Application of Wynn's epsilon algorithm to periodic continued fractions
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Application of Wynn's epsilon algorithm to periodic continued fractions
i + a2 + - in which am + i = am for m^n^O. (2) is periodic with period / and is equal to a quadratic surd if and only if the partial quotients, ak, are integers or rational ...
APPLICATION OF WYNN'S EPSILON ALGORITHM TO PERIODIC CONTINUED FRACTIONS by M. J. JAMIESON (Received 28th October 1985) 1. Introduction The infinite continued fraction (1) i+ a2 + in which (2) am + i = am for m ^ n ^ O is periodic with period / and is equal to a quadratic surd if and only if the partial quotients, ak, are integers or rational numbers [1]. We shall also assume that they are positive. The transformation discussed below applies only to pure periodic fractions where n is zero. Any fraction with n > 0 is equivalent [1] to a pure periodic fraction in the sense that numbers x and y are equivalent if x = (ay + b)/(cy + d) where
\ad — bc\ = l,
(3)
and such a fraction may be included in the discussion provided that it is first replaced by an equivalent pure periodic fraction. The convergents to the continued fraction (1) may be written [1] for
k^O
(4)
i
(5)
where pk and qk satisfy the recurrence relations
where u stands for p or q. We may extend the index k to negative values and start these recurrence relations with the pairs of values p _ 2 = 0,
