Journal of Applied Mathematics

Volume 2015 (2015), Article ID 376362, 8 pages

http://dx.doi.org/10.1155/2015/376362

## A Smoothening Method for the Piecewise Linear Interpolation

Department of Statistics and Computer Science, Kunsan National University, Gunsan 573-701, Republic of Korea

Received 11 May 2015; Accepted 2 July 2015

Academic Editor: Xiao-wei Gao

Copyright © 2015 Beong In Yun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose a method to smoothen a piecewise linear interpolation at a small number of nodes on a bounded interval. The method employs a sigmoidal type weight function having a property that clusters most points on the left side of the interval toward 0 and those on the right side toward 1. The proposed method results in a noninterpolatory approximation which is smooth over the whole interval. We provide an algorithm for implementing the presented smoothening method. To demonstrate usefulness of the presented method we introduce some numerical examples and investigate the results.

#### 1. Introduction

It is well known that, by Weierstrass approximation theorem, every continuous function on a bounded interval can be approximated arbitrarily accurately by polynomials. Nevertheless, it is also true that there is no fixed array of interpolation points that achieves convergence for all continuous functions as mentioned in the literature [1]. To overcome this problem, the piecewise polynomial interpolation or rational function approximation may be considered. In particular, piecewise polynomial functions such as spline functions have been used in various approximation fields including computer graphics, data fitting, numerical integration, and differential equations [2, 3]. However, it is not difficult to find troublesome examples for which the existing approximation methods will not suitably work when the number of interpolation nodes is not large enough.

In this paper, we propose a new noninterpolatory approximation method that is based on a smoothening process for the piecewise linear interpolation at a small number of nodes on a given interval. The main purpose of this work is summarized as follows:(i)Rendering the piecewise linear interpolant smooth, that is, infinitely differentiable over the whole interval.(ii)Improving the accuracy of the approximation over the interval except the interpolation nodes.To fulfill this purpose, we construct a rational function denoted by which smoothens all the vertices of the initial piecewise linear interpolant for nodes with a small integer . From numerical results for some examples one can see that the presented method is available and comparable with existing outstanding methods.

Contents of this paper are as follows. In Section 2, we employ a weight function in (1) of order whose prototype is a sigmoidal transformation introduced by Prössdorf and Rathsfeld [4]. In Section 3, for a set of equally spaced nodes on a bounded interval , we define a modified weight function of an integer order . Then, for each integer , we construct piecewise smooth functions , , in the th smoothening step so that each reflects the behavior of its precedents and , depending on order of the associated weight function . It is found that the resultant approximation is in and noninterpolatory with the approximation property at the given nodes as shown in Theorem 1. Section 4 includes numerical examples of some test functions whose results show the availability of the presented method.

#### 2. A Sigmoidal Type Weight Function

For an interval and a real number we set a real valued functionwhich we call a weight function of order . The derivative of with respect to issatisfying for all . In addition,

We summarize basic properties of below. (P1)Values of at the points are independently of the order . In addition, is strictly increasing from 0 to 1 on the interval since in (2). (P2)For sufficiently large asymptotic behavior of is (P3) satisfies for a central point . (P4) for any real .

By the change of variables the weight function becomes the so-called elementary sigmoidal transformation , defined in [4]. In general, traditional sigmoidal transformations have been used for numerical evaluation of the singular integrals [5–7].

#### 3. Smoothening the Piecewise Linear Interpolation

For a given interval , in this paper, we consider a set of equally spaced nodeswhere is an integer not too large. We generalize the weight function , defined in (1), on as follows. For an integer and for an integer , definewhere and are some nodes such as defined by (7).

The function with an integer order has inherited the properties (P1)–(P4) of on the restricted interval , including additional properties (P3)′ and (P4)′ on the whole interval below. (P3)′ satisfies for a central point . (P4)′.

General behavior of , with , for example, is shown in Figure 1. One can see that becomes flatter outside the interval as the order goes higher. This can be surmised from property (P2).