Approximation of piecewise continuous functions by a modification of piecewise hermite interpolation

Hertling, Jörg

doi:10.1007/bf02165511

Cited by 4 publications

(1 citation statement)

References 3 publications

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“…As part of an experimental process, it is common to face up to situations where only a limited amount of data is available and the estimation of values between consecutive data points is needed, either for predicting results or inferring conclusions from data. Although this problem is traditionally approached by function approximation methods based on polynomial interpolation, a preferred alternative can be found in the approximation techniques based on piecewise-linear models [7,16,17,5,4] which consists in connecting consecutive data points by straight line segments through a continuous function. However, piecewise-linear models have the shortcomings of having zero curvature between data points, exhibiting abrupt changes at the breakpoints, and having undefined derivatives.…”

Section: Introductionmentioning

confidence: 99%

Smoothing the High Level Canonical Piecewise-Linear Model by an Exponential Approximation of its Basis-Function

Jiménez‐Fernández

Vázquez-Leal

Filobello-Nino

et al. 2016

CyS

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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.

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