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
​x
 
(2 . 4)

 
int2
 
(3 . 6)

 
(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)
 
(0.6000000000000001 . 4.800000000000001)

 
(par2 int1 int2)
 
(1.2000000000000002 . 2.4000000000000004)

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)
 
(0.5 . 2)

 
(div-interval A B)
 
(0.3333333333333333 . 1.3333333333333333)

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)
 
(0.99 . 1.0101010101010102)

 
(div-interval AA BB)
 
(0.099 . 0.10010010010010009)

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)
 
(0.5 . 2)

 
(div-interval AAAA BBBB)
 
(0.3333333333333333 . 1.3333333333333333)

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
 
(3 . 0.3333333333333333)

 
(define BBB (make-c-p (center B) (percent B)))
BBB
 
(4.5 . 0.3333333333333333)

 
(div-interval AAA AAA)
 
(0.1111111111111111 . 9)

 
(div-interval AAA BBB)
 
(0.07407407407407407 . 9)