Research Article Shape Preserving Interpolation Using C 2 Rational Cubic Spline

Transcription

1 Applied Mahemaics Volume 1, Aricle ID 73, 1 pages hp://dx.doi.org/1.11/1/73 Research Aricle Shape Preserving Inerpolaion Using C Raional Cubic Spline Samsul Ariffin Abdul Karim 1 and Kong Voon Pang 1 Fundamenal and Applied Sciences Deparmen, Universii Teknologi PETRONAS, Bandar Seri Iskandar, 31 Seri Iskandar, Perak Darul Ridzuan, Malaysia School of Mahemaical Sciences, Universii Sains Malaysia (USM), 11 Minden, Penang, Malaysia Correspondence should be addressed o Samsul Ariffin Abdul Karim; samsul ariffin@peronas.com.my Received 1 January 1; Revised 1 April 1; Acceped 1 April 1 Academic Edior: Francisco J. Marcellán Copyrigh 1 S. A. Abdul Karim and K. Voon Pang. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. This paper discusses he consrucion of new C raional cubic spline inerpolan wih cubic numeraor and quadraic denominaor. The idea has been exended o shape preserving inerpolaion for posiive daa using he consruced raional cubic spline inerpolaion. The raional cubic spline has hree parameers α i, β i,andγ i. The sufficien condiions for he posiiviy are derived on one parameer γ i while he oher wo parameers α i and β i are free parameers ha can be used o change he final shape of he resuling inerpolaing curves. This will enable he user o produce many varieies of he posiive inerpolaing curves. Cubic spline inerpolaion wih C coninuiyisnoableopreserveheshapeofheposiivedaa.noablyourschemeiseasyouse and does no require knos inserion and C coninuiy can be achieved by solving ridiagonal sysems of linear equaions for he unknown firs derivaives d i, i=1,...,n 1. Comparisons wih exising schemes also have been done in deail. From all presened numerical resuls he new C raional cubic spline gives very smooh inerpolaing curves compared o some esablished raional cubic schemes. An error analysis when he funcion o be inerpolaed is f() C 3 [, n ] is also invesigaed in deail. 1. Inroducion Spline inerpolaion has been used exensively in many research disciplines such as in car design and airplane fuselage. Univariae and bivariae spline can be used o approximae or inerpolae he given finie daa ses. Even hough he cubic spline has second-order parameric coninuiy, C, i has some weakness such ha he inerpolaing curves may give few unwaned behavior of he original daa due o he exising wiggles along some inerval. This uncharacerisic behavior may desroy he daa. If he given daa is posiive cubic spline may give some negaive values along he whole inerval he inerpolaing curves will lie below xaxis. For some applicaion any negaiviy is unaccepable. For example, he wind speed, solar energy, and rainfall received are always having posiive values and any negaiviy values need o be avoided as i may desroy any imporan informaion ha may exisn he original daa. Similarly if he daa is monoone, hen he resuling inerpolaing curves also mus be monoone oo. Furhermore if he given daa is convex, he raional cubic spline inerpolaion should be able o mainain he shape of he original daa. Thus shape preserving inerpolaion is imporann compuer graphics and compuer aided geomeric design (CAGD). Due o he fac ha cubic spline is no able o produce compleely he posiive, monoone, and convex inerpolaing curves on enire given inerval, many researchers have proposed several mehods and idea o preserve he posiiviy, monooniciy, and convexiy of he daa. Frisch and Carlson [1] and Doughery e al. [] have discussed he monooniciy, posiiviy, and convexiy preserving by using cubic spline inerpolaion by modifying he firs derivaive values in which he shape violaion is found. Bu and Brodlie [3] and Brodlie and Bu [] have used cubic spline inerpolaion o preserve he posiiviy and convexiy of he finie daa by insering exra knos in he inerval in which he posiiviy and/or convexiy is no preserved by he cubic spline. Their mehods did no give any exra freedom o he user in conrolling he final shape of he inerpolaing curves. In order o change he final shape of he inerpolaing curves, he user needs o

2 Applied Mahemaics change he given daa. Anoher hing is ha heir mehods require he modificaion of he firs derivaive parameers. Sarfraz [], Sarfraz e al. [, 7], and Abbas [] sudied he use of raional cubic inerpolan for preserving he posiive daa. Meanwhile M. Z. Hussain and M. Hussain [9] sudied posiiviy preserving for curves and surfaces by uilizing raional cubic spline wih quadraic denominaor. In works by Hussain e al. [1] and Sarfraz e al. [11] he raional cubic spline wih quadraic denominaor has been used for posiiviy, monooniciy, and convexiy preserving wih C coninuiy. Hussain e al. [1] have only one free parameer meanwhile Sarfraz e al. [11] have no free parameer. Abbas [] and Abbas e al. [1] have discussed he posiiviy by using new C raional cubic spline wih wo free parameers. Anoher paper concerning C raionalsplinecanbefound by Delbourgo [13], Gregory [1], and Delbourgo and Gregory [1]. Karim and Kong [1 19] have proposed new C 1 raional cubic spline (cubic/quadraic) wih hree parameers wo of hem are free parameers. The raional cubic spline has been successfully applied o he local conrol of he inerpolaing funcions, posiiviy, monooniciy, and convexiy preserving as well as he derivaive conrol including an error analysis when he funcion o be inerpolaed is f() C 3 [, n ]. Moivaed by he works of Tian e al. [], Abbaseal.[1],Hussaineal.[1],andSarfrazeal.[11],in his paper he auhors will proposed new C raional cubic spline for posiiviy, monooniciy, and convexiy preserving and daa consrained modeling. Under some circumsances our C raional cubic spline will give new C raional cubic spline based on raional cubic spline defined by Tian e al. []. Numerical comparison beween C raional cubic spline and he works of Hussain e al. [1], Abbas [], Abbas e al. [1], and Sarfraz e al. [11] also has been made comprehensively. From all presened numerical resuls shape preserving inerpolaion by using he new C raional cubic spline gives comparable resuls wih exising raional cubic spline schemes. The main scienific conribuion of his paper is summarized as follows: (i) In his paper C raional cubic spline (cubic/ quadraic) wih hree parameers has been used for posiiviy, monooniciy, and convexiy preserving and consrained daa modeling while in works by Karim and Kong [17 19], M. Z. Hussain and M. Hussain[9],andSarfraz[]hedegreeofsmoohness aained is C 1. (ii) Hussain e al. [1] and Sarfraz e al. [11] discussed he posiiviy by using C raional cubic spline (cubic/quadraic) wih wo parameers wih one or no free parameer while our raional cubic spline has wo free parameers. Even hough Abbas e al. [1] also proposed C raional cubic spline (cubic/quadraic) wih wo parameers, heir raional spline is differen form our raional cubic spline. Furhermore i was noiced ha schemes of Abbas e al. [1] may no be able o produce compleely posiive inerpolaing curves wih C coninuiy. (iii) When γ i =,wemayobainhenewc raional cubic spline wih wo parameers, an exension o he original C 1 raional cubic spline of Tian e al. []. Thus our C raional cubic spline gives a larger class of raional cubic spline which also includes he raional cubicsplineoftianeal.[]. (iv)ourraionalschemeislocalwhileinlamberiand Manni [1] heir scheme is global. Furhermore our raional scheme works well for boh equally and unequally spaced daa while he raional spline inerpolan by Duan e al. [] and Bao e al. [3] only works for equally spaced daa. (v) Numerical comparisons beween he C raional cubic spline and he exising schemes such as Hussain e al. [1], Sarfraz e al. [11], and Abbas e al. [1] for posiiviy preserving and C 1 raional spline of Karim and Kong [17 19] also have been done comprehensively. (vi) Our mehod also does no require any knos inserion. Meanwhile he cubic spline inerpolaion by Bu and Brodlie[3],BrodlieandBu[],andFioroandTabka [] requires knos inserion in he inerval he inerpolaing curves produce he negaive values (lies below ) for posiive daa, nonmonoone inerpolaing curves (for monoone daa), and nonconvex inerpolaing curves for convex daa. (vii) This paper uilized he raional cubic spline meanwhile in Dube and Tiwari [], Pan and Wang [], and Ibraheem e al. [7] he raional rigonomeric spline is used in place of sandard raional cubic spline. Thus no rigonomeric funcions are involved. Therefore he mehod is no compuaionally expensive. The remainder of he paper is organized as follows. Secion inroduces he new C raional cubic spline wih hree parameers, wih some discussion on he mehods o esimae he firs derivaives values as well as shape conrols of he raional cubic spline inerpolaion. Meanwhile Secion 3 discusses he posiiviy preserving by using C raional cubic spline ogeher wih numerical demonsraions as well as comparison wih some exising schemes including error analysis. Secion is devoed for research discussion. Finally a summary and conclusions are given in Secion.. C Raional Cubic Spline Inerpolan This secion will inroduce C raional cubic spline inerpolan wih hree parameers. Originally his raional cubic spline has been iniiaed by Karim and Kong [1]. The main difference is han his paper he raional cubic spline has C coninuiywhileinworkbykarimandkong[1]ihas C 1 coninuiy. We begin wih he definiion of C raional cubic spline inerpolan, given se of daa poins (x i,f i ), i =,1,...,n}such ha x <x 1 < <x n.leh i =x i+1 x i, Δ i =(f i+1 f i )/h i and θ=(x x i )/h i, θ 1.For

3 Applied Mahemaics 3 x [x i,x i+1 ], i =, 1,,..., n 1, he raional cubic spline inerpolan wih hree parameers is defined as follows: s (x) s i (x) = A i (1 θ) 3 +A i1 θ (1 θ) +A i θ (1 θ) +A i3 θ 3 (1 θ) α i +θ(1 θ) (α i β i +γ i )+θ β i. The following condiions will assure ha he raional cubic spline inerpolann (1) has C coninuiy: s(x i )=f i, s(x i+1 )=f i+1, s (1) (x i )=d i, s (1) (x i+1 )=d i+1, (1) () Now C coninuiy, s () (x i+ )=s () (x i ), i = 1,,...,n 1, will give [γ i 1 (d i Δ i 1 ) α i 1 (Δ i 1 d i 1 β i 1 (d i Δ i 1 ))] h i 1 β i 1 = (γ i (Δ i d i ) β i (d i+1 Δ i ) α i β i (d i Δ i )) h i α i. Now () provides he following sysem of linear equaions ha can be used o compue he firs derivaive parameers, d i, i=1,,...,n 1,suchha wih () a i d i 1 +b i d i +c i d i+1 =e i, i=1,,...,n 1, (7) a i =h i α i 1 α i, b i =h i α i (γ i 1 +α i 1 β i 1 )+h i 1 β i 1 (γ i +α i β i ), s () (x i+ )=s () (x i ), c i =h i 1 β i 1 β i, () s (1) (x i ) and s () (x i ) denoe he firs- and second-order derivaive wih respec o x, respecively. Meanwhile he noaions s () (x i+ ) and s () (x i ) correspond o he righ and lef second derivaives values. Furhermore d i denoes he derivaive value which is given a he kno x i, i=,1,,...,n. By using (), he required C raional cubic spline inerpolan wih hree parameers defined by (1) has he unknowns A ij, j =, 1,, 3, and is given as follows: A i =α i f i, A i1 =(α i β i +α i +γ i )f i +α i h i d i, A i =(α i β i +β i +γ i )f i+1 β i h i d i+1, A i3 =β i f i+1. The parameers α i,β i >, γ i areusedoconrolhefinal shape of he inerpolaing curves. From work by Karim and Kong [19], he second-order derivaive s () (x) is given as s () (x) = 3 j= C ij (1 θ) 3 j θ j (3) h i [Q i (θ)] 3, () e i =h i α i (γ i 1 +α i 1 +α i 1 β i 1 )Δ i 1 +h i 1 β i 1 (γ i +β i +α i β i )Δ i. The sysem of linear equaions given by (7) is sricly ridiagonalandhasauniquesoluionforheunknownderivaive parameers d i, i=1,,...,n 1,forallα i,β i >, γ i.the sysem in (7) gives n 1linear equaions for n+1unknown derivaive values. Thus wo more equaions are required in order o obain he unique soluion in (7). The following is common choices for he end poins condiion, has, d and d n : s (1) (x )=d, s (1) (x n )=d n. By using (3), C raional cubic spline inerpolann (1) can be reformulaed and is given as follows: P i (θ) =α i f i (1 θ) 3 (9) s (x) s i (x) = P i (θ) Q i (θ), (1) C i =α i (γ i (Δ i d i ) β i (d i+1 Δ i ) α i β i (d i Δ i )), +((α i β i +α i +γ i )f i +α i h i d i )θ(1 θ) +((α i β i +β i +γ i )f i+1 β i h i d i+1 )θ (1 θ) (11) C i1 =α i β i (Δ i d i ), C i =α i β i (d i+1 Δ i ), C i3 =β i [γ i (d i+1 Δ i ) α i (Δ i d i β i (d i+1 Δ i ))]. () +β i f i+1 θ 3, Q i (θ) = (1 θ) α i +θ(1 θ) (α i β i +γ i )+θ β i. The daa-dependen sufficien condiions on parameers γ i will be developed in order o preserve he posiiviy, daa consrained, monooniciy, and convexiy on he enire inerval

4 Applied Mahemaics [x i,x i+1 ], i =,1,,...,n 1. The remaining parameers α i and β i canbeusedorefineheresulinginerpolaingcurves. Thus he raional cubic spline provides greaer flexibiliy o he user in conrolling he final shape of he inerpolaing curves. Theorem 1 (C raional cubic spline inerpolan). The raional cubic spline wih hree parameers defined by (1) is C [x,x n ] if here exis posiive parameers α i,β i >,γ i, and d i, i=1,,...,n 1,hasaisfy(7). The choice of he end poin derivaives d and d n depended on he original daa which are chosen as follows. Choice 1 (Geomeric Mean Mehod (GMM)). Consider d =, Δ =or Δ, = Δ (1+h /h 1 ) Δ ( h /h 1 ),1 oherwise, d n = Δ n 1 =or Δ n,n = Δ (1+h n 1/h n ) n 1 Δ ( h n 1/h n ) n,n oherwise. Choice (Arihmeic Mean Mehod (AMM)). Consider d =Δ +(Δ Δ 1 )( h h +h 1 ), h n 1 d n =Δ n 1 +(Δ n 1 Δ n )( ). h n 1 +h n (1) (13) Delbourgo and Gregory [9] give more deails abou he mehodhacanbeusedoesimaehefirsderivaivevalue. InhispaperheAMMwillbeusedoesimaeheendpoin derivaives d and d n,respecively. Some observaion and shape conrol analysis of he new C raional cubic spline inerpolan defined by (1) are given as follows: (1) When α i >, β i >, and γ i =,heraional inerpolann (1) reduces o he raional spline of he formcubic/quadraicbytianeal.[]andwemay obain C raional cubic spline wih wo parameers, an exension o C 1 raional cubic spline originally proposed by Tian e al. []. Thus by rewriing C condiion in (7), we can obain C raional cubic of Tian e al. []. () When α i = β i = 1; γ i =,heraionalcubic inerpolann (1) is jus a sandard cubic Hermie spline wih C 1 coninuiy ha may no be able o compleely preserve he posiiviy of he daa [1]: s (x) = (1 θ) (1 θ) f i +θ (3 θ) f i+1 +θ(1 θ) d i θ (1 θ) d i+1, for i=,1,...,n 1. (1) Table 1: A daa from work by Sarfraz e al. []. i x i f i h i 1 Δ i.. 1/3 d i (C 1 ) d i (C ) (3) Furhermore he raional inerpolann (1) can be wrien as [1] s (x) = (1 θ) f i +θf i+1 + h iθ (1 θ)[α i (d i Δ i ) (1 θ) +β i (Δ i d i+1 )θ]. Q i (θ) (1) () Obviously when α i, β i,orγ i, he raional inerpolann (1) converges o following sraigh line: lim α i,β i,γ i s (x) = (1 θ) f i +θf i+1. (1) Eiher he decrease of he parameers α i and β i or he increase of γ i will reduce he raional cubic spline o a linear inerpolan. Figure 1 shows his example. We es shape conrol analysis by using he daa from work by Sarfraz e al. [] given in Table 1. To show he difference beween C raional cubic spline wih hree parameers and C 1 raional cubic spline of Karim andkong[17 19],wechooseα i = β i = 1; γ i = for boh cases. The main difference is ha o generae C 1 raional inerpolaing curves he firs derivaive parameer d i, i =,1,,...,n,iscalculaedbyusingArihmeicMean Mehod (AMM); meanwhile o generae C raional cubic spline wih hree parameers, he firs derivaive parameer d i, i=1,,...,n 1, is calculaed by solving (7) wih suiable choices of he end poin derivaives d and d n. Remark. If Δ i d i =or d i+1 Δ i =,hend i =d i+1 =Δ i. In his case he raional inerpolann (1) will be linear in he corresponding inerval or region; has, s (x) = (1 θ) f i +θf i+1. (17) Figure shows he shape conrol using he raional cubic splineforhegivendaaintable1. Figure (f) shows he combinaion of Figures (a), (d), and (e). Meanwhile Figure (g) shows he combinaion of Figures (a), (b), and (c), respecively. Clearly he curves approach o he sraigh line if α i, β i,orγ i. Furhermore decreases in he value of β i will pull he curves upward and vice versa. This shape conrol will be useful for shape preserving inerpolaion as well as local conrol of he inerpolaing curves. MeanwhilefromFigure1(d),icanbeseenclearlyha C raional cubic spline inerpolaion (shown as red color) is

5 Applied Mahemaics (a) (b) (c) (d) Figure 1: Inerpolaing curve for daa in Table 1. (a) Defaul C 1 cubic spline wih α i =β i =1and γ i =.(b)c 1 raionalcubicsplinewih α i =β i =1and γ i =.(c)c raional cubic spline by using derivaive calculaed from ridiagonal equaion (7) wih α i =β i =1and γ i =. (d) The graphs of (b) C 1 raional cubic spline (blue) and (c) C raional cubic spline (red). smooher han C 1 raional cubic spline inerpolaion (shown as black color). Thus he new C raional cubic spline provides good alernaive o he exising C 1 and C raional cubic spline. Remark 3. Since he consruced C raional cubic spline has hree parameers, hen how do we choose he parameer values? To answer his quesion, he choices of he parameers oally depend on he daa ha are under consideraions by he main user. The main difference beween our C raional cubic spline and C cubic spline inerpolaion is ha, in order o change he final shape of he inerpolaing curves, our schemes are only required o change he parameers values wihou he need o change he daa poins iself. Bu here arenofreeparameersinhedescripionsofc cubic spline inerpolaion. 3. Posiiviy Preserving Using C Raional Cubic Spline Inerpolaion In his secion, he posiiviy preserving by using he proposed C raional cubic spline inerpolaion defined by (1) will be discussed in deail. We follow he same idea of KarimandKong[1]andAbbaseal.[1]suchhasimple daa-dependen condiions for posiiviy are derived on one parameer γ i while he remaining parameers α i and β i are free o be uilized. The main objecive is ha, in order o preserve he posiiviy of he posiive daa, he raional cubic spline inerpolan mus be posiive on he enire given inerval. The simple way o achieve is by finding he auomaed choice of he shape parameer γ i.webeginby giving he definiion of sricly posiive daa. Given he sricly posiive se of daa (x i,f i ), i =,1,...,n, x <x 1 < <x n,suchha f i >, i=,1,...,n. (1) Now from (1), he raional cubic spline will preserve he posiiviy of he daa if and only if P i (θ) > and Q i (θ) >. Since for all α i,β i >and γ i, he denominaor Q i (θ) >, i=,1,...,n 1.Thuss(x) > if and only if P i (θ) >, i =,1,...,n 1.The cubic polynomial P i (θ), i=,1,...,n 1 can be wrien as follows [1]: P i (θ) =B i θ 3 +C i θ +D i θ+e i, (19)

6 Applied Mahemaics (a) (b) (c) (d) (e) (f) (g) (h) Figure : Coninued.

7 Applied Mahemaics (i) (j) Figure : Shape conrol of raional cubic inerpolaing curves wih various values of shape parameers lisn Table. Table : Shape parameers value for Figure. Figure i 1 3 α i Figure (a) β i γ i α i Figure (b) β i γ i α i Figure (c) β i γ i α i Figure (d) β i γ i α i Figure (e) β i γ i α i Figure (h) β i γ i α i Figure (i) β i γ i α i Figure (j) β i γ i B i =α i h i d i +β i h i d i+1 +α i β i f i +γ i f i α i β i f i+1 γ i f i+1, C i = α i h i d i β i h i d i+1 +α i f i α i β i f i γ i f i +β i f i+1 +α i β i f i+1 +γ i f i+1, D i =α i h i d i α i f i +α i β i f i +γ i f, E i =α i f i. () From Schmid and Hess [3], wih variable subsiuion θ= s/(1 + s), s, P i (θ), i =,1,...,n 1,canberewrienas follows: P i (s) =As 3 +Bs +Cs+D, (1) A=β i f i+1, B=(α i β i +β i +γ i )f i+1 β i h i d i+1, C=(α i β i +α i +γ i )f i α i h i d i, D=α i f i. () Theorem. For sricly posiive daa defined in (1), C raional cubic inerpolan defined over he inerval [x,x n ] is posiive if in each subinerval [x i,x i+1 ], i =,1,...,n 1, he involving parameers α i,β i,andγ i saisfy he following sufficien condiions: α i,β i >, γ i > Max, α i [ h id i +(β i +1)f i f i ], β i [ h id i+1 (α i +1)f i+1 f i+1 ]}. (3) Remark. The sufficien condiion for posiiviy preserving by using C raional cubic inerpolans differen from he sufficien condiion for posiiviy preserving by using C 1 raional cubic inerpolan of Karim and Kong [1]. To achieve C coninuiy, he derivaive parameers d i, i=1,,...,n 1, mus be calculaed from C condiion given in (7); meanwhile o achieve C 1 coninuiy, he derivaive parameers d i, i =, 1,,..., n, areesimaedbyusingsandardapproximaion mehods such as Arihmeic Mean Mehod (AMM).

8 Applied Mahemaics Table 3: A posiive daa from work by Hussain e al. [1]. i 1 3 x i f i Table : A Posiive daa from work by Sarfraz e al. []. i 1 3 x i f i Table : Numerical resuls. i 1 3 d i (C 1 ) Δ i α i... β i... γ i d i (C ) Table : Numerical resuls. i 1 3 d i (C 1 ) Δ i α i β i γ i d i (C ) Remark. The sufficien condiion in (3) can be rewrien as α i,β i >, γ i =λ i + Max, α i [ h id i +(β i +1)f i f i ], β i [ h id i+1 (α i +1)f i+1 f i+1 ]}, λ i >. () The following is an algorihm ha can be used o generae C posiiviy-preserving curves. Algorihm 7 (C posiiviy preserving). Consider he following: (1) Inpu he daa poins x i,f i } n i=, d,andd n. () For i=,1,...,n 1, calculae he parameer γ i using () wih suiable choices of α i >, β i >,andλ i >. (3) For i=1,,...,n 1, calculae he value of derivaive, d i,bysolving(7)byusinglu decomposiion and so forh. () For i=,1,...,n 1, consruc he piecewise posiive C raional inerpolaing curves defined by (1) Numerical Demonsraions. In his secion several numerical resuls for posiiviy preserving by using C raional inerpolaing curves will be shown including comparison wih exising raional cubic spline schemes. Two ses of posiive daa aken from works by Hussain e al. [1] and Sarfraz e al. [] were used. Figures3andshowheposiiviypreservingbyusingC raional cubic spline for daa in Tables 3 and, respecively. Figures 3(a) and (a) show he defaul cubic Hermie spline polynomialfordaaintables3and,respecively.numerical resuls of Figure 3(d) were obained by using he parameers from he daa in Table. Meanwhile numerical resuls of Figure (e) were generaed by using he daa in Table. I canbeseenclearlyhaheposiiviypreservingbyusingour C raional cubic spline gives more smooh resuls compared wih he works of Hussain e al. [1] and Abbas e al. [1]. The graphical resuls in Figure 3(e) were very smooh and visually pleasing. Meanwhile for he posiive daa given in Table, our C raional cubic spline gives comparable resuls wih he works of Abbas e al. [1], Hussain e al. [1], and Sarfraz e al. [11]. The final resuling posiive curves by using he proposed C raional cubic spline inerpolan are slighly differen beween he works of Abbas e al. [1], Hussain eal.[1],andsarfrazeal.[11].finallyfigureshows he examples of posiive inerpolaing by using Frisch and Carlson [1] cubic spline schemes ha are well documened in MalabasPCHIP.Icanbeseenclearlyhashapepreserving by using PCHIP does no give smooh resuls and is no visually pleasing enough. Some of he inerpolaing curves end o oversho on some inerval ha he inerpolaing curves are igh when compared wih our work in his paper. For insance, in Figure (b), he inerpolaing curves are very igh and no visually pleasing. Error Analysis. Inhissecion,heerroranalysisforhe funcion o be inerpolaed is f() C 3 [, n ] using our C raional cubic spline which will be discussed in deail. Noe ha he consruced raional cubic spline wih hree parameers is a local inerpolan and wihou loss of generaliy, we may jus consider he error on he subinerval I i =[,+1 ]. By using Peano Kernel Theorem [31] he error of inerpolaion in each subinerval I i =[,+1 ] is defined as R [f] =f() P i () = 1 i+1 f (3) R [( τ) + ] dτ, () R [( τ) + ]= r (τ, ), <τ<, () s (τ, ), < τ < +1

9 Applied Mahemaics (a) (b) (c) (d) (e) Figure 3: Comparison of (a) cubic Hermie spline curve, (b) Hussain e al. [1], (c) Abbas e al. [1], and posiiviy-preserving raional cubic spline wih values of shape parameers: (d) α i =β i =.;(e)α i =β i =.

10 1 Applied Mahemaics (a) (b) (c) (d) (e) Figure : Comparison of (a) cubic Hermie spline curve, (b) Sarfraz e al. [11], (c) Abbas e al. [1], and posiiviy-preserving raional cubic spline wih values of shape parameers: (d) α i =β i =.;(e)α i =β i =..

11 Applied Mahemaics (a) (b) Figure wih r (τ, ) = ( τ) +s(τ, ), s (τ, ) = [(α iβ i +β i +γ i )ζ h i β i ζ} θ (1 θ) +β i ζ θ 3 ] Q i (θ) (7) wih ζ=(+1 τ). The absolue error in each subinerval I i = [x i,x i+1 ] is given as follows: f () P i () 1 f(3) (τ) i+1 R [( τ) + ]dτ. () In order o enable us o derive he error analysis of C raional cubic spline inerpolaion, we need o sudy he properies of he kernel funcions r(τ, ) and s(τ, ) andevaluaehe following definie inegrals: H= (ρ i β i )(α i +ρ i θ) α i ρ i θ. (31) The roos of s(τ, ) = are τ 3 =+1, τ =+1 β i h i (1 θ)/(β i +(1 θ)ρ i ). Thus he following hree cases can be obained. Case 1. For β i /ρ i 1,<θ<θ,hen()akeshe following form: f () P i () 1 f(3) (τ) i+1 R [( τ) + ]dτ = 1 f(3) (τ) G i (τ) (3) +1 R [( τ) + ]dτ= r (τ, x) dτ +1 + s (τ, x) dτ. (9) To simplify he inegrals in (9) we begin by finding he roos of r(, ) =, r(τ, ) =,ands(τ, ) =,respecively.iiseasy o see ha he roos of r(, ) = in [, 1] are θ=,1and θ =1 β i /ρ i,ρ i =α i β i +γ i. Meanwhile he roos of r(τ, ) = are wih Hence G i (τ) = r (τ, ) dτ s (τ, ) dτ}. τ τ s (τ, ) dτ (33) τ k =x θh i (θρ i + ( 1) k+1 H) α i +θρ i, k = 1,, (3) f () P i () f(3) (τ) ζ 1 (α i,β i,γ i,θ), (3)

12 1 Applied Mahemaics ζ 1 (α i,β i,γ i,θ)= h3 i θ3 3 + β ih 3 i θ3 3Q i (θ) (ρ i +β i ) (h 3 i θ (1 θ) ) 3(ρ i (1 θ) +β i ) 3 Q i (θ) Case 3. For β i /ρ i following form: > 1, < θ < 1,hen()akeshe f () P i () 1 f(3) (τ) i+1 R [( τ) + ]dτ = 1 f(3) (τ) i+1 r (τ, ) dτ + s (τ, ) dτ} () β i + (h3 i θ (1 θ) ) 3(ρ i (1 θ) +β i ) Q i (θ) 1β i h 3 i θ3 (1 θ) 3 3(ρ i (1 θ) +β i ) 3 Q i (θ) + h3 i (1 θ) θ 3Q i (θ) [ρ i β i ]. (3) = f(3) (τ) ζ 3 (α i,β i,γ i,θ), ζ 3 (α i,β i,γ i,θ)= h3 i θ3 3 (ρ i +β i )h 3 i (1 θ) θ 3Q i (θ) + β ih 3 i (1 θ) θ Q i (θ) β ih 3 i θ3 3Q i (θ). (1) Case. For β i /ρ i 1,θ <θ<1,hen()akeshe following form: Theorem. The error for he inerpolaing raional cubic spline inerpolan defined by (1) in each subinerval I i = [,+1 ],whenf() C 3 [, n ],isgivenby f () P i () 1 f(3) (τ) i+1 R [( τ) + ]dτ = 1 f(3) (τ) G (τ) (3) f () P i () 1 f(3) (τ) i+1 R [( τ) + ]dτ = 1 f(3) (τ) i+1 r (τ, ) dτ + s (τ, ) dτ} () wih = f(3) (τ) c i, Hence τ G (τ) = +1 + s (τ, ) dτ}. r (τ, ) dτ + r (τ, ) dτ τ (37) f () P i () f(3) (τ) ζ (α i,β i,γ i,θ), (3) ζ (α i,β i,γ i,θ)= h3 i θ3 (θρ i H) 3 3 β ih 3 i θ3 [G 3 (θ)] 3 3Q i (θ) (ρ i +β i )h 3 i (1 θ) θ [G 3 (θ)] 3 3Q i (θ) + β ih 3 i (1 θ) θ [G 3 (θ)] 3 Q i (θ) wih G 3 (θ) = (1 θ) + θ(θρ i H)/(α i +ρ i θ). (39) wih c i = max ζ(α i,β i,γ i,θ), ζ 1 (α i,β i,γ i,θ), ζ(α i,β i,γ i,θ)= ζ (α i,β i,γ i,θ), ζ 3 (α i,β i,γ i,θ), θ θ θ θ 1 θ 1. (3) Remark 9. When α i =1,β i =1,andγ i =, he inerpolaion funcion defined by (1) is reduced o he sandard cubic Hermie inerpolaion wih ζ 1 (α i,β i,γ i,θ)= θ (1 θ) 3 (3 (3 θ) ), θ 1, ζ (α i,β i,γ i,θ)= θ3 (1 θ) (3 (1+θ) ), 1 θ 1, ζ 3 (α i,β i,γ i, θ) =, θ 1, () and c i = 1/9. This is sandard error for cubic Hermie polynomial spline.. Discussions From all numerical resuls presened in Secion 3.1, i can be seen clearly ha he proposed C raional cubic spline

13 Applied Mahemaics 13 works very well and is comparable wih exising schemes such as Hussain e al. [1], Sarfraz e al. [11], and Abbas e al. [1] for posiiviy preserving. Furhermore similar o he consrucion of raional cubic of Abbas e al. [] our C raional cubic spline inerpolaion also has hree parameers wo are free parameers. Bu based on our numerical experimens, i was noiced ha schemes of Abbas e al. [1] may no be able o produce compleely C posiive inerpolaing curves. Meanwhile he sufficien condiion for posiiviy, monooniciy, and convexiy preserving and daa consrained are derived on he remaining parameer. One of he advanages by using our C raional cubic spline is ha when γ i =we may obain he new varian of C raional cubicsplineofhussainandali[3],m.z.hussainandm. Hussain [9], and Tian e al. [] for posiiviy- and convexiypreserving inerpolaion. Thus our C raional cubic spline has a larger spline class compared o he works by Abbas e al. [1]. Furhermore he error analysis when he funcion o be inerpolaed is f() C 3 [, n ] also has been derived in deail.. Conclusions In his paper he new C raional cubic spline wih hree parameers has been inroduced. Is an exension o he work of Karim and Pang [1]. To achieve C coninuiy a he join knos x i, i = 1,,...,n 1, he firs derivaive value, d i,is calculaed by solving sysems of linear equaion (ridiagonal) has sricly posiive and he soluion is unique. Shape conrol of he new C raional cubic inerpolaion wih numerical examples also was presened. Finally C raional cubic spline has been used for posiiviy preserving including a comparison wih exising schemes. From he numerical resuls clearly our C raional cubic spline gives comparable resuls wih exising schemes. Finally work in parameric shapepreservingisunderwaybyheauhors. Compeing Ineress The auhors declare ha here are no compeing ineress regarding he publicaion of his paper. Acknowledgmens Samsul Ariffin Abdul Karim would like o acknowledge Universii Teknologi PETRONAS (UTP) for he financial suppor received in he form of a research gran: Shor Term Inernal Research Funding (STIRF) no. 13AA-D91 including he Mahemaica Sofware. References [1] F. N. Frisch and R. E. Carlson, Monoone piecewise cubic inerpolaion, SIAM Journal on Numerical Analysis, vol. 17,no., pp. 3, 19. [] R.L.Doughery,A.S.Edelman,andJ.M.Hyman, Nonnegaiviy-, monooniciy-, or convexiy-preserving cubic and quinic Hermie inerpolaion, Mahemaics of Compuaion, vol.,no.1,pp.71 9,199. [3] S. Bu and K. W. Brodlie, Preserving posiiviy using piecewise cubic inerpolaion, Compuers and Graphics, vol.17,no.1,pp., [] K. W. Brodlie and S. Bu, Preserving convexiy using piecewise cubic inerpolaion, Compuers and Graphics,vol.1,no.1,pp. 1 3, [] M. Sarfraz, Visualizaion of posiive and convex daa by a raional cubic spline inerpolaion, Informaion Sciences, vol. 1, no. 1, pp. 39,. [] M. Sarfraz, S. Bu, and M. Z. Hussain, Visualizaion of shaped daa by a raional cubic spline inerpolaion, Compuers & Graphics,vol.,no.,pp.33,1. [7] M. Sarfraz, M. Z. Hussain, and A. Nisar, Posiive daa modeling using spline funcion, Applied Mahemaics and Compuaion, vol. 1, no. 7, pp. 3 9, 1. [] M. Abbas, Shape preserving daa visualizaion for curves and surfaces using raional cubic funcions [Ph.D. hesis], School of Mahemaical Sciences, Universii Sains Malaysia, Penang, Malaysia, 1. [9] M.Z.HussainandM.Hussain, Visualizaionofdaasubjec o posiive consrain, Informaion and Compuing Sciences,vol.1,no.3,pp.19 1,. [1] M. Z. Hussain, M. Sarfraz, and T. S. Shaikh, Shape preserving raional cubic spline for posiive and convex daa, Egypian Informaics Journal,vol.1,no.3,pp.31 3,11. [11] M. Sarfraz, M. Z. Hussain, T. S. Shaikh, and R. Iqbal, Daa visualizaion using shape preserving C raional spline, in Proceeding of h Inernaional Conference on Informaion Visualisaion, pp. 33, London, UK, July 11. [1] M. Abbas, A. A. Majid, M. N. H. Awang, and J. M. Ali, Posiiviy-preserving C raional cubic spline inerpolaion, ScienceAsia,vol.39,no.,pp. 13,13. [13] R. Delbourgo, Accurae C raional inerpolans in ension, SIAM Journal on Numerical Analysis,vol.3,no.,pp.9 7, [1] J. A. Gregory, Shape preserving spline inerpolaion, Compuer-Aided Design,vol.1,no.1,pp.3 7,19. [1] R.DelbourgoandJ.A.Gregory, C raional quadraic spline inerpolaion o monoonic daa, IMA Numerical Analysis,vol.3,no.,pp.11 1,193. [1] S. A. A. Karim and K. V. Pang, Local conrol of he curves using raional cubic spline, Applied Mahemaics,vol.1, Aricle ID 737, 1 pages, 1. [17] S. A. A. Karim and K. V. Pang, Monooniciy-preserving using raional cubic spline inerpolaion, Research Applied Sciences,vol.9,no.,pp.1 3,1. [1] S. A. A. Karim and V. P. Kong, Shape preserving inerpolaion using raional cubic spline, Research Applied Sciences, Engineering and Technology,vol.,no.,pp.17 1,1. [19] S. A. A. Karim and V. P. Kong, Convexiy-preserving using raional cubic spline inerpolaion, Research Applied Sciences, Engineering and Technology,vol.,no.3,pp.31 3, 1. [] M.Tian,Y.Zhang,J.Zhu,andQ.Duan, Convexiy-preserving piecewise raional cubic inerpolaion, Informaion and Compuaional Science,vol.,no.,pp.799 3,. [1] P. Lamberi and C. Manni, Shape-preserving C funcional inerpolaion via parameric cubics, Numerical Algorihms,vol.,no.1,pp.9,1.

14 1 Applied Mahemaics [] Q. Duan, L. Wang, and E. H. Twizell, A new C raional inerpolaionbasedonfuncionvaluesandconsrainedconrolof he inerpolan curves, Applied Mahemaics and Compuaion, vol.11,no.1,pp.311 3,. [3] F. Bao, Q. Sun, and Q. Duan, Poin conrol of he inerpolaing curve wih a raional cubic spline, JournalofVisualCommunicaion and Image Represenaion, vol.,no.,pp.7, 9. [] J.-C. Fioro and J. Tabka, Shape-preserving C cubic polynomial inerpolaing splines, Mahemaics of Compuaion,vol.7, no. 19, pp. 91 9, [] M. Dube and P. Tiwari, Convexiy preserving C raional quadraic rigonomeric spline, in Proceedings of he Inernaional Conference of Numerical Analysis and Applied Mahemaics (ICNAAM 1),vol.179,pp.99 99,Sepember1. [] Y.-J. Pan and G.-J. Wang, Convexiy-preserving inerpolaion of rigonomeric polynomial curves wih a shape parameer, Zhejiang Universiy SCIENCE A,vol.,no.,pp , 7. [7] F. Ibraheem, M. Hussain, M. Z. Hussain, and A. A. Bhai, Posiive daa visualizaion using rigonomeric funcion, Journal of Applied Mahemaics, vol.1,aricleid71,19pages, 1. [] M. Sarfraz, M. Z. Hussain, and F. S. Chaudary, Shape preserving cubic spline for daa visualizaion, Compuer Graphics and CAD/CAM,vol.1,pp.1 193,. [9] R. Delbourgo and J. A. Gregory, The deerminaion of derivaive parameers for a monoonic raional quadraic inerpolan, IMA Numerical Analysis, vol.,no.,pp.397, 19. [3] J. W. Schmid and W. Hess, Posiiviy of cubic polynomials on inervals and posiive spline inerpolaion, BIT Numerical Mahemaics, vol., no., pp. 3 3, 19. [31] M. H. Schulz, Spline Analysis, Prenice-Hall, Englewood-Cliffs, NJ, USA, [3] M. Z. Hussain and J. M. Ali, Posiiviy preserving piecewise raional cubic inerpolaion, Maemaika,vol.,no.,pp.17 13,.

15 Advances in Operaions Research Volume 1 Advances in Decision Sciences Volume 1 Applied Mahemaics Algebra Volume 1 Probabiliy and Saisics Volume 1 The Scienific World Journal Volume 1 Inernaional Differenial Equaions Volume 1 Volume 1 Submi your manuscrips a Inernaional Advances in Combinaorics Mahemaical Physics Volume 1 Complex Analysis Volume 1 Inernaional Mahemaics and Mahemaical Sciences Mahemaical Problems in Engineering Mahemaics Volume 1 Volume 1 Volume 1 Volume 1 Discree Mahemaics Volume 1 Discree Dynamics in Naure and Sociey Funcion Spaces Absrac and Applied Analysis Volume 1 Volume 1 Volume 1 Inernaional Sochasic Analysis Opimizaion Volume 1 Volume 1