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Sigmoid functions for the smooth approximation to $ \vert{x}\vert $
Version 1
: Received: 11 March 2019 / Approved: 13 March 2019 / Online: 13 March 2019 (09:12:38 CET)
A peer-reviewed article of this Preprint also exists.
Journal reference: Moroccan Journal of pure and applied analysis 2020, 7, 12-19
DOI: 10.2478/mjpaa-2021-0002
Abstract
We present smooth approximations to $ \vert{x} \vert $ using sigmoid functions. In particular $ x\,erf(x/\mu) $ is proved to be better smooth approximation for $ \vert{x} \vert $ than $ x\,tanh(x/\mu) $ and $ \sqrt{x^2 + \mu} $ with respect to accuracy. To accomplish our goal we also provide sharp hyperbolic bounds for error function.
Keywords
Sigmoid functions; error function; smooth approximation; Hyperbolic tangent
Subject
MATHEMATICS & COMPUTER SCIENCE, Analysis
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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