70 research outputs found
Effective Genericity and Differentiability
We prove that a real x is 1-generic if and only if every differentiable
computable function has continuous derivative at x. This provides a counterpart
to recent results connecting effective notions of randomness with
differentiability. We also consider multiply differentiable computable
functions and polynomial time computable functions.Comment: Revision: added sections 6-8; minor correction
Time- and Space-Efficient Evaluation of Some Hypergeometric Constants
The currently best known algorithms for the numerical evaluation of
hypergeometric constants such as to decimal digits have time
complexity and space complexity of or .
Following work from Cheng, Gergel, Kim and Zima, we present a new algorithm
with the same asymptotic complexity, but more efficient in practice. Our
implementation of this algorithm improves slightly over existing programs for
the computation of , and we announce a new record of 2 billion digits for
Efficient Multiple-Precision Evaluation of Elementary Functions
Let M(t) denote the time required to multiply two t-digit numbers using base b arithmetic. Methods are presented for computing the elementary functions in O(t1/3 M(t)) time
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