**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 section 1.1.7 and the fixed-point procedure of section 1.3.3 in terms of **iterative-improve**.

The first two functions were written in the form of earlier exercises. The -iter functions are implemented with iterative-improve.

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(define (sqrt guess x) (define (improve guess x) (define (average a b) (/ (+ a b) 2)) (average guess (/ x guess))) (define (great-enough? oldguess guess) (< (abs (- 1 (/ oldguess guess))) 0.0001)) (if (great-enough? guess (improve guess x)) (improve guess x) (sqrt (improve guess x) x))) (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)) (define (iterative-improve close-enough? improve) (define (iter guess) (let ((new-guess (improve guess))) (if (close-enough? guess new-guess) new-guess (iter new-guess)))) iter) (define (sqrt-iter x) (define (improve guess) (define (average a b) (/ (+ a b) 2)) (average guess (/ x guess))) (define (great-enough? oldguess guess) (< (abs (- 1 (/ oldguess guess))) 0.0001)) ((iterative-improve great-enough? improve) 1.0)) (define (fixed-point-iter f) (define tolerance 0.0001) (define (close-enough? v1 v2) (< (abs (- 1 (/ v1 v2))) tolerance)) (define (improve guess) (f guess)) ((iterative-improve close-enough? improve) 1.0)) |

Old implementations:

> (sqrt 1.0 2)

*1.4142135623746899*

> (fixed-point (lambda (y) (* 0.5 (+ (/ 2 y) y))) 1.0)

*1.4142135623746899*

New implementations:

> (fixed-point-iter (lambda (y) (* 0.5 (+ (/ 2 y) y))))

*1.4142135623746899*

> (sqrt-iter 2)

*1.4142135623746899*