23 hours ago

my attempt to do the exercises in sicp.

## Friday, July 4, 2008

### sicp exercise 1.45

; Exercise 1.45. We saw in section 1.3.3 that attempting to compute square roots by naively finding a

; fixed point of y -> x/y does not converge, and that this can be fixed by average damping. The same

; method works for finding cube roots as fixed points of the average-damped y -> x/y^2. Unfortunately,

; the process does not work for fourth roots -- a single average damp is not enough to make a fixed-point

; search for y -> x/y^3 converge. On the other hand, if we average damp twice (i.e., use the average damp

; of the average damp of y -> x/y^3) the fixed-point search does converge. Do some experiments to

; determine how many average damps are required to compute nth roots as a fixed-point search based upon

; repeated average damping of y -> x/y^(n-1). Use this to implement a simple procedure for computing nth

; roots using fixed-point, average-damp, and the repeated procedure of exercise 1.43. Assume that any

; arithmetic operations you need are available as primitives.

(define tolerance 0.00001)

(define (fixed-point f first-guess)

(define (close-enough? v1 v2)

(< (abs (- v1 v2)) tolerance))

(define (try guess)

(display guess)(newline)

(let ((next (f guess)))

(if (close-enough? guess next)

next

(try next))))

(try first-guess))

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

(define (compose f g)

(lambda(x)

(f (g x))))

(define (repeated f n)

(define (iter i res)

(if (>= i n)

res

(iter (+ i 1) (compose f res))))

(iter 1 f))

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

(define (fast-exp-iter base exp)

(define (func n b res)

(cond ((= n 0) res)

((even? n) (func (/ n 2) (square b) res))

(else (func (- n 1) b (* res b)))))

(func exp base 1))

(define (average-damp f)

(lambda(x)

(average x (f x))))

(define (func x n)

(lambda(y)

(/ x (fast-exp-iter y (- n 1)))))

; nth root is the fixed-point of the function func

(define (nth-root x n)

(fixed-point (repeated (average-damp (func x n)) 5) 3.0))

(display (nth-root 256 8))(newline)

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