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The EH interpolation spline and its approximation.

1. Introduction

Spline interpolation is a useful and powerful tool for curves and surfaces modeling. Standard cubic Hermite spline is one of those interpolation functions. But, for the given interpolation condition, the cubic Hermite interpolation spline is fixed; that is to say, the shape of the interpolation curve or surface is fixed for the given interpolation data [16]. Since the interpolation function is unique for the given interpolation data, to modify the shape of the interpolation curve to approximate the given curve seems to be impossible and it is contradictory to the uniqueness of the interpolation function for the given interpolation data. For the given interpolation condition, how to improve the approximation accuracy of the interpolation spline is an important problem in the computer aided geometric design. In recent years, many authors have presented some new method to modify the shape of the interpolation curve to satisfy the industrial product design with several kinds of new interpolation splines with parameters [7-20].

These new splines all have similar properties of the standard cubic Hermite spline. For example, for the given interpolation data, if interpolation interval approaches zero, theoretically speaking, these splines can approximate the given curve and surface well. However, there exists a problem, in the process of the actual calculation; the amount of computation will increase dramatically if the length of the interpolating intervals decreases. On the other hand, the approximation accuracy of these new splines may not be better than the standard cubic Hermite spline.

In [7-17], many rational form interpolation splines with multiple parameters were presented. For the given interpolation data, the change of the parameters causes the change of the interpolation curve. Nevertheless, the computation of the splines with multiple parameters is very complicated. Several kinds of rational splines with a single parameter were presented in the papers [18,19], which is simple to compute, but its approximation accuracy is not good for the given curves and surfaces. In general, polynomial-form splines are suitable to be used to design and compute. In [20], a polynomial-form spline, called quartic Hermite spline with single parameter, is presented as the extension of the standard Hermite spline. The quartic spline has a simple form, and its approximation rate to the given curves and surfaces is not high.

In this paper, a class of new quartic splines with two parameters is developed, which is the extension of the standard cubic Hermite interpolation spline and inherits the same properties of the standard cubic Hermite interpolation spline. For the given interpolation conditions, the shape of the proposed splines can be adjusted by changing the values of the parameters. Furthermore, the introduced splines could approximate to the interpolated functions better than the standard cubic Hermite interpolation splines and the quartic Hermite interpolation splines with single parameter.

The remainder of the paper is organized as follows. Section 2 introduces the standard cubic Hermite spline and some of its properties. A kind of interpolation spline with two parameters is presented in Section 3. Section 4 discusses the approximation of the introduced spline curve with numerical examples. In the end, a summary and conclusions are given.

2. The Standard Cubic Hermite Spline and Its Basis Functions

Generally, for t [member of] [0,1], the following four basis functions,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

are called the standard cubic Hermite bases.

These bases satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

For given knots a = [x.sub.0] < [x.sub.1] < ... < [x.sub.n] = b and data {([x.sub.i], [y.sub.i], [d.sub.i]), i = 0, 1, ..., n}, where [y.sub.i] and [d.sub.i] are the values of the function value and the first-order derivative value of the function being interpolated, let [h.sub.i] = [x.sub.i+1] - [x.sub.i], t = (x - [x.sub.i])/[h.sub.i] and then the standard cubic Hermite spline in the interval [[x.sub.1], [x.sub.i+1]] can be defined as follows:

[H.sub.i](x) = [[alpha].sub.0] (t) [y.sub.i] + [[alpha].sub.1] (t) [y.sub.i+1] [[beta].sub.0](t)[h.sub.i][d.sub.i+1], i = 0, 1, ..., n - 1. (3)

Obviously, we have [H.sub.i]([x.sub.i]) = [y.sub.i], [H.sub.i]([x.sub.i+1]) = [y.sub.i+1], [H'.sub.i],([x.sub.i]) = [d.sub.i], [H'.sub.i]([x.sub.i+1]) = [d.sub.i+1].

The standard cubic Hermite spline is [C.sup.1] continuous. However, ifinterpolation data is given, the shape and approximation of the spline cannot be changed.

3. The EH Interpolation Spline

In order to overcome the disadvantage of the standard cubic Hermite spline, we extend its basis functions firstly.

3.1. The Basis Functions of the EH Interpolation Spline

Definition 1. For arbitrary real number [[lambda].sub.i], [[mu].sub.i] and 0 [less than or equal to] t [less than or equal to] 1, the following four functions with parameters [[lambda].sub.i], [[mu].sub.i],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

are calledbasis functions of the EH interpolation spline, briefly EH bases.

The EH bases are the extension of the standard cubic Hermite bases. When [[lambda].sub.i] = [[mu].sub.i] = 0, the bases are the standard cubic Hermite bases. The bases have the similar properties of the standard cubic Hermite bases.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

When [[lambda].sub.i] = [[mu].sub.i], the EH bases 4) are basis functions with single parameter in [20].

Figure 1 shows the four EH bases, where the solid lines are the standard Hermite bases, the parameters [[lambda].sub.i] = 2, [[mu].sub.i] = -2 are for the dot-dash lines, and [[lambda].sub.i] = -2, [[mu].sub.i] = 2 are for dashed line.

So, we may construct the Ferguson curve with two parameters based on the EH bases as follows:

[EH.sub.i] (f) = e[[alpha].sub.0] (t) [P.sub.i] + e[[alpha].sub.1] (t) [p.sub.i+1] + e[[beta].sub.0][p'.sub.i] + e[[beta].sub.1](t) [p'.sub.i+1], (5)

where [p.sub.i], [p.sub.i+1] and [p'.sub.i], [p'.sub.i+1] are two interpolation points and their tangent vectors, respectively.

For the given two interpolation points and tangent vectors, with the different parameters [[lambda].sub.i], [[mu].sub.i], we may obtain different shape of the Ferguson curve with two parameters accordingly.

Figure 2 shows the Ferguson curves with different parameters, where the solid line is the standard Ferguson spline curve with [[lambda].sub.i] = [[mu].sub.i] = 0, the parameters [[lambda].sub.i] = 2, [[mu].sub.i] = -2 are for the dot-dash line, and [[lambda].sub.i] = -2, [[mu].sub.i] = 2 are for dashed line.

3.2. The EH Interpolation Spline

Definition 2. Given a data set {([x.sub.i], [y.sub.i], [d.sub.i]), i = 0, 1, ..., n}, where [y.sub.i] and [d.sub.i] are the values of the function value and the first-order derivative value of the function being interpolated and a = [x.sub.0] < [x.sub.1] < ... < [x.sub.n] = b is the knot spacing, let [h.sub.1] = [x.sub.i+1] - [x.sub.i,t] = (x - [x.sub.i])/[h.sub.i]; then the EH interpolation spline in the interval [[x.sub.t], [x.sub.i+1]] can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

where e[[alpha].sub.0](t), e[[alpha].sub.1](t), e[[beta].sub.0](t), e[[beta].sub.1](t) are the EH bases.

Obviously, for the data set {([x.sub.i], [y.sub.i], [d.sub.i]), i = 0, 1, ..., n}, EH(v) satisfies

RH ([x.sub.i]) = [y.sub.i], RH' ([x.sub.i]) = [d.sub.i], i = 0, 1, ..., n. (7)

If [[lambda].sub.i] = [[mu].sub.i], = 0, it is just the standard cubic Hermite spline. It is of interest that, for suitable selected parameters [[lambda].sub.i], [[mu].sub.i], the piecewise interpolation function EH(x) can be [C.sup.2]-continuous in the interval [[x.sub.0], [x.sub.n]]. In fact, denote [[DELTA].sub.i] = ([y.sub.i+1] - [y.sub.i])/[h.sub.i], and let

EH" (x+) = EH" ([x.sub.i]-), I = 1, 2, ..., n - 1; (8)

then the equations connecting the parameters [[lambda].sub.i] and [[mu].sub.i],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

may be obtained. If the successive parameters ([[lambda].sub.i-1], [[mu].sub.i- 1]) and ([[lambda].sub.i], [[mu].sub.i]) satisfy (9) at i = 1, 2, ..., n - 1, then EH(x) [member of] [C.sup.2]([x.sub.0], [x.sub.n]). Furthermore, if the knots are equally spaced and [[lambda].sub.i] = [[mu].sub.i] = 0, then (9) becomes the well-known tridiagonal system for a cubic spline

[d.sub.i-1] + 4[d.sub.i] + [d.sub.i+1] = 3 ([[DELTA].sub.i-1] + [[DELTA].sub.i]) , i = 1, 2, ..., n - 1. (10)

Hence, if given the parameter values [[lambda].sub.0], [[mu].sub.0] in the interval [[x.sub.0], [x.sub.1]], by (9), we may obtain the [[lambda].sub.1] and [[mu].sub.1] and so on. Thus, we can construct a [C.sup.2]-continuous interpolation curve.

4. The Approximation of the EH Interpolation Spline

According to the interpolation reminder of cubic Hermite spline, when interpolation interval approaches zero, the cubic Hermite spline curve can approximate well to the function being interpolated. However, for the EH interpolation spline we constructed, it can approximate well to the function being interpolated without interpolation interval approaching zero, and it can approximate to the interpolated functions better than the standard cubic Hermite interpolation spline.

Firstly, we give the definition of the "good approximation"

Definition 3. Let [H.sub.i](x) be the standard cubic Hermite spline, E[H.sub.i](x) be the EH interpolation spline, and y(x) be the function being interpolated. Denoting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then if EH[[epsilon].sub.i] < H[[epsilon].sub.i], we can call R[H.sub.i](v) has "good approximation" to the interpolated functiony(v) better than [H.sub.i](x).

According to the Definition 3, if EH[[epsilon].sub.i] < H[[epsilon].sub.i], we can get the range of the parameters value, [[lambda].sub.i] and [[mu].sub.i]. In the range of the parameters value, selecting the arbitrary values of the parameters [[lambda].sub.i] and [[mu].sub.i], we have a "good approximation" curve.

Example 4. Given the function y(x) = x + cos(([pi]/2)x) and knots [x.sub.i] = (i/2) (i = 0,1,5), namely, [h.sub.i] = (i/2) (i = 0,..., 4). According to the inequality EH[[epsilon].sub.i] < H[[epsilon].sub.i], we may get the range of the parameters [[lambda].sub.i], and [[mu].sub.i]. For the fixed interpolation condition, the max error and the parameters [[lambda].sub.i] and [[mu].sub.i] are given for every interval [[x.sub.i], [x.sub.i+1]] in Table 1. The error curves of the EH(v) and H(x) to y(x) are shown in Figure 3.

By using the tensor product method, we can construct the EH interpolation spline surfaces, which has the similar EH interpolation spline curve.

Definition 5. Let [OMEGA]: [a,b] x [c,d] be the plane region and f(x, y) a bivariate function defined in the region Q and let a = [x.sub.0] < [x.sub.1] < ... < [x.sub.m] = b and c = [y.sub.0] < [y.sub.1] < ... < [y.sub.n] = d be the knot sequences. Denote [h.sub.i] = [x.sub.i+1] - [x.sub.i], [h.sub.j] = [y.sub.j+1] - [y.sub.j], u = (x - [x.sub.i])/[h.sub.i], v = (y - [y.sub.j])/[h.sub.j]; then the EH interpolation spline surface on the region [[x.sub.i], [x.sub.i+1]] x [[y.sub.i], [y.sub.i+1]] can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12

Given the end-points, the first order partial derivative and the second-order blending partial derivative of the function interpolated, with proper parameters, the EH interpolation spline surfaces could approximate to the bivariate functions being interpolated better than the standard cubic Hermite spline surfaces.

Example 6. Given the bivariate function being interpolated f(x, y) = sin([pi]/2)x cos([pi]/2)y, let a = 0 < 1 < 2 = b and c = -1 < 0 < 1 = d be the knot sequences. Denote [h.sub.i] = [x.sub.i+1] - [x.sub.i], [h.sub.j] = [y.sub.j+1] -[y.sub.j],u = (x-[x.sub.i])/h, and v = (y - [y.sub.j])/[h.sub.j]. By selecting [[lambda].sub.0] = [[lambda].sub.1] = 0.3208, [[mu].sub.0] = [[mu].sub.1] = 0.6995, we can work out that the max error of the EH(v, y) - f(x, y) equals 0.5069 x [10.sup.-3], but the max error of the H(x, y)-f(x, y) equals 0.1061 x [10.sup.-1].

Figure 4 shows the error surface of the EH(x, y)- f(x, y). Figure 5 shows the error surface of the H(x, y) - f(x, y).

5. Conclusion

This paper introduced a kind of EH interpolation spline, which is the extension of the standard cubic Hermite interpolation spline. The shape of the proposed splines can be adjusted by changing the values of the parameters for the fixed interpolation conditions. Also, the introduced spline could approximate to the interpolated function better than the standard cubic Hermite interpolation spline.

http://dx.doi.org/10.1155/2014/745765

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work was funded by the Natural Science Foundation of Anhui Province of China under Grant no. 1208085MA15, the Key Project Foundation of Scientific Research, Education Department of Anhui Province under Grant no. KJ2014ZD30, and the Key Construction Disciplines Foundation of Hefei University under Grant no. 2014XK08.

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Jin Xie (1) and Xiaoyan Liu (2)

(1) Department of Mathematics and Physics, Hefei University, Hefei 230601, China

(2) Department of Mathematics, University of La Verne, La Verne, CA 91750, USA

Correspondence should be addressed to Jin Xie; hfuuxiejin@126.com

Received 28 February 2014; Accepted 15 June 2014; Published 30 June 2014

Academic Editor: Grzegorz Nowak

TABLE 1: The parameters and for EH interpolation spline and the max
error.

[x.sub.i]   [y.sub.i]   [d.sub.i]   [[lambda].sub.i]   [[mu].sub.i]

0.0000       1.0000      1.0000          0.0421           0.0412
0.5000       1.2071      -0.1107         0.0146           0.0129
1.0000       1.0000      -0.5708         0.2451           0.2783
1.5000       0.7955      -0.1451         0.0188           0.0192
2.0000       1.0000      1.0000          0.0108           0.0108
2.5000       1.7929      2.1107

[x.sub.i]   RH[[epsilon].sub.i]     H[[epsilon].sub.i]

0.0000      0.2569 x [10.sup.-4]   0.9062 x [10.sup.-3]
0.5000      0.2111 x [10.sup.-4]   0.2569 x [10.sup.-3]
1.0000      0.3769 x [10.sup.-4]   0.2569 x [10.sup.-3]
1.5000      0.2974 x [10.sup.-4]   0.1069 x [10.sup.-2]
2.0000      0.1735 x [10.sup.-4]   0.8647 x [10.sup.-3]
2.5000
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Title Annotation:Research Article
Author:Xie, Jin; Liu, Xiaoyan
Publication:Abstract and Applied Analysis
Article Type:Report
Date:Jan 1, 2014
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