Jurnal Teknologi POSITIVITY PRESERVING INTERPOLATION BY USING RATIONAL CUBIC BALL SPLINE. Full Paper. Samsul Ariffin Abdul Karim *

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1 Jurnal Teknolog POSITIVITY PRESERVING INTERPOLATION BY USING RATIONAL UBI BALL SPLINE Samsul Arffn Abdul Karm * Fundamental Appled Scences Department, Unverst Teknolog PETRONAS, 360 Ser Iskar, Perak Darul Rdzuan, Malaysa Full Paper Artcle hstory Receved 0 Aprl 06 Receved n revsed form 6 June 06 Accepted 8 October 06 *orrespondng author samsul_arffn@petronas.com.my Graphcal abstract Abstract Ths paper dscusses the postvty preservng by usng ratonal cubc Ball nterpolant of the form cubc/quadratc wth two parameters. The suffcent condton for the postvty s derved on one parameter meanwhle the other one s a free parameter to control the fnal shape of the nterpolatng curves. The degree smoothness acheved s. From numercal results, the ratonal cubc Ball splne wth two parameters gves smooth nterpolatng postve curves as well as vsually pleasng for computer graphcs vsualzaton. Furthermore the scheme s better than exstng schemes.e. ts easness to use less computaton. All numercal results are produced by usng Mathematca. Keywords: Ratonal cubc Ball nterpolant, postvty, dependent, vsualzaton Abstrak Makalah n membncangkan pengekalan kepostfan dengan menggunakan pengnterpolas kubk nsbah Ball dalam bentuk kubk/kuadratk dengan dua parameter. Syarat cukup untuk kepostfan akan dhaslkan pada satu parameter manakala yang lag satu adalah parameter bebas untuk mengawal bentuk akhr lengkung nterpolas. Darjah keselanjaran yang dcapa adalah. Darpada keputusan berangka, kubk spln nsbah Ball dengan dua parameter memberkan lengkung nterpolas yang postf serta gambaran jelas untuk grafk komputer. Dsampng tu skema n lebh bak darpada skema seda ada.e. mudah untuk dgunakan dan juga kurang pengraan. Semua keputusan berangka dhaslkan dengan menggunakan persan Mathematca. Kata kunc: Pengnterpolas kubk nsbah Ball, kepostfan, kebergantungan, pemaparan 06 Penerbt UTM Press. All rghts reserved.0 INTRODUTION Shape preservng approxmaton nterpolaton are mportant n scences engneerng based applcatons. One of the requrements s to preserves the postvty of the data sets. For nstance the level of Sodum Hydroxde (NaOH) are always postve any nterpolant must be able to produce the nterpolatng curves wth postvty preservng. The ranfall dstrbutons also show a postve value any negatvty values just smply unrealstc. The common shape preservng nterpolaton schemes usng ratonal cubc splne of the form cubc numerator denomnator functon can be lnear or quadratc or cubc wth up to four parameters n the descrpton of the ratonal cubc nterpolant. Recently, several researchers have proposed shape preservng nterpolaton by usng ratonal cubc Ball nterpolant. For nstance Karm [4] proposed ratonal cubc Ball nterpolant wth two parameters. Meanwhle Karm [5, 6] proposed new ratonal cubc nterpolant wth four parameters for postvty monotoncty preservng nterpolaton. Karm [7] 78: (06) eissn 80 37

2 4 Samsul Arffn Abdul Karm / Jurnal Teknolog (Scences & Engneerng) 78: (06) 4 48 dscussed the postvty monotoncty preservng nterpolaton usng ratonal cubc Ball nterpolant (cubc numerator quadratc denomnator) wth three parameters. Most recently Jaafar et al. [3] dscussed the postvty preservng nterpolaton usng ratonal cubc Ball nterpolant of the form cubc numerator cubc denomnator wth four parameters. Ther work s n lne wth Karm [5, 6]. Tahat et al. [3] also dscussed the postvty usng dfferent form of ratonal cubc Ball nterpolant wth four parameters. Besdes the use of ratonal cubc Ball nterpolant, most researchers have used the ratonal cubc splne of the form of cubc numerator wth lnear, quadratc or cubc functon as denomnator wth up to four parameters n the descrpton of the ratonal cubc nterpolant. For nstance Abbas et al. [] dscussed the postvty preservng nterpolaton usng ratonal cubc splne (quadratc denomnator) wth three parameters. The suffcent are derved on one parameter meanwhle the remanng two are free parameters. They clamed that ther schemes s. But ther schemes suffer from the fact that t may not generate the postve nterpolatng curves wth contnuty. Brodle Butt [] Butt Brodle [5] dscussed the convexty postvty preservng nterpolaton usng cubc Hermte splne. The postvty are acheved by nsertng one or two extra knots along the nterval n whch shape volaton are found. Bordle et al., (995) extend the dea n [5] for postve surface nterpolaton. Brodle et al. [4] also dscussed the postvty preservng but usng the Shepard nterpolaton famly. The postvty data constraned are acheved by solvng some optmzaton problem. Hussan et al. [8] dscussed the postvty convexty shape preservng wth contnuty. In general there are many researchers nvestgated the postvty preservng nterpolaton, for nstance [9, 0,, -3, 8-]. Some good surveys on shape preservng approxmaton nterpolaton can be found n [6], [7] [4]. In ths paper new ratonal cubc Ball splne nterpolant (cubc denomnator) wth two parameters s constructed for postvty preservng nterpolaton. Ths work s nspred by the work of Sarfraz et al., (03). We dentfy several nce features of our ratonal cubc splne for postvty preservng. It s summarzed below: () () A new ratonal cubc Ball nterpolaton (cubc/quadratc) wth two parameters has been used for postvty preservng. Meanwhle Jaafar et al. [3] utlzed the ratonal cubc Ball nterpolant (cubc denomnator) wth four parameters that appear prevously n [5, 6]. The ratonal cubc Ball splne scheme does not requred any extra knots but Brodle Butt [3], Brodle et al. [4] Butt Brodle [5] requres the extra knots (one or two) to preserves the postvty of the data. () Our ratonal scheme s based from cubc Ball functon whle n Hussan et al. [0] Ibraheem et al. [] the nterpolant are based from ratonal trgonometrc splne. Snce n the defnton of cubc Ball, there are two bass functons wth degree two (quadratc), the proposed ratonal cubc Ball may requre less computaton compare to the ratonal cubc splne ratonal cubc trgonometrc splne respectvely. (v) The proposed ratonal cubc Ball s smple to use are better than the exstng schemes such as Jaafar et al. [3] Tahat et al. [3] n the sense of easness to use vsually pleasng of the resultng postve nterpolatng curves. Furthermore the schemes s not nvolvng any radal bass functons (RBF) as appear n the works of [4] [5]. Thus no need to fnd sutable bass functons sets of parameters by solvng any optmzaton problem. Ths paper s organzed as follows. The new ratonal cubc Ball nterpolaton s dscussed n Secton ncludng shape control analyss. Arthmetc Mean Method (AMM) formula to estmate the frst dervatve at the respectve knots the constructon of the suffcent condtons for postvty preservng are dscussed n detals through ths Secton. Numercal graphcal results for postvty preservng preservng are gven n Secton 3. A conclusons are gven n the fnal secton..0 METHODOLOGY Ths secton ntroduces the new ratonal cubc Ball wth two parameters,,,..., n. The shape control of the ratonal cubc nterpolant also wll be dscussed n detals wth numercal examples.. Ratonal ubc Ball Interpolant Gven that the scalar (or functonal) data s gven such that x, f,,,..., n where x x xn. hoosng h x x, f f / h where 0. For x x / h x, x,,,..., x n the ratonal cubc Ball nterpolant s defned by P S x S x Q, () where P f V W f Q

3 43 Samsul Arffn Abdul Karm / Jurnal Teknolog (Scences & Engneerng) 78: (06) 4 48 The parameters,,,..., n are free parameters. The ratonal cubc Ball nterpolant n () satsfes the followng S condtons: S x f, S x d x f, S x d From condtons n () the unknown varables,,..., n s gven as follows: V When () V, W f h d, W f h d 0, the ratonal cubc Ball nterpolant defned by () s reducng to stard cubc Ball polynomal gven as follows: S x f f h d f f h d Furthermore wth S x S can be rewrtten as follows: x E d d (3) h E f f (4) Q When or 0, from (4) the ratonal cubc nterpolant converges to straght lne gven below: lm S, x f f. Shape ontrol of Ratonal ubc Ball Interpolant Fgure shows the examples of shape control of the ratonal cubc Ball nterpolant defned n () by usng data sets from Hussan Sarfraz [9] gven n Table. (5) Fgure Shape control of ratonal cubc Ball nterpolaton for data n Table wth: (black), (gray), 0 (dashed) 0, (dashed) (black),, 5 5, Table A data from Hussan Sarfraz [9] x Dervatve Estmaton f d (gray) Snce we are dealng wth scalar data sets.e. functonal nterpolaton, thus the frst dervatve values need to be estmated by usng some method. In ths paper, the arthmetc mean method (AMM) are used. The formulaton of AMM s further elaborated as follows: At the end ponts x x n h d (6) h h h d (7) n n n n n hn hn At the nteror ponts, x,,3,..., n, the values of d are gven as h d h h h (8)

4 44 Samsul Arffn Abdul Karm / Jurnal Teknolog (Scences & Engneerng) 78: (06) Postvty Preservng Usng Ratonal ubc Ball Interpolant The suffcent condton for the postvty of the ratonal cubc Ball nterpolant, defned by () wll S x be developed n ths secton. We assume that the strctly postve set of data x, f,,,..., n are gven, where f 0,,,..., n (9) The ratonal cubc Ball nterpolant S x 0 f both denomnator P 0 Q 0 Q 0 f only. learly the for all, 0,,,..., n. Thus the postvty of the ratonal nterpolant Sx depend to the postvty of the numerator P 0,,,..., n. What we need to do s that fnd the suffcent condtons on shape parameters that satsfy P 0. Theorem (Postvty of ubc polynomal []) For the strct nequalty postve data n (9), P 0 f only f where P 0, P R R (0) 3P 0 3P R a, b : a, b, h h a b f f a b ab a b, : R 3 fa fbh ab 3 fa 3 fb 3 3 4h fa fb h a b 0 wth a P 0, b P P f P f 0,. s () () For strctly postve data sets n (9), the followng theorem gves the suffcent condtons for the postvty of the ratonal cubc Ball nterpolant. It s data dependent has one free parameter to alter the fnal postve nterpolatng curves. Theorem. For a strctly postve data defned n (9), the ratonal cubc Ball nterpolant defned on x, xn s postve f n each subnterval x, x,,,..., n the followng suffcent condtons are satsfed: 0, Proof. Max 0, h d, h d f f (3) By usng (), the followng two nequaltes wll be obtaned: f V 3 f (4) h h f W 3 f (5) h h Smple algebrac manpulaton to (4) (5) gves the followng condtons: f h d 3 f (6) (7) f h d 3 f For 0, then the smplest condtons that wll guarantee the postvty of the cubc polynomal P,,,..., n s f h d (8) f h d (9) Thus combnng (8) (9) leads us to the followng suffcent condtons for the postvty of ratonal cubc splne defned by (): wth 0. Max 0, h d, h d f f (0) For the purpose of computer mplementaton, the above condton can be rewrte as: h d h d Max 0,,,, 0 () f f Algorthm for computer mplementaton: Input: Data ponts Output: Parameters postve nterpolatng curves Step : Input data ponts x, f,,,..., n Step : For,..., n alculate the frst dervatve values by usng AMM Step 3: For,..., n Intalze free parameter 0 alculate the parameter values, 0 by usng the suffcent condton gven n ().

5 45 Samsul Arffn Abdul Karm / Jurnal Teknolog (Scences & Engneerng) 78: (06) 4 48 Repeat by choosng dfferent values of free parameter 0 Step 4: For onstruct the postve nterpolatng curves wth contnuty.,..., n 3.0 RESULTS AND DISUSSION ratonal cubc Ball nterpolant wth default curves already preserves the postvty of the data sets. The nterpolatng curves also very smooth comparable wth the work of Jaafar et al. [3]. For data sets n Table 3, the user may use the proposed ratonal cubc Ball nterpolant snce wth stard cubc Ball polynomal the postvty of the data are already preserved the nterpolatng curves also very smooth even though wth contnuty. In order to llustrate the shape preservng nterpolaton by usng the proposed ratonal cubc Ball nterpolaton (cubc/quadratc), three postve data are taken from Tahat et al. [3], Jaafar et al. [3] Abbas et al. [] respectvely. Table data from Tahat et al. [3] x f d Table 3 data from Jaafar et al. [3] x f d Table 4 data from Abbas et al. [] x f d For postve data n Table 3, t was notced that the stard cubc Ball polynomal nterpolaton already preserves the shape of the data. In Jaafar et al. [3], the authors shows that the polynomal cubc Ball nterpolaton does not preserves the postvty of the data sets. From Fgure clearly our proposed (c) Fgure Default cubc Ball nterpolaton wth, 0 for data n: Table Table 3 (c) Table 4 respectvely

6 46 Samsul Arffn Abdul Karm / Jurnal Teknolog (Scences & Engneerng) 78: (06) 4 48 (c) Fgure 3 Postvty preservng usng the ratonal cubc Ball nterpolaton for data n Table wth: 0. (c) the postvty usng Tahat et al. [3] (c) Fgure 4 Postvty preservng usng the ratonal cubc Ball nterpolaton for data n Table 4 wth:.5 (c) the schemes by Jaafar et al. [3]

7 47 Samsul Arffn Abdul Karm / Jurnal Teknolog (Scences & Engneerng) 78: (06) 4 48 Fgure 3 Fgure 4 show the examples of postvty preservng by usng the proposed schemes ncludng comparson wth exstng schemes. Fnally Fgure 5 shows the examples that the ratonal cubc Ball nterpolant defned by () converges to the straght lne when or respectvely. 0 Acknowledgement Ths research s fully supported by Unverst Teknolog PETRONAS (UTP) through STIRF grant: 053AA-D9 ncludng Mathematca software. References Fgure 5 Lnear reproducng for ratonal cubc Ball nterpolant 4.0 ONLUSION In ths study, new ratonal cubc Ball splne wth two parameters has been constructed. The suffcent condtons for postvty are derved on one parameter, meanwhle the other parameter s use to alter the resultng postve nterpolatng curves. Thus the proposed scheme has one degree of freedom. omparson wth the works of Tahat et al. [3] Jaafar et al. [3] also have been done n detals. From the results t can be seen clearly that the proposed ratonal cubc Ball splne works very well gve smooth results as well as vsually pleasng are better than both schemes presented n [3] [3] respectvely. Work on constraned data modelng, convexty monotoncty preservng nterpolaton as well as 3D problems are underway. Furthermore constraned data nterpolaton ([4] [6]) also very nterestng topcs all the results wll be reported soon. [] Abbas, M., Majd, A. A., Awang, M. N. Hj. Al, J. M. 03. Postvty-preservng Ratonal ubc Splne Interpolaton. Scence Asa. 39: [] Brodle, K. W. Butt, S. 99. Preservng onvexty Usng Pecewse ubc Interpolaton. omputers Graphcs. 5: 5-3. [3] Brodle, K. W., Mashwama, P. Butt, S Vsualzaton Of Surface Data To Preserve Postvty And Other Smple onstrants. omputers Graphcs. 9(4): [4] Brodle, K. W., Asm, M.R. Unsworth, K onstraned Vsualzaton Usng Shepard Interpolaton Famly. omputers Graphcs Forum. 4(4): [5] Butt, S. Brodle, K. W Preservng Postvty Usng Pecewse ubc Interpolaton. omputer Graphcs. 7(): [6] Goodman, T. N. T., Ong, B.H. Unsworth, K. 99. onstraned nterpolaton usng ratonal cubc splnes. NURBS for urve Surface Desgn. G. Farn (Ed.), SIAM, Phladelpha, [7] Grener, G. 99. A survey on unvarate data nterpolaton approxmaton by splnes of gven shape. Mathematcal omputer Modelng. 5(0): [8] Hussan, M.Z., Sarfraz, M. Shakh, T.S., 0. Shape preservng ratonal cubc splne for postve convex data. Egyptan Informatcs Journal. : [9] Hussan, M.Z. Sarfraz, M Postvty-Preservng nterpolaton of postve data by ratonal cubcs, Journal of omputatonal Appled Mathematcs. 8: [0] Hussan, M.Z., Hussan, M., Waseem, A. 04. Shapepreservng trgonometrc functons. omputatonal Appled Mathematcs. 33: [] Hussan, M.Z., Hussan, M. Aqeel, B. 04. Shapepreservng surfaces wth constrants on tenson parameters. Appled Mathematcs omputaton, 47: [] Ibraheem, F., M. Hussan, M.Z. Hussan A.A. Bhatt. 0. Postve data vsualzaton usng trgonometrc polynomals, Journal of Appled Mathematcs. Artcle ID 470. [3] Jaafar, W.N.W., Pah, A.R.Mt. Abbas, M. 05. Data vsualzaton for postve data usng ratonal cubc Ball functon. AIP onf. Proc [4] Karm, S.A.A. 03. Ratonal ubc Ball Functons for Postvty Preservng. Far East Journal of Mathematcal Scences (FJMS). Vol. 8, No., pp [5] Karm, S.A.A. 04. Monotonc Interpolatng urves by Usng Ratonal ubc Ball Interpolaton. Appled Mathematcal Scences, vol. 8, no. 46, [6] Karm, S.A.A. 05. Postvty preservng by usng ratonal cubc Ball functon. AIP onf. Proc [7] Karm, S.A.A. 05. Shape Preservng by Usng Ratonal ubc Ball Interpolant. Far East Journal of Mathematcal Scences (FJMS). 96(): -30. [8] M. Sarfraz. 0. Vsualzaton of postve convex data by a ratonal cubc splne nterpolaton, Informaton Scences. 46(-4): [9] Sarfraz, M., Hussan, M.Z. Nsar. 00. A. Postve data modelng usng splne functon. Appled Mathematcs omputaton. 6:

8 48 Samsul Arffn Abdul Karm / Jurnal Teknolog (Scences & Engneerng) 78: (06) 4 48 [0] Sarfraz, M., Hussan, M.Z. M Hussan. 0. Shapepreservng curve nterpolaton, Internatonal Journal of omputer Mathematc. 89(): [] Sarfraz, M., Hussan, M.Z. Hussan, M. 03. Modelng ratonal splne for vsualzaton of shaped data. Journal of Numercal Mathematcs. (): [] Schmdt, J.W. Hess, W Postvty of cubc polynomals on ntervals postve splne nterpolaton. BIT. 8: [3] Tahat, A.N.H., Pah, A.R.Mt. Yahya, Z.R. 05. Postvty preservng curves usng ratonal cubc Ball nterpolant. AIP onf. Proc [4] Walther, M.B.- Schmdt, J.W Range Restrcted Interpolaton usng Gregory s Ratonal ubc Splnes. Journal of omputatonal Appled Mathematcs. 03: -37. [5] Wu, J., Zhang, X. Peng, L. 00. Postve Approxmaton Interpolaton Usng ompactly Supported Radal Bass Functons. Mathematcal Problems n Engneerng. Artcle ID 96458, 0 pages.

Cubic Spline Interpolation for. Petroleum Engineering Data

Cubic Spline Interpolation for. Petroleum Engineering Data Appled Mathematcal Scences, Vol. 8, 014, no. 10, 5083-5098 HIKARI Ltd, www.m-hkar.com http://dx.do.org/10.1988/ams.014.4484 Cubc Splne Interpolaton for Petroleum Engneerng Data * Samsul Arffn Abdul Karm