Manprit Kaur and Arun Kumar

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CUBIC X-SPLINE INTERPOLATORY FUNCTIONS Manprit Kaur and Arun Kumar manpreet2410@gmail.com, arun04@rediffmail.com Department of Mathematic and Computer Science, R. D. Univerity, Jabalpur, INDIA.

Here we tudy exitence and error of approximation of cubic pline interpolation having interpolatory condition area matching in each ub-interval.

1 CUBIC X-SPLINE INTERPOLATORY FUNCTIONS Manprit Kaur and Arun Kumar Department of Mathematic and Computer Science, R. D. Univerity, Jabalpur, INDIA. Abtract: Cubic X-pline function wa introduced to generalize cubic pline function. Here we tudy exitence and error of approximation of cubic pline interpolation having interpolatory condition area matching in each ub-interval. Keyword: Cubic X-pline function, Cubic pline function, Interpolation. AMS Claification: MCS A15-1. DEFINITIONS AND NOTATIONS : Let : 0 = x 0 < x 1 < x 2 <...< x n = 1, be a knot equence in the interval [0,1]. We denote the ub-interval of the partition by I i = [, ], i = 1,2,...,n. Let α = {α i } n 1 i=1 be a parameter of n-1 real number. A cubic polynomial function C 1 [0,1] i termed a a Cubic X- Spline on the interval [0,1] with knot equence, parameter α = {α i } n 1 i=1, if it atifie the following three condition: i i a Cubic polynomial on each ub-interval [, ], i=1,2,...,n and it i denoted by i. ii C 1 [0,1], that i, for i = 1,2,...,n, i = i+1, i = i+1. and iii i+1 - i = α i [ i+1 - i ], i = 1,2,...,n. We denote uch cla of pline function by 4,,α

2 The X-Spline wa conidered to generalize the conventional pline. Here the econd derivative i allowed to poe dicontinuitie at the knot. The magnitude of thee dicontinuitie are related to thoe of the dicontinuitie in the third derivative by mean of a imple relationhip which introduce one free parameter to each ub-interval. Thi give a choice of the parameter for meeting certain requirement, for example to chooe it to adjut error. The importance of pline function have been conidered in e.g. [1],[3] and [4]. An error analyi of finding the optimal etimate for the uniform norm of approximation of continuou function by interpolating cubic and quadratic X-pline with arbitrary paced knot reult wa obtained in [2] and [5]. It i intereting to have a reult on Cubic X-pline, o that it include the correponding reult of Cubic Spline a a particular cae. Subequently, the reult concerning X-Spline wa obtained by Da [4]. The error of approximation wa alo tudied. In thi paper we conider X-Spline the ame a conidered by Da [4] and replacing the condition of point interpolation by Area Matching. We firt prove the following theorem and it error of approximation i obtained ubequently. THEOREM : Let f be a 1- periodic bounded continuou function and let { } n i=0 be knot point of the partition with - = h for all i. For a given α = {α i } n 1 i=1 the et of non-negative real number, there exit a unique 1-periodic Cubic X-Spline atifying the interpolating condition : 1 xi xdx = fxdx, 2 if h 4α i α i 1 δ6δ + δ α i δ + α i 1 δ α i 1 + 2δ + α i δ δ. Proof of the theorem : We write = M i, i = 0,1,...,n-1, and let i = y i, i = 1,2,...,n value of the pline defined at on the i-th interval i.e., in [, ]. It can be een that uch pline can be written on the interval [, ], i = 1,2,...,n a 2 140

3 3 h 3 x = M i 1 hx x 2 M i hx 2 x +y i 1 x 2 [2x + h] +y i x 2 [2 x + h]. We have 4 h 3 x = M i 1 h[ x 2 2 xx ] M i h[2x x x 2 ] +2y i 1 [ x 2 x{2x + h}] +2y i [ x 2 + x {2 x + h}]. We find that i = i+1 = M i, i = 0, 1,..., n 1. Thi how that continuou at the knot,i.e., x C 1. Further, we ee that 5 h 3 i x = 6M i 1h + 6M i h + 12y i 1 12y i. Thu pline conidered here are X-pline. i.e., we have i+1 i = α i [ i+1 i ], for i = 1, 2,..., n. On ubtitution in the above X-pline equation we find that 6 2hM i+1 h + 3α i 8h 2 M i 2hM i 1 h 3α i = 6[h + 2α i y i+1 y i + h 2α i y i y i 1 ]. Now applying area matching interpolatory condition 1, we have xi fxdx = M i 1 h 2 x x 2 dx M i h 2 x 2 xdx +y i 1 h 3 { x 2 [2x + h]}dx 3 141

4 i.e., 7 ay. Thi yield, +y i h 3 {x 2 [2 x + h]}dx, xi h 2 fxdx = M i 1 12 M h 2 i 12 + h 2 y i 1 + h 2 y i = F i, 8 F i+1 F i = h2 6 M i h2 12 M i+1 h2 12 M i 1 + h 2 y i+1 y i + h 2 y i y i 1. After eliminating y i in the above equation from 6 and 7, we have 9 W i = pm i 2 + qm i 1 + rm i + M i+1, i = 1, 2,..., n, where W i = 12h 2 [α i 1 h + 2α i F i+1 F i + α i h 2α i 1 F i F i 1 ], p = hα i 4α i α i 1, q = hα i 1 12α i α i hα i, r = 10hα i α i α i 1 + hα i, = hα i 1 + 4α i α i 1. The ytem of equation have unique olution if the coefficient matrix of the above ytem of equation i non-ingular. In order to etablih it non-ingularity, we decompoe the matrix a the following : B = , C = p λ p λ p λ λ p, 4 142

5 where and λ are determined by the et of the following equation : and λ + p = q + λ = r. Subtituting the value of λ, and r we ee that + λ r = 0. The value of are given by the equation αi α i α i 2 + α i 1 10α i 1 + α i h 12α i α i 1 + 4α i α i α i α i 1 + 4α i α i 1 2 = 0. Let u conider = 1 + δ, for mall poitive δ, i.e., δ < 1 2. The value of the above expreion i 12α i 1 1 δ + α i 1 δ α i 1 2δ + α i δ 2 13 δh +24α i α i 1 δ1 δ + 4α i α i 1 δ, which i poitive. And at = 1 δ, it become 12αi δ + α i 1 δ α i 1 + 2δ + α i δ δ h 4α i α i 1 δ6δ + δ 2 + 6, which i negative if 4α i α i 1 δ6δ + δ h 12α i δ + α i 1 δ α i 1 + 2δ + α i δ δ. on δ. Thu, one of the root of lie in [ 1 δ, 1+δ] under the condition Now we how that the matri diagonally dominant with the diagonal element λ. We have λ = α i α i α i h + 4α i α i 1 3 and thi i negative if h 4α iα i 1 3 α i α i α i

6 Thi i true for [ 1 δ, 1 + δ] if 11 h 12α iα i 1 4α i α i δ α i α i α i 1 δ. therefore Next we ee that from the hypothei 2 p i alo negative. Obviouly i poitive. We have λ p λ p 1 + δ 12 D = 2α i 1 + α i 1 δ 10α i + 11α i δ α i δ 2 h 1 + δ Thi i poitive if 13 h < 8α iα i 1 16α i α i 1 δ + 4α i α i 1 δ δ 4α i α i 1 2 4δ + δ 2 α i 10 + δ 2 11δ + 2 δα i 1. We ee that the expreion on the right hand ide of the inequality 10 i maller than the expreion on the right hand ide of the inequalitie 11 and 13. Hence the reult i true if h 4α i α i 1 δ6δ + δ α i δ + α i 1 δ α i 1 + 2δ + α i δ δ. To viualize the above condition we take, for example, δ = 0.1 in thi cae the above condition become h 2.644α i α i α i α i. ERROR OF APPROXIMATION : We write ex = x fx. Since = M i, i = 1,2,...,n, we get e i = M i f i

7 From 9 we find 14 pe i 2 + qe i 1 + re i + e i+1 = W i pf i 2 qf i 1 rf i f i+1, i = 1,2,...,n. We denote the right hand ide of the above expreion by U i, ay. We have U i = 12h 2 [α i 1 h + 2α i F i+1 F i +α i h 2α i 1 F i F i 1 ] hα i 4α i α i 1 f i 2 hα i 1 12α i α i hα i f i 1 10hα i α i α i 1 + hα i f i hα i 1 + 4α i α i 1 f i+1. By Taylor erie expanion, we ee that F i+1 F i = +1 x Hence f ξ i+1 x f i+1 + x f η i x f i dx + h2 U i =h12h 2 +1 x + x f η i x f i dx 2 f i+1 + h2 2 f i. dx f ξ i+1 x f i+1 dxα i 1 + x f ξ i x f i dx + 2 x f η i 1 x f i 1 dxα i +5α i 1 f i+1 4α i 1f i + 5α if i 4α if i 1 α if i 2 α i 1f i 1 +24h 2 α i α i 1 +1 x f ξ i+1 x f i+1 dx + x f η i x f i dx x f ξ i x f i dx 2 x f η i 1 x f i 1 dx +12α i α i 1 f i+1 f i + 4α i α i 1 f i 2 f i

8 U i h 12α i 1ω f ; h +4α i f i f i α i α i 1 ω f ; h, + 12α i ω + α i 1 f i+1 f i 1 18α i 1 + α i h + 4α i α i 1 ω f ; h. f ; h + 4α i 1 f i+1 f i + α i f i f i 2 If we take α = max{α i } then Hence We have U i 36α h + 2α ω f ; h. e i = up e i A 1 U i e i 18α i 1 + α i h + 4α i α i 1 D 1 ω f ; h where D i defined in 12. From the condition 1 there i at leat one point x i 1 in [, ] uch that ex i 1 = 0. Thi give that ex = x x e xdx. i 1 Therefore ex h 18α i 1 + α i h + 4α i α i 1 D 1 ω f ; h. Reference : 1. Ahlberg, J.H. Nilon, E.N and Walh, J.L., The theory of pline and their application, London, Academic Pre. 2. Bujalka A. and Smarzewki R., Quadratic X-Spline, IMA, J.Numer. Analyi, 1982, 2, Clenhaw C.W.; Negu B., The Cubic X-pline and it application to interpolation, J. Int. Math. Appl., , Da, V.B., Spline interpolation by lower degree polynomial uing area matching condition, Ph.D. Thei, Rani Durgawati Uni. 2004, Jabalpur. 5. Smarzewki R. and Bujalka A.,Uniform Convergence of Cubic and Quadratic X-pline interpolant, IMA, J. Numer. Analyi, ,

One Class of Splitting Iterative Schemes

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