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ch1.3.4.rkt
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#lang racket
;; chapter 1.3.4 프로시저를 만드는 프로시저
(define (average x y)
(/ (+ x y) 2))
(define (fixed-point f guess)
(define (close-enough? a b)
(< (abs (- a b)) 0.00001))
(define (try g)
(let ((next (f g)))
(if (close-enough? g next)
next
(try next))))
(try guess))
;; (fixed-point (lambda (x) (cos x)) 1.0)
;; (fixed-point (lambda (x) (average x (cos x))) 1.0)
(define (average-damp f)
(lambda (x) (average x (f x))))
(define (sqrt x)
(fixed-point (lambda (y) (average y (/ x y))) 1.0))
;; (sqrt 2)
(define (sqrt-2 x)
(fixed-point (average-damp (lambda (y) (/ x y))) 1.0))
;; (sqrt-2 2)
;; newton-method
;; f(x)의 접선식이 0이 되는 점을 두번째 x값으로 하여 풀이하는 방법
;; http://en.wikipedia.org/wiki/Newton%27s_method
(define dx 0.0001)
(define (derive f)
(lambda (x) (/ (- (f (+ x dx)) (f x)) dx)))
;(define (newton-method f s)
; (fixed-point (lambda (x) (- x (/ (f x) ((derive f) x)))) s))
(define (newton-transform f)
(lambda (x) (- x (/ (f x) ((derive f) x)))))
(define (newton-method f s)
(fixed-point (newton-transform f) s))
(define (sqrt-nm x)
(newton-method (lambda (y) (- (* y y) x)) 10.0))
;; (sqrt-nm 2)
;; (newton-method (lambda (x) (- (* x x) 1)) -2.0)
;; sqrt를 계산하는데
;; y => x / y 의 고정점 찾기 문제로 푸는 방법과
;; y^2 - x = 0 를 newton method로 푸는
;; 2가지 방법을 봤음
;; 하지만 두번째 newton method는 결국 y => y - (y^2 - x) / (2 * dy) 의 고정점 찾기 문제임
;; 이를 일반화 하여 fixed-point-of-transform 프로시저로 짜면 아래와 같다.
(define (fixed-point-of-transform f transform guess)
(fixed-point (transform f) guess))
;; sqrt x / y 고정점 찾기
(define (sqrt-fpt-1 x)
(fixed-point-of-transform (lambda (y) (/ x y)) average-damp 1.0))
;; sqrt newton method
(define (sqrt-fpt-2 x)
(fixed-point-of-transform (lambda (y) (- (* y y) x)) newton-transform 1.0))
;; ex 1.40
;; newton-method로 x^3 + a*x^2 + b*x + c = 0 의 해 구하기
(define (cubic a b c)
(lambda (x) (+ (* x x x) (* a x x) (* b x) c)))
;; (x-1)^3
;; (newton-method (cubic -3 3 -1) 2)
;; ex 1.41
(define (double f)
(lambda (x) (f (f x))))
(define (inc x)
(+ x 1))
(define (square x)
(* x x))
;; 맞바꿈 계산법으로 풀기
;
;(((double (double double)) inc) 5)
;
;(((double (lambda (x) (double (double x)))) inc) 5)
;
;(((lambda (y) ((lambda (x) (double (double x))) ((lambda (z) (double (double z))) y))) inc) 5)
;
;(((lambda (x) (double (double x))) ((lambda (z) (double (double z))) inc)) 5)
;
;(((lambda (x) (double (double x))) (double (double inc))) 5)
;
;((double (double (double (double inc)))) 5)
;
;((double (double (double (lambda (x) (inc (inc x)))))) 5)
;
;((double (double (lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a))))) 5)
;
;((double (lambda (b)
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a))) b)))) 5)
;((lambda (c)
; ((lambda (b)
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a))) b)))
; ((lambda (b)
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a))) b))) c))) 5)
;
;((lambda (b)
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a))) b)))
; ((lambda (b)
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a))) b))) 5))
;
;((lambda (b)
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a))) b)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a))) 5)))
;
;((lambda (b)
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a))) b)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) 5))))
;
;((lambda (b)
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a))) b)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (x) (inc (inc x)))
; (inc (inc 5)))))
;
;((lambda (b)
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a))) b)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; (inc (inc (inc (inc 5))))))
;
;((lambda (b)
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a))) b)))
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) (inc (inc (inc (inc 5)))))))
;
;((lambda (b)
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a))) b)))
; ((lambda (x) (inc (inc x)))
; (inc (inc (inc (inc (inc (inc 5))))))))
;
;((lambda (b)
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a))) b)))
; (inc (inc (inc (inc (inc (inc (inc (inc 5)))))))))
;
;((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a))) (inc (inc (inc (inc (inc (inc (inc (inc 5))))))))))
;
;((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) (inc (inc (inc (inc (inc (inc (inc (inc 5)))))))))))
;
;((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; ((lambda (x) (inc (inc x)))
; (inc (inc (inc (inc (inc (inc (inc (inc (inc (inc 5))))))))))))
;
;((lambda (a)
; ((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) a)))
; (inc (inc (inc (inc (inc (inc (inc (inc (inc (inc (inc (inc 5)))))))))))))
;
;((lambda (x) (inc (inc x)))
; ((lambda (x) (inc (inc x))) (inc (inc (inc (inc (inc (inc (inc (inc (inc (inc (inc (inc 5))))))))))))))
;
;(inc (inc (inc (inc (inc (inc (inc (inc (inc (inc (inc (inc (inc (inc (inc (inc 5))))))))))))))))
;
;
;; ex 1.42
(define (compose f g)
(lambda (x) (f (g x))))
;; ((compose square inc) 6) ;; => 49
;; ex 1.43
(define (repeated f i)
(define (iter g n)
(if (= n 1)
g
(iter (compose g f) (- n 1))))
(iter f i))
;; ((repeated inc 10) 0) ;; => 10
;; ((repeated square 2) 5) ;; => (square (square 5)) => 625
;; ex 1.44
(define (smooth f)
(lambda (x) (/ (+ (f (- x dx)) (f x) (f (+ x dx))) 3)))
(define (smooth-n f n)
((repeated smooth n) f))
;; ex 1.45
(define (cube-root x)
(fixed-point (average-damp (lambda (y) (/ x (square y)))) 1.0))
;; (cube-root 27)
(define (4th-root x)
(fixed-point (average-damp (lambda (y) (/ x (* y y y)))) 1.0))
;; (4th-root 16) => infinite loop
;; y -> x/y^(n-1) 은 몇번 average-damp 해야 하는지 실험하기
(define (nth-root x n)
(fixed-point ((repeated average-damp n) (lambda (y) (/ x (expt y (- n 1)))))
1.0))
;; 실험 결과.. 패턴이 안보인다
;; n = 4 > 2
;; n = 5 > 2
;; n = 13 > 3
;; n = 22 > 2
;; ex 1.46 - iterative-improve
(define (iterative-improve good? improve)
(lambda (guess)
(define (iter v)
(if (good? v)
v
(iter (improve v))))
(iter guess)))
;; 참고 솔루션과 틀린점은
;; guess 프로시저가 하나의 인자만을 받는다는 점이다.
;; 대신 improve를 정의하는 scope에서 자유변수를 사용하게 된다
;; 이렇게 하지 않고, 2개의 인자를 받는 경우(참고 솔루션) sqrt의 경우 결국 문제를 fixed-point로 변환해야만 한다.
;; 뭐가 맞는건지 ????
;; sqrt
(define (sqrt-ii x)
(define (good? guess)
(< (abs (- (square guess) x)) 0.001))
(define (improve guess)
(/ (+ guess (/ x guess)) 2))
((iterative-improve good? improve) 1.0))
(display "iterative-improve sqrt")
(newline)
(sqrt-ii 9)
;; fixed-point
(define (fixed-point-ii f g)
(define (good? guess)
(< (abs (- guess (f guess)) 0.00001)))
((iterative-improve good? f) g))
(display "iterative-improve fixed-point cos")
(newline)
(fixed-point cos 1.0)