A Cumulative Averaging Method for Piecewise Polynomial Approximation to Discrete Data

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1 Applied Mathematical Sciences, Vol. 1, 16, no. 7, HIKARI Ltd, A Cumulative Averaging Method for Piecewise Polynomial Approximation to Discrete Data Beong In Yun Department of Statistics and Computer Science Kunsan National University, Gunsan, Republic of Korea Copyright c 15 Beong In Yun. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. Abstract We present a new family of weight functions to develop a cumulative averaging method for piecewise polynomial interpolations when a discrete data is given. As a result, we have a suitable smooth approximation over the whole region, though the interpolation property becomes loose at most interior nodes. Mathematics Subject Classification: 65H5, 68W5 Keywords: piecewise polynomial interpolation, sigmoidal type weight function, cumulative averaging method 1 Introduction Approximation methods to a given data, for example, Hermite interpolation and spline interpolation have been used in various approximation field such as data fitting, numerical integration and computer graphics [1, 5, 7]. However, we are often in the face of troublesome when the given data is pathological or the number of data is not large enough. Recently, to resolve this problem for a small number of data, the author proposed a non-interpolatory approximation

2 33 Beong In Yun method based on a smoothening process for piecewise linear interpolation, employing the following weight function [9]. h m (α, β; x) = (x α) m (x α) m + (β x) m, α x β (1) for m 1. Though this method results in satisfactory approximation for n +1 data, it requires quite large number of computations as the method includes 1 smoothening processes in each -th step, = 1,,, n. In this wor, motivated by the aforementioned method, we propose a new smooth approximation method which uses weight functions and piecewise interpolating polynomials. We are aiming an approximation method which is available for any type of data and requires preferably small number of computations. In fact, in implementing the presented method we need not solve any system of equations for unnown coefficients unlie the spline approximation. In Section we construct a family of sigmoidal type weight functions in (3) which tae a generalized form of the weight function h m (α, β; x) in (1). Then, in Section 3, we propose a so called cumulative averaging method using the developed weight functions and the piecewise interpolating polynomials. The method results in a plausible approximation that is smooth over the whole region, while the interpolation property becomes loose at most interior nodes. Furthermore, in Section 4, we extend the cumulative averaging method to the piecewise interpolating polynomials of any degree. Numerical results for some examples show that the presented method is competitive with existing approximation methods such as the Hermite and the cubic spline interpolations. Construction of weight functions Suppose that for an integer N a set of nodes x < x 1 < x < < x N () is given. For each subinterval [x, x ], =, 3, 4,, N, we define a real valued function v (x) = (x x ) { m } m, x x x, (3) (x x ) m ξ + x x ξ (x x) where m 1 and ξ is a point such that x < ξ < x. We call it a sigmoidal type weight function of order m as its archetype is the sigmoidal transformation

3 A cumulative averaging method for piecewise polynomial approximation 333 introduced in [6] which is mainly used for numerical evaluation of the wealy singular integrals [ 4, 8]. The derivative of v (x) is v (x) = m(x x )(x ξ ) m (ξ x ) m (x x) m 1 (x x ) m 1 {(ξ x ) m (x x) m + (x ξ ) m (x x ) m } (4) which satisfies v (x) > for all x < x < x. In addition, v (ξ ) = We can observe main properties of v (x) as follows. (i) Values of v m (x) at the points x = x, ξ, x are m(x x ) 4(ξ x )(x ξ ). (5) v (x ) =, v (ξ ) = 1, v (x ) = 1 (6) for each =, 3, 4,, N. In addition, v (x) is strictly increasing on the interval [x, x ] because v (x) > on the interval (x, x ) from (4). (ii) For m large enough v (x) has the asymptotic behavior of ([ ] m ) x x O ξ x, x x < ξ v (x) = ([ ] m ) (7) x 1 + O x x ξ, ξ < x x (iii) v (x) C (x, x ) for any integer m 1. Figure 1 shows, for example, the graph of v 4 (x), x x x 4, with ξ 4 = x 3 for m =, 4, 8 to illustrate the above properties of the proposed weight function. It is noticed that the graph of v 4 (x) becomes flatter near endpoints of the interval [x, x 4 ] as the value of m goes higher. 3 A cumulative averaging method From now on we assume that a data {(x j, y j ) j =, 1,,, N} is given for the nodes {x j } N j= in (). Denote by l (x) a linear function interpolating two consecutive points (x 1, y 1 ) and (x, y ) and set a piecewise linear interpolation, l 1 (x), if x x x 1 l P [1] (x), if x 1 x x (x) := (8). l N (x), if x n 1 x x N

4 334 Beong In Yun 1 m= m=4 m=8.5 x x 1 x x 3 x 4 Figure 1. Graphs of a weight function v 4 (x) with ξ 4 = x 3, x x x 4, for m =, 4, 8. Referring to the function v (x) with ξ = x 1 in (3), we define a weight function w [1] (x) = (x x ) { m } m, x x x. (9) (x x ) m + x 1 x x x 1 (x x) To smoothen the above piecewise linear interpolation P [1] (x) we propose a cumulative (weighted) averaging method using w [1] (x) as follows. Q [1] 1 (x) = l 1 (x), x x x 1 { } (x) = 1 w [1] (x) Q [1] 1 (x) + w[1] (x)l (1) (x), x x x Q [1] for each =, 3, 4,, N. The property (ii) of the weight function v (x) in the previous section implies that, in each th process of (1), the reflection of the approximation Q [1] 1 (x) and linear interpolation l (x) are restricted on the subintervals [x, x 1 ] and [x 1, x ], respectively. It can be seen that from the property (iii) the final cumulative average Q [1] N (x) is smooth on the whole interval x < x < x N instead of losing the interpolation property at the interior nodes. In addition, referring to the property (i), one can see that the remaining interpolations are Q [1] n (x j ) = y j, j =, N 1, N. (11)

5 A cumulative averaging method for piecewise polynomial approximation 335 The first three cumulative averaging processes to obtain Q [1] (x), Q [1] 3 (x) 4 (x) are illustrated in Figure. The thic blue lines indicate the -th and Q [1] cumulative averages Q [1] l (x)(:blac lines) and w [1] (x), =, 3, 4, associated with Q[1] 1 (x)(:purple lines), (x)(:thin blue lines). Additionally, Figure 3 shows the consecutive process of the presented method with m = in the case of N = 6, where a real data is given as {(, 3.1), (1, 3.3), (,.5), (3, 3.7), (4, 3.5), (5, 3.5), (6, 4.5)}. l (x) [1] Q 1 (x) [1] Q (x) l 3 (x) [1] Q 3 (x) l 4 (x) 1.5 [1] w (x) x x 1 x x 3 x 4 [1] (a) Q (x) 1.5 [1] w 3 (x) x x 1 x x 3 x 4 [1] (b) Q 3 (x) 1.5 [1] w 4 (x) x x 1 x x 3 x 4 [1] (c) Q 4 (x) Figure. Illustration for construction of the cumulative averages Q [1] (x) associated with the weight function w [1] (x), =, 3, 4. Similarly, assuming N is an even number, we denote by q (x) a quadratic polynomial which interpolates three consecutive points (x, y ), (x 1, y 1 ) and (x, y ) for each = 1,, 3,, N. Then we consider a piecewise quadratic interpolation, q 1 (x), if x x x q P [] (x), if x x x 4 (x) := (1). q N/ (x), if x N x x N

6 336 Beong In Yun [1] (a) Q (x) 4 6 [1] (b) Q 3 (x) 4 6 [1] (c) Q 4 (x) [1] (d) Q 5 (x) 4 6 [1] (e) Q 6 (x) Figure 3. Cumulative averages Q [1] (x), =, 3, 4, 5, 6, for a given data(n = 6). Modify the weight function v (x) in (3) as w [] (x) = (x x ) { m } (x x ) m x( 1) x m, x x x. (13) + x x ( 1) (x x) Using this weight function, to smoothen the piecewise quadratic polynomial P [] (x) we set a cumulative averaging process as follows. Q [] 1 (x) = q 1 (x), x x x { } (x) = 1 w [] (x) Q [] 1 (x) + w[] (x)q (14) (x), x x x Q [] for each =, 3, 4,, N. The final cumulative average Q[] N/ (x) is smooth on the whole interval x < x < x N. Instead, its interpolation property is loosen at every interior node with the following exceptional cases. Q [] N/ (x j) = y j, j =, N, N. (15) It should be noticed that the total number of averaging processes to obtain the final cumulative average Q [] N/ (x) is N 1.

7 A cumulative averaging method for piecewise polynomial approximation 337 If N is not an even integer, we may tae an additional data as, for example, where x N 1 = x N x N 1. (x N+1, y N+1 ) = (x N + x N 1, y N 1 ) (16) Example 3.1 For a function, we set a data f(x) = exp(x + cos(πx)), 1 x 1 D 1 = {(x j, f(x j )) j =, 1,,, N} with N = 8. Cumulative averages Q [] (x) with m = 4, = 1,, 3, 4, are included in Figure 4 and the last cumulative average Q [] N/ (x) = Q[] 4 (x) is compared with the Hermite interpolation H(x) and the cubic spline interpolation S(x) in Figure 5. Therein, the dotted lines denote the graph of f(x) generating the data D 1. In Figure 5 it is observed that both H(x) and S(x) have some deviation or distortion on the subintervals (x 5, x 6 ) and (x 7, x 8 ) while Q [] 4 (x) has some deviation on (x, x 5 ). Overall, approximation of the proposed cumulative average Q [] 4 (x) seems to be a little better than H(x) and S(x). Figure 4. Cumulative averages Q [] (x), = 1,, 3, 4, for the given data D 1 in Example 3.1(N = 8).

8 338 Beong In Yun (a) Q [] N/ (x) (b) H(x) (c) S(x) Figure 5. Comparison of the presented approximation Q [] N/ (x), the Hermite interpolation H(x) and the cubic spline interpolation S(x) for the data D 1 in Example 3.1(N = 8). Example 3. For a parametric function (x(t), y(t)) = (sin t sin(t/3), cos t + cos(t/3)), representing a smooth curve in the plane we set a data π 3 t 4π D = {(x(t j ), y(t j )) j =, 1,,, N} with N = 8. In this case we separate the given data into x-variable data and y-variable data as D x = {(t j, x(t j )) j =, 1,,, N}, D y = {(t j, y(t j )) j =, 1,,, N}. Then we apply the proposed cumulative averaging method to each data D x and D, y individually. Figure 6 shows cumulative averages Q [] (x) with m = 6, = 1,, 3, 4 and, in Figure 7, the last cumulative average Q [] N/ (x) = Q[] 4 (x) is compared with the Hermite interpolation H(x) and the cubic spline interpolation S(x). Lie the previous example, we can see that the approximation of the proposed cumulative average Q [] 4 (x) is better than H(x) and S(x) over all. 4 Generalized scheme Suppose that a data D = {(x j, y j ) j =, 1,,, N} is given and let d 1 be an integer. In this section we extend the presented cumulative averaging

9 A cumulative averaging method for piecewise polynomial approximation 339 Figure 6. Cumulative averages Q [] (x), = 1,, 3, 4, for the given data D in Example 3.(N = 8). method associated with the linear and quadratic interpolants to any case of degree d of the interpolants. Set N = nd l, l d 1 for some integer n. If l 1 then we may choose additional data such as, for example, (x N+i, y N+i ) = (x N + x N i, y N i ), i = 1,, l, (17) where x N i = x N x N i. This modifies the given data D as D = {(x j, y j ) j =, 1,,, N = nd}. Then, for each = 1,, 3,, n, denote by r (x) a polynomial of degree d which interpolates (d+1) consecutive points on the interval x ( 1)d x x d. Defining a weight function w [d] (x) = (x x ) { m } (x x ) m x( 1)d x m, x x x d, (18) + x d x ( 1)d (x d x)

10 34 Beong In Yun (a) Q [] N/ (x) (b) H(x) (c) S(x) Figure 7. Comparison of the presented approximation Q [] N/ (x), the Hermite interpolation H(x) and the cubic spline interpolation S(x) for the data D in Example 3.(N = 8). we set a cumulative averaging process associated with w [d] (x) as Q [d] 1 (x) = r 1 (x), x x x d { } Q [d] (x) = 1 w [d] (x) Q [d] 1 (x) + w[d] (x)r (19) (x), x x x d for each =, 3, 4,, n. The number of the averaging processes to obtain the final cumulative average Q [d] n (x) is n 1 = N+l 1. d Particulary, for an even integer N, if tae d = N/ as the degree of the interpolating polynomials, then since n = just two polynomials of degree d, r 1 (x) and r (x) are required in the following simple averaging process. { } Q [d] (x) = 1 w [d] (x) r 1 (x) + w [d] (x)r (x), x x x d = x N. () It should be noted that, in using the proposed method (19) associated with the interpolants of higher degree d lie (), the order m of the weight function w [d] (x) should be large enough to prevent excessive loss of the accuracy in approximation to the given data. For example, we chose the data D 1 and D with N = 16 given in the previous section. We too d = 8, that is, n =. Numerical results of Q [d] (x) in () for the data D 1 and D are given in Figure 8 and Figure 9, respectively,

11 A cumulative averaging method for piecewise polynomial approximation 341 to demonstrate availability of the proposed method. Therein, we used the weight function w [d] (x) of order m = 3. In can be seen that partial interpolating polynomials r 1 (x) and r (x) are properly reflected on the half regions [x, x 8 ] and [x 8, x N ], respectively and that their approximation errors in opposite regions are sufficiently reduced via the weight function w [d] (x) of higher degree. (a) r 1 (x), r (x) (b) Q [d] (x) Figure 8. Interpolating polynomials r 1 (x) and r (x) of degree d = N/ in (a) and the presented average Q [d] (x) composed of r 1(x) and r (x) in (b) for the data D 1 in Example 3.1(N = 16). 5 Conclusion In this paper a new family of sigmoidal type weight functions is proposed. Using these weight functions, we developed the so called cumulative averaging method to smoothen the piecewise polynomial interpolation for a given discrete data. Though the interpolation property becomes loose at most interior nodes, we have a plausible approximation which is smooth over the whole region. In practice the approximation errors are negligible at every interior node as long as the order m of the used weight function is chosen large enough. Numerical results of some selected examples illustrate availability of the presented method. Acnowledgements. 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-13R1A1A4A3579).

12 34 Beong In Yun (a) r 1 (x), r (x) (b) Q [d] Figure 9. Interpolating polynomials r 1 (x) and r (x) of degree d = N/ in (a) and the presented average Q [d] (x) composed of r 1(x) and r (x) in (b) for the data D in Example 3.(N = 16). References [1] K.E. Atinson, An Introduction to Numerial Anaysis, second ed., John Wiley and Sons, New Yor, [] D. Elliott, The Euler Maclaurin formula revised, Journal of the Australian Mathematical Society Series B, 4 (E) (1998), E7 - E76. [3] D. Elliott, Sigmoidal transformations and the trapezoidal rule, Journal of the Australian Mathematical Society Series B, 4 (E) (1998), E77 - E137. [4] P.R. Johnston, Application of sigmoidal transformations to wealy singular and near singular boundary element integrals, International Journal for Numerical Methods in Engineering, 45 (1999), [5] D. Kincaid, W. Cheney, Numerial Anaysis, third ed., Broos/Cole, Singapore,. [6] S. Prössdorf, A. Rathsfeld, On an integral equation of the first ind arising from a cruciform crac problem, in Integral Equations and Inverse Problems, V. Petov, R. Lazarov (ed.), Longman: Coventry, 1991, 1-19.

13 A cumulative averaging method for piecewise polynomial approximation 343 [7] L.N. Trefethen, Approximation Theory and Approximation Practice, SIAM, Philadelphia, 13. [8] B.I. Yun, An extended sigmoidal transformation technique for evaluating wealy singular integrals without splitting the integration interval, SIAM Journal on Scientific Computing, 5 (3), [9] B.I. Yun, A smoothening method for the piecewise linear interpolation, Journal of Applied Mathematics, 15 (15), 1-8, Article ID Received: December 8, 15; Published: February, 16