Positivity Preserving Interpolation by Using

Transcription

1 Appled Matematcal Scences, Vol. 8, 04, no. 4, HIKARI Ltd, ttp://dx.do.org/0.988/ams Postvty Preservng Interpolaton by Usng GC Ratonal Cubc Splne Samsul Arffn Abdul Karm Department of Fundamental and Appled Scences Unverst Teknolog PETRONAS Bandar Ser Iskandar, 3750 Trono Perak Darul Rdzuan, Malaysa Copyrgt 04 Samsul Arffn Abdul Karm. Ts s an open access artcle dstrbuted under te Creatve Commons Attrbuton Lcense, wc permts unrestrcted use, dstrbuton, and reproducton n any medum, provded te orgnal work s properly cted. Abstract. Ts paper proposed new ratonal cubc splne wt GC contnuty for data nterpolaton. It as cubc numerator and ratonal as denomnator wt two parameters tat can be used to refne te fnal sape of te nterpolatng curves. Data dependent constrants are derved on one sape parameter, r wle te second parameter e s a free parameter tat can be canged to obtan dfferent forms of nterpolaton curve. Numercal results are gven to test te applcablty of te proposed sceme. Keywords: Sape preservng; GC contnuty; ratonal cubc splne; postvty Introducton Interpolatng and approxmatng te data are mportant n computer grapcs (CG), geometrc modelng and reverse engneerng (RE) problems. Cubc splne nterpolaton s a common metod to nterpolate te gven data sets. One of man Ts work s supported by Unverst Teknolog PETRONAS troug ts Sort Term Internal Researc Fundng (STIRF) No. 35/0.

2 054 Samsul Arffn Abdul Karm advantage of cubc splne s tat t as C contnuty. Meanwle cubc Hermte splne also as been use for data nterpolaton purpose wt contnuous. Even toug bot metods can be used to solve many scence and engneerng problems, tey suffer from te exstence of unwanted oscllaton (especally for cubc splne) and also does not preserves te sape of te data []. Tus researcers keep tryng to fnd best possble functon tat can nterpolate te data wt sape preservng property. Postvty s present n our real lfe problems. For examples, levels of gas dscarge n certan cemcal reactons, volume as well as wnd energy are examples of postve data [7]. Postvty preservng as been dscussed by many researcers. For examples, Butt and Brodle [] ave used cubc splne to preserve te postve data by nsertng one or two extra knots on te regon n wc te postvty are volated. In Ref. [4], te autors dscussed te postvty preservng by usng cubc and quntc splne by adjustng te frst dervatve untl te nterpolant are postve. Tey proposed local algortm. Due to te lmtaton of cubc and quntc splne polynomal for sape preservng, many researcers ave proposed ratonal splne nterpolant to overcome ts lmtaton. For example Ref. [7] ave use new cubc/lnear sceme for postvty and monotoncty preservng. Ter nterpolatng curves looks lke ave only C 0 even toug te autors clam ter metods produces C nterpolatng curves. Ref. [8] ave used cubc/cubc sceme for postvty preservng (bot for curves and surfaces). In [9], te autor as proposed new cubc/cubc sceme by utlzng cubc Ball functon for postvty preservng. Sape preservng for postve data by usng dfferent types of ratonal splne are dscussed n [3], [4] and [5]. For certan applcatons, te user may need dfferent types of contnuty. For example, tey may want te fnal nterpolatng curves wll be GC nstead of C contnuous. Ref. [6] as proposed GC cubc splne for postve, monotone and convexty preservng. Ter dea as been extended by Karm [] for monotoncty preservng by usng cubc Ball polynomal wt GC contnuty. Motvated by ts development, Ref. [0] and Ref. [] ave proposed a new GC ratonal quartc splne (quartc/lnear) to preserves te monotone and postve data sets respectvely. Te proposed ratonal cubc splne wt GC contnuty n ts paper s totally new and as several features. Frstly, te curves wc preserve te postvty of data ave an explct representaton wt two sape parameters. Secondly, tere are no knots nsertons to preserve postvty of te data sets as appear n te work by []. Furtermore, an algortm to mplement te sceme s gven and from te numercal results, t sows tat te obtaned nterpolatng curves are smoot. Te remander of te paper s organzed as follows; Secton we revew te GC ratonal cubc splne nterpolant. Te dervatve parameter estmaton s dscussed n Secton 3. Te postvty problem s dscussed n Secton 4 for te generaton postvty-preservng wt algortm for te computer mplementaton. Te numercal results wll be presented n Secton 5. Fnally, Secton 6 concludes te paper.

3 Postvty preservng nterpolaton 055 GC Ratonal Cubc Splne Interpolant In ts secton, te new GC ratonal cubc splne nterpolant wt lnear denomnator wll be dscussed. It started wt te followng assumpton: Suppose { ( x, f), =,..., n} s a gven set of data ponts, were x < x <... < x n. Let ( f+ f) = x x, Δ = ( x x ) and a local varable, θ =, ence 0 θ. + x x, x+, =,,..., n, For P ( θ ) s( x) s( x θ) S ( θ) ( θ ), were P( θ ) = e fφ0( θ) + Vφ( θ) + Vφ( θ) + f+ φ3( θ) Q( θ ) = e( θ) + θ were 3 3 φ0( θ) ( θ) φ( θ) θ( θ) φ( θ) θ ( θ) φ3( θ) θ ed d + and V = ( e + ) f + V = ( e + ) f. = + = () Q =, =, =, =. (), r 0, r 0 + e > > are sape parameters, and d denotes a gven dervatve value at knot x. Te ratonal cubc nterpolant () satsfes te followng propertes: s( x ) = f, s( x ) = f, ence, s ( x) s + + () ( ) () ( ) r + d d s x, s x, + = + = r r+ GC n x, xn. Obvously wen e = and standard cubc Hermte polynomal gven as follows s ( x) = ( θ) ( + θ) f ( 3 ) ( ) + θ θ f+ + θ θ d θ ( θ) d +. (4) Meanwle wen r = te ratonal nterpolant n () wll reduces to te ratonal (3) r = te ratonal nterpolant n () wll reduces to te cubc splne (cubc/lnear) wt one sape parameter orgnally proposed by [5, 6]. 3 Dervatve Estmaton In most applcatons, te frst dervatve values, d are not gven and t must be determned troug some matematcal metods. One of te estmaton metods s artmetc mean metod (AMM) [3]. Ts metod s use to estmate te frst dervatves d, =,,..., n. AMM s sutable for postvty preservng nterpolaton. ( ) d = Δ + Δ Δ (5) +

4 056 Samsul Arffn Abdul Karm ( ) n d n = Δ n + Δ n Δ n (6) n + n Meanwle d,,3,..., n = are estmated by usng te followng formula: 0, f Δ = 0 or Δ = 0 d = Δ + Δ, oterwse =,3,..., n + (7) 4 Postvty-Preservng ratonal cubc splne Interpolant In general, te GC ratonal cubc splne nterpolant wt lnear denomnator descrbed n te Secton, does not always preserves te postvty of te gven data sets. As mentoned n te Introducton Secton, te ordnary cubc splne nterpolaton does not guarantee to preserve te postvty of te data sets lsted n Table and Table respectvely. Fg. and Fg. sow ts fact clearly. We may fnd te sutable value of sape parameters e, =,,..., n and r j, j =,,..., n by tral and error strategy. But, ts s not an effcent metod and not easy to be mplemented. Tus, te best possble ways to overcome tose problems s to derve te suffcent condton for postvty on one of te sape parameter wle te oter parameter can be used to refne te fnal sape of te curves. Our man strategy s to mpose certan condtons on a ratonal cubc defned n () to ensure tat t wll preserve te postvty of a data set. Let us assume a strctly postve data set s gven {( x,f ),( x,f ),...,( x n,fn) } (8) were x < x <... < x n, (9) and f > 0: =,,...,n. (0) Snce te sape parameter e s postve, te denomnator n () s always postve, tus te ratonal nterpolant () wll be postve f te cubc polynomal (n numerator) s postve. Determnaton of sutable value of rj, j =,,..., n s sgnfcant to ensure te exstng of postve ratonal cubc nterpolant (). Now by usng te man results from Scmdt and Hess [7], te followng s te suffcent condton for te postvty of te cubc splne polynomal. Proposton (Postvty of cubc splne polynomal) * For te strct nequalty postve data n (0), P ( θ ) > 0 f and only f: ( P ( 0, ) P ( ) ) R R were

5 Postvty preservng nterpolaton 057 were ( ) ( ) a = P 0, b= P. 3ef 3f+ R = ( a, b) : a >, b<, ( + + Δ + + Δ ) ( ab) f f a b ab ( a b), : R = + 3( f+ a fb)( ab 3 f+ a+ 3 fb) ( f+ a fb ) a b > 0 Now, by dfferentatng P ( θ ) w.r.t. θ, te results are ( ) 3ef + e + f + ed / r P ( 0 ) =, and ( e ) f+ d + / r+ 3f P () =. ( P P ) R R Now from Proposton, t can be sown tat ( 0, ) ( ) and f () () ( ) > (3) 3ef+ e+ f+ ed/ r 3ef ( e ) f+ d + / r+ 3f+ 3 f+ < (4) Te nequalty (3) and (4) leads to te followng relatons: and ed r >, =,,..., n f ( e + ) + ( ) e + d r + >, =,,..., n f + (5) (6) Care must be taken snce for =,3,..., n tere exst two sets of sape parameter value r. Now, let * ed d r = Max 0,,, =,3,..., n. ( e + ) f ( e + ) f Tus, (5), (6) and (7) can be summarzed as Proposton below. (7)

6 058 Samsul Arffn Abdul Karm Proposton (Postvty of Ratonal cubc splne nterpolant wt lnear denomnator) For a strctly postve data, te ratonal cubc nterpolant wt GC (cubc numerator and lnear denomnator) (defned over te nterval [ x, x n ] s postve n eac subnterval [ x, x+ ], =,,..., n f te followng suffcent condton s satsfed: and r > ed n ( e + ) f ( e + ) { * }, r > n n n r Max 0, r,,3,..., n. d f n, (8) > = (9) Proof. Rearrange Eq. (5), (6) and (7). Te fnal results n (8) and (9) wll be obtaned. Remarks : After te sape parameter r, =,,..., n ave been calculated from (8) and (9), te postve nterpolant for postve data can be generated usng (). Remarks : Te parameter, =,,..., n can be used to refne te postve nterpolatng curves. e Remarks 3: Te suffcent condton n (8) and (9) can be re-wrtten as follows: and were λ > 0. ed r = λ+ Max 0,, r d = λ + fn n n Max 0,, n n ( e+ ) f ( en + ) { * } (0) r = λ + Max 0, r, =,3,..., n. () Remarks 4: Wen te data are constant on certan nterval,.e. Δ = 0, ten t s necessary to set d = d + = 0, ence S ( x) = f = f+ s a constant on te nterval [ x, x ], =,,..., + n. Suffcent condtons for postvty can also be determned from (). But f we proceed n ts drecton, t wll requre a lot of calculatons and more complcated. Terefore, te condtons n (0) and () are more practcal and more sutable for our purposes. An algortm to generate GC postvty-preservng curves usng te results n Proposton s gven below.

7 Postvty preservng nterpolaton 059 Algortm for postvty-preservng usng GC ratonal cubc splne. Input te number of data ponts, n, and data ponts ( ) x, f, =,..., n. For =,,..., n, estmate d usng artmetc mean metod (AMM). 3. Calculate 4. For =,,..., n ed r = λ+ Max 0,, r d = λ + n n Max 0,, fn n n ( e+ ) f ( en + ) Calculate, Δ and assgned any sutable values of > 0 Calculate te sape parameter r gven as follows: { * } r = λ + Max 0, r, λ > 0. e. λ, λ n > 0 Calculate te nner control ordnates V and V and generate te pecewse postve nterpolatng curves usng (). 5 Results and dscusson In order to llustrate te sape preservng nterpolaton by usng GC ratonal cubc splne nterpolaton (cubc numerator and lnear denomnator), two sets of data taken from Brodle and Butt [] and Sarfraz et al. [4] were used. Table. A postve data from [] x f

8 060 Samsul Arffn Abdul Karm Table. A postve data from [4] x f Fg.. Default cubc splne for data n Table wt e =, r = Fg.. Default cubc splne for data n Table wt e =, r =.

9 Postvty preservng nterpolaton Fg. 3. Ratonal Cubc splne for data n Table wt e =, λ = Fg. 4. Ratonal Cubc splne for data n Table wt e = 0.5, λ =, were λ = 0., λ = Fg. 5. Ratonal Cubc splne for data n Table wt e =, r = 3.

10 06 Samsul Arffn Abdul Karm Fg. 6. Ratonal Cubc splne for data n Table wt e = 5, λ = Fg. 7. Ratonal Cubc splne for data n Table wt λ = and e = (sold), e = (dased) Fg. 8. Ratonal Cubc splne for data n Table wt λ = 0. 3 and e = 0. 5 (sold), e = 0. (dased) Fg. 3 and Fg. 4 sow te resultant sape preservng nterpolaton usng proposed ratonal cubc splne wt lnear denomnator for data n Table. Smlarly, Fg. 5

11 Postvty preservng nterpolaton 063 and Fg. 6 sow te postve nterpolatng curves usng te proposed sceme for postve data n Table. Meanwle, Fg. 7 and Fg. 8 sow te postve nterpolatng curves by coosng dfferent values of e for data n Table and Table respectvely. From Fg. 3 untl Fg. 6, we can concluded tat te proposed GC ratonal cubc splne (cubc/lnear) are capable to mantan te geometrc features of te gven data sets. Furtermore, te suffcent condtons for postvty of ratonal cubc splne derved n Proposton are easy to use and can be modfy accordngly. An algortm to mplement te proposed ratonal sceme s also beng gven. Tus, te ratonal cubc splne wt GC contnuty provdes alternatve to te exstng sape preservng metod. 6 CONCLUSIONS Ts paper dscussed te problem of postvty-preservng nterpolaton by usng GC ratonal cubc splne nterpolant (cubc numerator and lnear denomnator). Te smple data dependent constrants on one of te sape parameter r for postvty are derved. Te sape parameter r wll determne te fnal sape of postve nterpolatng curves by manpulatng te oter sape parameter e. Proposton guarantee te exstence of postve GC ratonal cubc splne nterpolant. Te work n ts paper can be extended to te monotoncty and convexty preservng. Ts wll be our man subject for future researc. Acknowledgment Te autor would lke to acknowledge Unverst Teknolog PETRONAS (UTP) for te fnancal support receved n te form of a researc grant: Sort Term Internal Researc Fundng (STIRF) No. 35/0. Specal tanks are due to Dr Kong Voon Pang from Unverst Sans Malaysa, Penang for s knd advce and encouragement. References. Brodle, K. W. and Butt, S. Preservng convexty usng pecewse cubc nterpolaton. Computers and Grapcs 5, 5-3 (99).. Butt, S. and Brodle. K.W. Preservng postvty usng pecewse cubc nterpolaton. Computer and Grapcs 7 ():55-64 (993)

12 064 Samsul Arffn Abdul Karm 3. Delbourgo, R. and Gregory, J.A. Te Determnaton of Dervatve Parameters for a Monotonc Ratonal Quadratc Interpolant. IMA Journal of Numercal Analyss, 5: (985) 4. Dougerty, R.L., Edelman, A. and Hyman, J.M. Nonnegatvty-, Monotoncty-, or Convexty-Preservng Cubc and Quntc Hermte Interpolaton. Matematcs of Computaton, Volume 5(86) (989) 5. Duan, Q., Wang, X. and Ceng, F. Constraned nterpolaton usng ratonal cubc splne curve wt lnear denomnators. Korean J. Computer and Appl. Mat. : 03-5 (999). 6. Duan, Q., Djdjel, K., Prce, W.G. and Twzell, E.H. Constraned control and approxmaton propertes of a ratonal nterpolatng curve. Informaton Scences 5:8-94 (003) 7. Hussan, M.Z., Sarfraz, M. and Hussan, M. Scentfc Data Vsualzaton wt Sape Preservng C Ratonal Cubc Interpolaton. European Journal of Pure and Appled Matematcs. Vol. 3, No., 94- (00) 8. Hussan, M.Z. and Sarfraz, M. Postvty-Preservng nterpolaton of postve data by ratonal cubcs, Journal of Computatonal and Appled Matematcs 8, (008) 9. Karm, S.A.A. Ratonal Cubc Ball Functons For Postve Interpolatng Curves. 0. Preprnt. 0. Karm, S.A.A. and Pang, K.V. Monotoncty Preservng usng GC Ratonal Quartc Splne. AIP Conf. Proc. 48:6-3 (0). ttp://dx.do.org/0.063/ Karm, S.A.A., Kong, P.V. and Hasm, I. Postvty Preservng usng GC Ratonal Quartc Splne. AIP Conf. Proc. 5, (03); ttp://do:0.063/ Karm, S.A.A. GC Monotoncty Preservng usng Ball Cubc Interpolaton. Australan Journal of Basc and Appled Scences, 7(): (03) 3. Sarfraz, M., Hussan, M.Z. and Nsar, A. Postve data modelng usng splne functon. Appled Matematcs and Computaton, 6, , 00.

13 Postvty preservng nterpolaton Sarfraz, M., Butt, S. and Hussan, M.Z. Vsualzaton of saped data by a ratonal cubc splne nterpolaton. Computers & Grapcs 5: (00) 5. Sarfraz, M. Vsualzaton of postve and convex data by a ratonal cubc splne nterpolaton. Informaton Scences 46:-4, (00) 6. Sarfraz, M., Hussan, M.Z. and Caudary, F.S. Sape Preservng Cubc Splne for Data Vsualzaton. Computer Grapcs and CAD/CAM 0, (005) 7. Scmdt, J.W. and Hess, W. Postvty of cubc polynomals on ntervals and postve splne nterpolaton. BIT 8, (988). Receved: August, 04