sicp1.29 通用的高阶求和算法 + 辛普森积分算法

simson-rule.scm

(define (sum term a next b)
    (if (> a b)
        0
        (+ (term a)
           (sum term (next a) next b)
        )
    )
)

; 求立方.
(define (cube x) (* x x x))

(define (integral f a b dx)
    (define (add-dx x) (+ x dx))
    (* (sum f (+ a (/ dx 2.0)) add-dx b)
        dx)
)

#|
> (integral cube 0 1 0.01)
0.24998750000000042
> (integral cube 0 1 0.001)
0.249999875000001
|#

; simson-rule
(define (simson-rule f a b n)
    (define h (/ (- b a) n))

    (define (factor k)
        (cond ((or (= k 0) (= k n)) 1)
              ((even? k) 2)
              (else 4)
        )
    )

    (define (term k)
        (* (factor k)
           (f (+ a (* k h)))
        )
    )

    (define (next k)
        (+ k  1)
    )

    (* (/ h 3.0)
       (sum term 0 next n)
    )
)

#|
> (simson-rule cube 0 1 100)
0.24666666666666667
> (simson-rule cube 0 1 1000)
0.24966666666666665
|#

关键点:

• term 用来计算真正要相加的 项.
• next 用来计算下一个 "a" 值.

相较于社区参考答案, 我认为我的 实现, 可读性相对更好些.
ref: http://community.schemewiki.org/?sicp-ex-1.29