected 0.01 we get 0.0323 … because the latter’s square is so close to 0.0001 that the difference is less than 0.001 - the error margin used in good-enough?.

If we replace good-enough? with:

(define (good-enough? guess x)
  (< (abs (- (square guess) x)) (/ guess 1e4)))

todo