Sic 1.45 1.46

Solution: 1.45

In theory, it is hard to prove that we have to conduct experiments. The initial idea is to find out the number of times the n root needs to calculate the average damping, and find out the law.

1.46

(define (iterative-improve good-enough? improve)  (define (try x)    (let ((next (improve x)))      (if (good-enough? x next)          next          (try next))))  try)(define tolerance 0.00001)(define (close-enough? v1 v2)  (< (abs (- v1 v2)) tolerance))(define (fixed-point f first-guess)  ((iterative-improve close-enough? f) first-guess))(define (sqrt x)  (define (good-enough? dummy guess)    (< (abs (- (* guess guess) x)) tolerance))  (define (improve guess)    (/ (+ ( / x guess) guess) 2.0))  ((iterative-improve good-enough? improve) 1.0))

To sum up, lisp is used for higher-order function calculus. If these functions are implemented in C, we do not know how to write them. It is ugly to write them out.

Sic 1.45 1.46