Sigmoid Functions Cut Function Generalized Cut Function of Degree P+1 Step Function Logistic Function Shifted Logistic Function Uniform and Hausdorff Approximation
Issue Date:
2015
Publisher:
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation:
Serdica Journal of Computing, Vol. 9, No 1, (2015), 93p-104p
Abstract:
We introduce a modification of the familiar cut function by
replacing the linear part in its definition by a polynomial of degree p + 1
obtaining thus a sigmoid function called generalized cut function of degree
p + 1 (GCFP). We then study the uniform approximation of the (GCFP)
by smooth sigmoid functions such as the logistic and the shifted logistic
functions. The limiting case of the interval-valued Heaviside step function
is also discussed which imposes the use of Hausdorff metric. Numerical
examples are presented using CAS MATHEMATICA.