Manprit Kaur and Arun Kumar

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. 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: MCS2010 41A15-1. 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. 1 139

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 2 + 6 12α i 1 1 + δ + α i 1 δ 2 + 12α i 1 + 2δ + α i δ 2 13 + 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 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

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 1 + 10hα i, r = 10hα i 1 + 12α 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 = 1 0... 0 0 0 1... 0 0 0 0 1.. 0 0........................ 0 0 0.. 0 1, C = p λ 0 0... 0 0 0 p λ 0... 0 0 0 0 p λ 0.. 0 0.............................. λ 0 0.... 0 p, 4 142

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 1 + 10α i α i 2 + α i 1 10α i 1 + α i h 12α i α i 1 + 4α i α i 1 + 12α 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 2 + 12α 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 1 1 + δ + α i 1 δ 2 + 12α i 1 + 2δ + α i δ 2 13 + δ h 4α i α i 1 δ6δ + δ 2 + 6, which i negative if 4α i α i 1 δ6δ + δ 2 + 6 10 h 12α i 1 1 + δ + α i 1 δ 2 + 12α i 1 + 2δ + α i δ 2 13 + 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 1 + 10α i α i h + 4α i α i 1 3 and thi i negative if h 4α iα i 1 3 α i 1 + 10α i α i. 5 143

Thi i true for [ 1 δ, 1 + δ] if 11 h 12α iα i 1 4α i α i 1 1 + δ α i 1 + 10α 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 2. 1 + 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 2 + 6 12α i 1 1 + δ + α i 1 δ 2 + 12α i 1 + 2δ + α i δ 2 13 + 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 1 13.21α i 1 + 14.531α i. ERROR OF APPROXIMATION : We write ex = x fx. Since = M i, i = 1,2,...,n, we get e i = M i f i. 6 144

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 1 + 10hα i f i 1 10hα i 1 + 12α 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+1. 7 145

U i h 12α i 1ω f ; h +4α i f i f i 1 + 72α 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., 1967. 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, 37-47. 3. Clenhaw C.W.; Negu B., The Cubic X-pline and it application to interpolation, J. Int. Math. Appl., 221978, 109-119. 4. 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, 1983 3, 353-372. 8 146