SICP 1.3节习题(1)

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第一章 构造过程抽象

1.3 用高阶函数做抽象

EX 1.29

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;38页递归版本sum
(define (sum term a next b)
  (if (> a b)
      0
      (+ (term a) (sum term (next a) next b))))
(define (simpson f a b n)
  (define h (/ (- b a) n))
  (define (yk k) (f (+ a (* k h))))
  (define (next k) (+ k 1))
  (define (coefficient k)
    (cond ((or (= k 0) (= k n)) 1)
          ((odd? k) 4)
          (else 2)))
  (define (term k)
    (* (coefficient k) (yk k)))
  (* (sum term 0 next n) (/ h 3)))

结果
“1-29”

EX 1.30

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;sum的迭代版本
(define (sum term a next b)
  (define (iter a result)
    (if (> a b)
        result
        (iter (next a) (+ (term a) result))))
  (iter a 0))

EX 1.31

a)

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;类似于sum的过程product,递归版本
(define (product f a next b)
  (if (> a b)
      1
      (* (f a) (product f (next a) next b))))
;用product定义阶乘过程factorial
(define (factorial n)
  (define (f x) x)
  (define (next i) (+ i 1))
  (product f
           1
           next
           n))
;计算pi值,首先计算每一个分数
(define (fraction i)
  (if (odd? i)
      (/ (+ i 1) (+ i 2))
      (/ (+ i 2) (+ i 1))))
;计算pi值,n是精确度
(define (pi n)
  (define (next i) (+ i 1))
  (* (product fraction 1.0 next n) 4.0))

结果
“1-31a”

b)

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;product的迭代版本
(define (product f a next b)
  (define (iter a result)
    (if (> a b)
        result
        (iter (next a) (* (f a) result))))
  (iter a 1))