2.14

练习 2.14 请确认 Len 是对的。请你用各种不同的算术表达式来检查这一系统的行为。请做出两个区间 A 和 B，并用它们计算表达式 A/A 和 A/B。如果所用区间的宽度相对于中心值取很小百分数，你将会得到更多的认识。请检查对于中心-百分比形式（见练习2.12）进行计算的结果。

先做个实验：

(define (make-interval a b) (cons a b))
(define upper-bound cdr)
(define lower-bound car)
(define (add-interval x y) 
    (make-interval (+ (lower-bound x) (lower-bound y))
                    (+ (upper-bound x) (upper-bound y))))
(define (sub-interval x y) (add-interval x (make-interval (- 0 (upper-bound y)) (- 0 (lower-bound y)))))
(define (subtract-interval x y)
    (define p1 (- (lower-bound x) (lower-bound y)))
    (define p2 (- (lower-bound x) (upper-bound y)))
    (define p3 (- (upper-bound x) (lower-bound y)))
    (define p4 (- (upper-bound x) (upper-bound y)))

    (make-interval (min p1 p2 p3 p4) (max p1 p2 p3 p4)))

(define (mul-interval x y)
    (define p1 (* (lower-bound x) (lower-bound y)))
    (define p2 (* (lower-bound x) (upper-bound y)))
    (define p3 (* (upper-bound x) (lower-bound y)))
    (define p4 (* (upper-bound x) (upper-bound y)))

    (make-interval (min p1 p2 p3 p4) (max p1 p2 p3 p4))
)

(define (div-interval x y)
    (if (<= (* (lower-bound y) (upper-bound y)) 0) "Not allowed!"
    (mul-interval x
                (make-interval (/ 1.0 (upper-bound y))
                                (/ 1.0 (lower-bound y))))))

(define int1 (make-interval 2 4))
(define int2 (make-interval 3 6))
int1

int2

(define (par1 r1 r2) 
    (div-interval (mul-interval r1 r2) (add-interval r1 r2)))
(define (par2 r1 r2)
    (define one (make-interval 1 1))
    (div-interval one (add-interval (div-interval one r1) (div-interval one r2))))
(par1 int1 int2)

(par2 int1 int2)

果然，结果相差甚远。再看一下 A/A 和 A/B 的结果。

(define A int1)
(define B int2)

(div-interval A A)

(div-interval A B)

然后，看一下对于宽度相对于区间中心值取很小的百分数的计算结果。

(define AA (make-interval 99 100))
(define BB (make-interval 999 1000))
(div-interval AA AA)

(div-interval AA BB)

下面再看一下使用中心-百分比的形式来计算的结果。

(define (make-center-percent center percent)
  (make-interval (- center (* center percent))
                 (+ center (* center percent))))

(define (center x) (/ (+ (lower-bound x) (upper-bound x)) 2))
(define (width x) (/ (- (upper-bound x) (lower-bound x)) 2))

(define (percent x)
  (/ (width x) (center x)))

(define AAAA (make-center-percent (center A) (percent A)))
(define BBBB (make-center-percent (center B) (percent B)))

(div-interval AAAA AAAA)

(div-interval AAAA BBBB)

结果和 (make-interval) 生成的区间计算的结果一致。

再看一下另一种中心-百分比的形式计算的结果。

(define make-c-p cons)
(define c car)
(define p cdr)
(define (w x) (* (c x) (p x)))

(define (center x) (/ (+ (lower-bound x) (upper-bound x)) 2))
(define (width x) (/ (- (upper-bound x) (lower-bound x)) 2))
(define (percent x) (/ (width x) (center x)))

(define AAA (make-c-p (center A) (percent A)))
AAA

(define BBB (make-c-p (center B) (percent B)))
BBB

(div-interval AAA AAA)

(div-interval AAA BBB)