SICP - Solution: Exercise 1.46

October 29, 2018

Exercise 1.46 #

Several of the numerical methods described in this chapter are instances of an extremely general computational strategy known as iterative improvement. Iterative improvement says that, to compute something, we start with an initial guess for the answer, test if the guess is good enough, and otherwise improve the guess and continue the process using the improved guess as the new guess. Write a procedure iterative-improve that takes two procedures as arguments: a method for telling whether a guess is good enough and a method for improving a guess. iterative-improve should return as its value a procedure that takes a guess as argument and keeps improving the guess until it is good enough. Rewrite the sqrt procedure of 1.1.7 and the fixed-point procedure of 1.3.3 in terms of iterative-improve.

Solution #

Rewriting sqrt using iterative-improve #

This exercise is not just about algorithms, but also how to handle abstraction. If you want to solve ｀ iterative-improve ｀ for only the first question, you arrive at something like this, where good-enough? takes only guess as parameter:

(define (iterative-improve good-enough? improve)
  (define (iter guess)
    (if (good-enough? guess)
        guess
        (iter (improve guess))))
  (lambda (guess) (iter guess)))

Then you can rewrite sqrt:

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

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

(define (sqrt x)
  (define (improve guess)
    (average guess (/ x guess)))
  (define (good-enough? guess)
    (< (abs (- (square guess) x)) 0.001))
  ((iterative-improve good-enough? improve) 1.0))

(display (sqrt 11))(newline)

Rewriting fixed-point using iterative-improve #

The problem here is that the function to check if a guess is good enough requires the next value. The only option that I found was to rewrite iterative-improve so that it takes two arguments:

(define (iterative-improve good-enough? improve-guess)
  (define (iter guess)
    (let ((next (improve-guess guess)))
      (if (good-enough? guess next)
          next
          (iter next))))
  (lambda (guess) (iter guess)))

; --- fixed-point ---
(newline)
(define tolerance 0.00001)

(define (fixed-point f first-guess)
  (define (close-enough? v1 v2)
    (< (abs (- v1 v2))
       tolerance))
  (define (improve-guess guess)
    (f guess))
  (define (try guess)
    (let ((next (improve-guess guess)))
      (if (close-enough? guess next)
          next
          (try next))))
  (try first-guess))

Then you can use fixed-point as the previous exercise:

(define (average-damp f)
  (lambda (x)
    (average x (f x))))

(define (sqrt-fixed-point x)
  (fixed-point
   (average-damp
    (lambda (y) (/ x y)))
   1.0))

(display (sqrt-fixed-point 11))(newline)

The last step is to rewrite sqrt using the directly iterative-improve:

(define (sqrt x)
  (define (improve guess)
    (average guess (/ x guess)))
  (define (good-enough? next guess)
    (< (abs (- (square next) x)) 0.001))
  ((iterative-improve good-enough? improve) 1.0))