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Journal of Applied Mathematics
Volume 2015, Article ID 376362, 8 pages
http://dx.doi.org/10.1155/2015/376362
Research Article

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 areindependently 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) satisfiesfor 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 [57].

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)′ satisfiesfor 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).

Figure 1: Graphs of the functions with and for each on a subinterval in (a) and those on the whole interval in (b).

From now on we set for an integer . Then, for a continuous function , we define linear functionseach of which interpolates two points and . Then we consider a piecewise linear interpolation,To smoothen the piecewise linear interpolation over the whole interval we propose an -step strategy using the weight function as follows.

Step  0. Setting initial lines:for each .

Step  1. Smoothening vertices of at the nodes : setfor each , and define a function

Step  2. Smoothening vertices of at the nodes : setfor each , and define a function

Step. Smoothening vertices of at the nodes , : set, for each , and define a function

Step. Smoothening a vertex of at the node : define a final function.

In each th smoothening step () above, function is defined by (8) with an appropriate integer order and nodes set byfor each .

As a result, every vertex of the initial piecewise linear interpolation has been smoothened by , whereas the interpolation property may be weakened as for all ’s except , , and as shown in Theorem 1. In fact, property implies that .

For intuitional comprehension on the presented method, Figure 2 is included. It shows the effect of in the first smoothening step () by comparing results for the orders of the associated weight function . One can see that reflects the behavior of its precedents and over the larger region as order goes higher, preserving the interpolation and the smoothness on the doubled local interval . In general, each th piecewise smooth function , has the same property on the local interval .

Figure 2: Effect of the first step () smoothening for each of the orders of the weight function .

Referring to the aforementioned comments, we may consider that, for small   (i.e., for the small subintervals ), order of the weight function might be low in order to sustain the natural behavior in the range between interpolation nodes. For large   (i.e., for the large subintervals), higher orders are required to prevent loss of the accuracy at the nodes except the endpoints and the midpoint of the subinterval. Thus, for practical performance of the presented method, we suggest integer orders , as

Theorem 1. For integer orders , satisfying (21), of the weight functions , , with in (8), we havewhere .

Proof. For each and , we havefrom the definition of and property (P1) of the weight function . Similarly, for the midpoint ,To prove the approximation property of at interior points , we first consider the case of . Definition of with the related nodes and in (20) and the assumption (21) for the order of imply that as goes to the infinity, where the last equality results from . Thus, it follows thatIn fact, since and in (20) for definition of ,Then from property (P2) of the weight function of order with and we havewhere . This results inBy a similar way, we have, for any odd integer ,For the case of , an even number, where the last equality results from . ThusSince and in (20) for definition of ,Then from property (P2) of the weight function of order with and we haveThusBy a similar way, we have, for any even integer ,The proof is completed.

The concept of the presented method for a case of is illustrated in Figure 3. Therein one can see that nonsmooth vertices at the nodes appear in the initial piecewise linear interpolation. Figure 3 explains that the vertices at the nodes , and are blunted in the first smoothening step () and those at the nodes and are blunted in the second step (). Finally, the remaining vertex at is blunted in the last step ().

Figure 3: Illustration of the presented method for the case of . The symbol denotes an interpolation point at the th node which is smoothened by in the th smoothening step.

For implementation of the presented method, we write the following algorithm with a given data and properly chosen orders , , based on condition (21):While Set While While Set the th smoothening interpolation,

4. Examples

To perform the presented algorithm we used a Mathematica (V.10) programming code, given in Appendix, where we have taken linearly increasing orders for the weight functions .

We choose a test function below. Considerwith nodes ()Results of the presented method are given in Figure 4, which shows consistency with the illustration given in Figure 3 and Theorem 1. The dotted curve indicates the original graph of , and it is observed that , , have highly improved the accuracy of the initial piecewise linear approximation . In Figure 5, the resultant approximation is compared with two existing approximations, the Hermite interpolation and the cubic spline interpolation . One can see that is better than and in approximation over the whole interval, while the interpolation property at the given nodes is weakened as shown in Theorem 1. In fact, computation of the -norm error of the presented approximation at the nodes results in .

Figure 4: Smoothened piecewise linear interpolation , , for the example with . The dotted curve indicates the original graph of .
Figure 5: Comparison of the presented approximation , Hermite interpolation , and spline interpolation for with . Dotted curves indicate the original graph of .

For another example, we take a parametric functionwith the nodesIn this case, the presented method is applied to both functions and . Figures 6 and 7 include the results of the presented method, which shows the same availability and efficiency of the method as the case of the previous test function.

Figure 6: Smoothened piecewise linear interpolations , , for the parametric function with . The dotted curve indicates the original graph of .
Figure 7: Comparison of the presented approximation , Hermite interpolation , and spline interpolation for the parametric function with . Dotted curves indicate the original graph of .

5. Conclusions

In this paper, we proposed a method which results in the noninterpolatory approximations , , and the last one smoothens all the vertices of the initial piecewise linear interpolant . Although the interpolation property of at most nodes is weakened by as the approximation property in Theorem 1, we have obtained smoothness over the whole interval. Moreover, in the results of the numerical implementation for some examples, we have found that the accuracy of the presented approximation is somewhat better than that of the Hermite interpolation and the spline interpolation over the whole interval.

The performance of the presented method depends on the orders of the associated weight functions, and analysis for the problem of choosing optimal orders is left as a further work. In addition, the idea proposed in this work may be extended to the higher order piecewise polynomial interpolation.

Appendix

See Algorithm 1.

Algorithm 1: Mathematica code for the presented method.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to show sincere gratitude to reviewers for their helpful comments and valuable suggestions on the first draft of this paper. This research was supported by Basic Science Research program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2013R1A1A4A03005079).

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