Petar Kapriš
1b2b2dfec1
To be noted: the last two exercises, 2.15 and 2.16 were left unfinished, and will be left for another time.
72 lines
No EOL
2.1 KiB
Scheme
72 lines
No EOL
2.1 KiB
Scheme
#lang sicp
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(define make-interval cons)
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(define (lower-bound i)
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(car i))
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(define (upper-bound i)
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(cdr i))
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(define (sub-interval a b)
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(make-interval (- (lower-bound a)
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(upper-bound b))
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(- (upper-bound a)
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(lower-bound b))))
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(define (mul-interval x y)
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(let ((p1 (* (lower-bound x) (lower-bound y)))
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(p2 (* (lower-bound x) (upper-bound y)))
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(p3 (* (upper-bound x) (lower-bound y)))
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(p4 (* (upper-bound x) (upper-bound y))))
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(make-interval (min p1 p2 p3 p4)
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(max p1 p2 p3 p4))))
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(define (add-interval x y)
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(make-interval (+ (lower-bound x) (lower-bound y))
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(+ (upper-bound x) (upper-bound y))))
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(define (div-interval x y)
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(if (<= (* (lower-bound y) (upper-bound y)) 0)
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(error "division by zero")
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(mul-interval
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x
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(make-interval (/ 1.0 (upper-bound y))
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(/ 1.0 (lower-bound y))))))
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(define (par1 r1 r2)
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(div-interval (mul-interval r1 r2)
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(add-interval r1 r2)))
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(define (par2 r1 r2)
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(let ((one (make-interval 1 1)))
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(div-interval
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one (add-interval (div-interval one r1)
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(div-interval one r2)))))
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(define example-1 (make-interval 1 2))
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(define example-2 (make-interval 0.999 1.001))
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(div-interval example-1 example-1)
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;(0.5 . 2.0)
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(div-interval example-2 example-2)
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;(0.9980019980019982 . 1.002002002002002)
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; It's simplest to explain this using the previous result
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; that the tolerance of a product of a*b, is roughly the
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; sum of the tolerances (if the tolerances are small
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; relative to the center point)
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; We can also derive that the tolerance of 1/a is equal to
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; the tolerance of a.
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; Proof:
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; a = [x-w,x+w], t_a = w/x
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; 1/a = [1/(x+w), 1/(x-w)] =
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; [(1/(x+w)) * (x-w)/(x-w), (1/(x-w)) * (x+w)/(x+w)] =
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; [(x-w)/(x^2-w^2), (x+w)/(x^2-w^2)] =
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; x/(x^2-w^2) +- w/(x^2-w^2)
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; the tolerance of the product is then:
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; (w/(x^2-w^2)) / (x/(x^2-w^2)) = w/x, same as t_a.
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;
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; This means, using the previous approximation, that the
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; expression a/a = 1, rather than having a tolerance of 0
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; is going to have a tolerance doubling that of a. |