Sigmoid functions for the smooth approximation to the absolute value function

Abstract

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msqrt> <m:mrow> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>+</m:mo> <m:mi>μ</m:mi> </m:mrow> </m:msqrt> </m:mrow> </m:math> \sqrt {{x^2} + \mu } with respect to accuracy. To accomplish our goal we also provide sharp hyperbolic bounds for the error function.

Topics & Concepts