A Smoothening Method for the Piecewise Linear Interpolation

We propose a method to smoothen a piecewise linear interpolation at a small number of nodes on a bounded interval.Themethod 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.


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 2  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  ≥ 1 whose prototype is a sigmoidal transformation introduced by Prössdorf and Rathsfeld [4].In Section 3, for a set of equally spaced nodes {  } 2  =0 on a bounded interval [, ], we define a modified weight function V , () of an integer order   ≥ 1.Then, for each integer 1 ≤  ≤ , we construct piecewise smooth functions  , (),  = 1, 2, . . ., 2 − , in the th smoothening step so that each  , () reflects the behavior of its precedents  −1,2−1 () and  −1,2 (), depending on order   of the associated weight function V , ().It is found that the resultant approximation   () :=  ,1 () 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.

A Sigmoidal Type Weight Function
For an interval [, ] and a real number  ≥ 1 we set a real valued function Journal of Applied Mathematics which we call a weight function of order .The derivative of   (, ; ) with respect to  is satisfying    (, ; ) > 0 for all  <  < .In addition, We summarize basic properties of   () :=   (, ; ) below.

Smoothening the Piecewise Linear Interpolation
For a given interval [, ], in this paper, we consider a set of equally spaced nodes where  ≥ 2 is an integer not too large.We generalize the weight function   , defined in (1), on [, ] as follows.For an integer  ≥ 1 and for an integer   ≥ 1, define where   and   are some nodes such as  ≤   <   ≤  defined by (7).
The function V , () 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.4], for example, is shown in Figure 1.One can see that V , () becomes flatter outside the interval [  ,   ] as the order   goes higher.This can be surmised from property (P2).
Referring to the aforementioned comments, we may consider that, for small  (i.e., for the small subintervals  2  (−1) ≤  ≤  2   ), order   of the weight function V , () 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   ,  = 1, 2, . . ., , as Theorem 1.For integer orders   , satisfying (21), of the weight functions V , (),  = 1, 2, . . ., , with  = 2  in (8), we have where 0 < ℎ  < 1.

Examples
To perform the presented algorithm we used a Mathematica (V.10) programming code, given in Appendix, where we have taken linearly increasing orders   = 3 − 2 for the weight functions V , ().
We choose a test function below.Consider with 2  + 1 nodes ( = 3) 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   (),  = 1, 2, 3, have highly improved the accuracy of the initial piecewise linear approximation  0 ().In Figure 5, the resultant approximation  3 () is compared with two existing approximations, the Hermite interpolation () and the cubic spline interpolation ().
One can see that  3 () 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  2 -norm error of the presented approximation  3 () at the nodes {  } 8 =0 results in √ ∑ 8 =0 {(  ) −  3 (  )} 2 ≈ 0.068.
In 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.

Conclusions
In this paper, we proposed a method which results in the noninterpolatory approximations   (),  = 1, 2, . . ., , and q 0 (x) q 1 (x)   the last one   () smoothens all the vertices of the initial piecewise linear interpolant  0 ().Although the interpolation property of  0 () 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.

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