42 lines
1.3 KiB
Racket
42 lines
1.3 KiB
Racket
#lang racket
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(define (cont-frac nf df k)
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(define (recurse i)
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(if (>= i k)
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0
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(/ (nf i) (+ (df i) (recurse (+ 1 i))))))
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(recurse 0))
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; produces the same result as cont-frac, but
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; generates an iterative process.
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(define (cont-frac-iterative nf df k)
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(define (iterate i numer denom)
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(if (<= i 0)
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(/ numer denom)
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(iterate (- i 1) (nf (- i 1)) (+ (df (- i 1)) (/ numer denom)))))
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(iterate (- k 1) (nf k) (df k)))
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(define (golden-ratio k)
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"cont-frac generates 1/golden ratio, so we just inverse it to get the
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real thing. call with k=1000 or something to get an accurate result"
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(/ 1.0 (cont-frac-iterative (λ (i) 1.0) (λ (i) 1.0) k)))
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;; example 1-38
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(define (e-helper i)
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(let ((r (remainder (- i 1) 3)))
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(if (= 0 r)
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(* 2.0 (/ (+ i 2) 3))
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1.0)))
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(define (eulers-constant k)
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"Finds euler's constant. the continued fraction gives e - 2, so we add 2."
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(+ 2 (cont-frac-iterative (λ (i) 1.0) e-helper k)))
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;; example 1-39
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(define (tan-cf x k)
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"Finds an approximation of the tangent function. The first numerator is not negative,
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even though our numerator function calculates them all as negative, so we need to negate
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the result at the end."
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(-
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(cont-frac-iterative (λ (i) (- (if (<= i 0) x (* x x))))
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(λ (i) (- (* 2 (+ i 1)) 1))
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k))) |