<abstract xmlns="http://www.w3.org/1999/xhtml"><p>We present smooth approximations to the absolute value function |<italic>x</italic>| using sigmoid functions. In particular, <italic>x</italic> erf(<italic>x</italic>/<italic>μ</italic>) is proved to be a better smooth approximation for |<italic>x</italic>| than <italic>x</italic> tanh(<italic>x</italic>/<italic>μ</italic>) and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0002_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mrow><msup><mrow><mi>x</mi></mrow><mn>2</mn></msup><mo>+</mo><mi>μ</mi></mrow></msqrt></mrow></math><tex-math>\sqrt {{x^2} + \mu }</tex-math></alternatives></inline-formula> with respect to accuracy. To accomplish our goal we also provide sharp hyperbolic bounds for the error function.</p></abstract>

We present smooth approximations to the absolute value function |x| using sigmoid functions. In particular, x erf(x/μ) is proved to be a better smooth approximation for |x| than x tanh(x/μ) and x2+μ\sqrt {{x^2} + \mu } with respect to accuracy. To accomplish our goal we also provide sharp hyperbolic bounds for the error function.

Sigmoid functions for the smooth approximation to the absolute value function

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Sigmoid functions for the smooth approximation to the absolute value function"}

We present smooth approximations to the absolute value function |x| using sigmoid functions. In particular, x erf(x/μ) is proved to be a better smooth...

Sigmoid functions for the smooth approximation to the absolute value function

Published Online: Nov 22, 2020

Page range: 12 - 19

Received: Jul 26, 2020

Accepted: Oct 17, 2020

DOI: https://doi.org/10.2478/mjpaa-2021-0002

Keywords
Sigmoid functions, Error function, Smooth approximation, Hyperbolic tangent

© 2020 Yogesh J. Bagul et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Sigmoid functions for the smooth approximation to the absolute value function

Published Online: Nov 22, 2020

Page range: 12 - 19

Received: Jul 26, 2020

Accepted: Oct 17, 2020

DOI: https://doi.org/10.2478/mjpaa-2021-0002

KeywordsSigmoid functions, Error function, Smooth approximation, Hyperbolic tangent

© 2020 Yogesh J. Bagul et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Sigmoid functions, Error function, Smooth approximation, Hyperbolic tangent