37 lines
1.3 KiB
Racket
37 lines
1.3 KiB
Racket
#lang sicp
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;; Yeah okay, this one was extremely satisfying.
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;; I just did the calculations on-paper.
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;; it's some simple algebra anyway, but I'll type it here:
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;; assuming Tpq on (a0, b0):
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;; a1 = b0q + a0 * ( p + q)
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;; b1 = b0p + a0q
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;; a2 = (b0p + a0q) * p + (b0q + a0 * ( p + q)) * (p + q)
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;; b2 = (b0p + a0q) * p + (b0q + a0 * ( p + q)) * q
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;;
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;; rearrange a2 and b2 into a similar form to the definition of a1 and b1:
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;; (i.e. define a2 and b2 in terms of a0 and b0 and p and q)
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;; a2 = b0 * (q^2 + 2*p*q) + a0 * (2*q^2 + 2*p*q + p^2)
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;; b2 = b0 * (p^2 + q^2) + a0 * (q^2 + 2*p*q)
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;;
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;; from here we can see that p'= p^2 + q^2, and q'= q^2 + 2 * p * q
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;; and as a result, we have logarithmic fibonacci!
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;; printing the resulting number takes longer than calculating it lmao.
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(define (fib n)
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(fib-iter 1 0 0 1 n))
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(define (fib-iter a b p q count)
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(cond ((= count 0) b)
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((even? count)
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(fib-iter a
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b
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(+ (* p p) (* q q)) ; compute p′
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(+ (* q q) (* 2 (* p q))) ; compute q′
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(/ count 2)))
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(else (fib-iter (+ (* b q) (* a q) (* a p))
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(+ (* b p) (* a q))
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p
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q
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(- count 1)))))
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