sicp每日一題[2.10]

Exercise 2.11

In passing, Ben also cryptically comments: “By testing the signs of the endpoints of the intervals, it is possible to break mul-interval into nine cases, only one of which requires more than two multiplications.” Rewrite this procedure using Ben's suggestion.
After debugging her program, Alyssa shows it to a potential user, who complains that her program solves the wrong problem. He wants a program that can deal with numbers represented as a center value and an additive tolerance; for example, he wants to work with intervals such as 3.5±0.15 rather than [3.35, 3.65]. Alyssa returns to her desk and fixes this problem by supplying an alternate constructor and alternate selectors:

(define (make-center-width c w)
  (make-interval (- c w) (+ c w)))
(define (center i)
  (/ (+ (lower-bound i) (upper-bound i)) 2))
(define (width i)
  (/ (- (upper-bound i) (lower-bound i)) 2))

Unfortunately, most of Alyssa's users are engineers. Real engineering situations usually involve measurements with only a small uncertainty, measured as the ratio of the width of the interval to the midpoint of the interval. Engineers usually specify percentage tolerances on the parameters of devices, as in the resistor specifications given earlier.

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這道題目難度並不大，就是有點麻煩，要把每一種情況的上界和下界都分析出來。至於區間採用置信區間的表示方法，只要用寬度的一半除以中間值就行了。

; 把 w 改為百分比形式
(define (make-center-width center percent)
  (let ((tolerance (* center (/ percent 100))))
    (make-interval (- center tolerance) (+ center tolerance))))

(define (center i)
  (average (lower-bound i) (upper-bound i)))

(define (percent i)
  (* (/ (/ (- (upper-bound i) (lower-bound i)) 2)
        (center i))
     100))

(define (display-interval-tolerance i) 
  (newline)
  (display (center i))
  (display " ± ")
  (display (percent i))
  (display "%"))

; 記 x 的下界為 x1，上界為 x2；記 y 的下界為 y1，上界為 y2
; 將他們按 x1 x2 y1 y2 從左到右排序，並按位置記為 1, 2, 3, 4
; 則按照 x 和 y 的區間端點數字的符號，可以分成以下9種情況：
; - - - -   min: 2*4, max: 1*3
; - - - +   min: 1*4, max: 1*3
; - - + +   min: 1*4, max: 2*3
; - + - -   min: 2*3, max: 1*3
; - + - +   min: min(1*4, 2*3), max: max(1*3, 2*4)
; - + + +   min: 1*4, max: 2*4
; + + - -   min: 2*3, max: 1*4
; + + - +   min: 2*3, max: 2*4
; + + + +   min: 1*3, max: 2*4
(define (mul-interval x y)
  (let ((x1 (lower-bound x))
        (x2 (upper-bound x))
        (y1 (lower-bound y))
        (y2 (upper-bound y))
        (p1 (* (lower-bound x) (lower-bound y)))    ; 即 1*3
        (p2 (* (lower-bound x) (upper-bound y)))    ; 即 1*4
        (p3 (* (upper-bound x) (lower-bound y)))    ; 即 2*3
        (p4 (* (upper-bound x) (upper-bound y))))   ; 即 2*4
    (cond ((and (negative? x2) (negative? y2)) (make-interval p4 p1))
          ((and (negative? x2) (negative? y1) (positive? y2)) (make-interval p2 p1))
          ((and (negative? x2) (positive? y1)) (make-interval p2 p3))
          ((and (negative? x1) (positive? x2) (negative? y2)) (make-interval p3 p1))
          ((and (negative? x1) (positive? x2) (negative? y1) (positive? y2)) (make-interval (min p2 p3) (max p1 p4)))
          ((and (negative? x1) (positive? x2) (positive? y1)) (make-interval p2 p4))
          ((and (positive? x1) (negative? y2)) (make-interval p3 p2))
          ((and (positive? x1) (negative? y1) (positive? y2)) (make-interval p3 p4))
          ((and (positive? x1) (positive? y1)) (make-interval p1 p4)))))


(define a (make-interval -6.12 7.48))
(define b (make-interval -4.465 4.935))
(display-interval (mul-interval a b))
(display-interval-tolerance (mul-interval a b))

; 執行結果
[-33.3982, 36.9138]
1.7577999999999996 ± 2000.0000000000007%