Fast computation of continued fractions
نویسندگان
چکیده
منابع مشابه
Computation of Continued Fractions without Input Values
An algorithm for the computation of the continued fraction expansions of numbers which are zeros of differentiable functions is given. The method is direct in the sense that it requires function evaluations at appropriate steps, rather than the value of the number as input in order to deliver the expansion. Statistical data on the first 10000 partial quotients for various real numbers are also ...
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For x ∈ [0, 1), let x = [a 1 (x), a 2 (x), · · · ] be its continued fraction expansion with partial quotients {a n (x), n ≥ 1}. E(ψ) := x ∈ [0, 1) : lim n→∞ 1 ψ(n) n j=1 log a j (x) = 1 is completely determined without any extra condition on ψ.
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For an irrational number x ∈ [0, 1), let x = [a1(x), a2(x), · · · ] be its continued fraction expansion. Let ψ : N → N be a function with ψ(n)/n → ∞ as n → ∞. The (upper, lower) fast Khintchine spectrum for ψ is defined as the Hausdorff dimension of the set of numbers x ∈ (0, 1) for which the (upper, lower) limit of 1 ψ(n) ∑n j=1 log aj(x) is equal to 1. The fast Khintchine spectrum was determi...
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We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et. al., we focus on a new generalization to an arbitrary integer base b. We show that all of our so-called type III continued logarithms converge and all rational numbers have fin...
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ژورنال
عنوان ژورنال: Computers & Mathematics with Applications
سال: 1991
ISSN: 0898-1221
DOI: 10.1016/0898-1221(91)90095-l