Calculating the optimal continued fraction of a quadratic irrational
This program finds the optimal continued fraction expansion
as far as the end of the first period
of a quadratic irrational x = (u+t√d)/v, where d,t,u,v are integers, d >1 and nonsquare, t and v nonzero:
where the first least period is printed in boldface.
We first convert x to (P0+√d)/Q0 where Q0 divides d - P02.
The algorithm is based on the paper Optimal continued fractions by Wieb Bosma, Indag. Math. 49 (1987) 353-379 (see pseudo-code).
We find the first occurrence of equality of complete quotients ξj=ξj+k, j < k. We know that if sj > |Q0|, this gives either a period or half-period. We then look for an inequality ξj+t ≠ ξj+k+t, 1 ≤ t < k. If no such inequality is detected, the OCF period length = l = 2 x NSCF period length = 2k; otherwise the OCF period length = l = NSCF period length = k.
We then progressively work back, checking to see if
etc. to get the first least period.
We output the partial numerators and denominators εk and ak, the complete quotients ξk=(Pk+√d)/Qk and the convergents rk/sk, with the first least period highlighted in boldface.
Here ak=[ξk] or ak=[ξk]+1 and 0< vk/(2vk+sk) <1/2.
e=1 prints the bk and uk/vk.
Warning. The algorithm can get slowed down by the way I calculate ak=[ξk+vk/(2vk+sk)]. e.g. ξ0=(943+√57)/1681, which has period-length 2730. The author has written a faster BC OCF algorithm based on the NSCF algorithm.
Last modified 12th November 2009
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