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## Computación y Sistemas

*Print version* ISSN 1405-5546

#### Abstract

JIMENEZ-FERNANDEZ, Victor M. et al. Smoothing the High Level Canonical Piecewise-Linear Model by an Exponential Approximation of its Basis-Function.* Comp. y Sist.* [online]. 2016, vol.20, n.2, pp.227-237.
ISSN 1405-5546. http://dx.doi.org/10.13053/cys-20-2-2290.

Piecewise-linear models constitute an attractive alternative to construct a function whose graph fits a finite set of discrete points. These models are preferably selected over other approximation strategies like polynomials or splines. Although there are several piecewise-linear models reported in literature, the so-called High Level Canonical has the remarkable advantage of emerging from a well-structured algorithmic methodology to efficiently determine the parameters of any given piecewise-linear function. However, as it happens in all other piecewise-linear models, it also has the problem of lack of differentiability at the breakpoints. In order to solve this problem, an approach based on an exponential approximation of the basis-function is proposed as a strategy to transform the High Level Canonical piecewise-linear model into a smooth-piecewise one. This mathematical transformation ensures the existence and continuity of the nth-order derivatives of the resulting smooth model. Besides of this, it is observed that by applying the piecewise-linear to smooth transformation, the number of terms of the resulting smooth representation can significantly be reduced due to a great number of them can be approximated by a line equation. In order to verify the effectiveness of this proposal, numerical simulations performed on one-dimensional and two-dimensional functions are reported.

**Keywords
:
**High-Level-Canonical; piecewise-linear; smoothing; basis-function; approximation.