1 #lang racket
  2 
  3 (define (sum term a next b)
  4   (if (> a b)
  5       0
  6       (+ (term a)
  7          (sum term (next a) next b))))
  8 
  9 (define (product term a next b)
 10   (if (> a b)
 11       1
 12       (* (term a)
 13          (product term (next a) next b))))
 14 
 15
 
 (define (product-iter term a next b)
 16   (define (iter a result)
 17     (if (> a b)
 18         result
 19         (iter (next a) (* (term a) result))))
 20   (iter a 1))
 21 
 22 (define (factorial x)
 23   (define (identity x) x)
 24   (define (next k) (+ k 1))
 25   (product-iter identity 1
 
 next x))
 26 
 27 (define (accumulate combiner null-value term a next b)
 28   (if (> a b)
 29       null-value
 30       (combiner (term a)
 31                 (accumulate combiner null-value term (next a) next b))))
 32 
 33 (define (accumulate-iter combiner null-value term a next b)
 34   (define (iter a result)
 
 35     (if (> a b)
 36         result
 37         (iter (next a) (combiner (term a) result))))
 38   (iter a null-value))
 39 
 40 #|
 41 (define (sum term a next b)
 42   (define (combiner x y) (+ x y))
 43   (accumulate-iter combiner 0 term a next b))
 44 |#
 45 
 46 (define (filtered-accumulate-iter combiner filtered null-value term a next b)
 47   (define (iter a result)
 48     (if (> a b)
 49         result
 50         (iter (next a) (combiner (filtered (term a)) result))))
 51   (iter a null-value))
 52 
 53 (define (small-divisor n)
 54   (define (find-divisor n test-divisor)
 55     (cond ((> (square test-divisor) n) n)
 56           ((divides? test-divisor n) test-divisor)
 57           (else (find-divisor n (+ test-divisor 1)))))
 58   (define  (divides? a b)
 59     (= (remainder b a) 0))
 60   (define (square x) (* x x))
 61   (find-divisor n 2))
 62 
 63 (define (prime? n)
 64   (= n (small-divisor n)))
 65 
 66 (define (prime-filter n)
 67   (if (prime? n)
 68       n
 69       0))
 70 
 71 (define (sum-of-prime term a next b)
 72   (define (combiner x y) (+ x y))
 73   (filtered-accumulate-iter combiner prime-filter 0 term a next b))
 74 
 75 
 76 (define (solution-1 a b)
 77   (define (identity x) x)
 78   (define (next x) (+ x 1))
 79   (sum-of-prime identity a next b))
 80 
 81 ; 區間折半方法尋找方程的根
 82 
 83 (define (abs x)
 84   (if (< x 0)
 85       (- x)
 86       x))
 87 
 88 (define (close-enough? x y)
 89   (< (abs (- x y)) 0.001))
 90 
 91 (define (average x y)
 92   (/ (+ x y) 2))
 93 
 94 (define (positive? x)
 95   (> x 0))
 96 
 97 (define (negative? x)
 98   (< x 0))
 99 
100 (define (search f neg-point pos-point)
101   (let ((midpoint (average neg-point pos-point)))
102     (if (close-enough? neg-point pos-point)
103         midpoint
104         (let ((test-value (f midpoint)))
105           (cond ((positive? test-value)
106                  (search f neg-point midpoint))
107                 ((negative? test-value)
108                  (search f midpoint pos-point))
109                 (else midpoint))))))
110 
111 (define (half-interval-method f a b)
112   (let ((a-value (f a))
113         (b-value (f b)))
114     (cond ((and (negative? a-value) (positive? b-value))
115            (search f a b))
116           ((and (negative? b-value) (positive? a-value))
117            (search f b a))
118           (else
119            (error "Values are not of opposite sign" a b)))))
120 
121 (half-interval-method sin 2.0 4.0)
122 
123 (half-interval-method (lambda (x) (- (* x x x) (* 2 x) 3))
124                       1.0
125                       2.0)