Monotonic Interpolating Curves by Using Rational. Cubic Ball Interpolation

Transcription

1 Appled Mathematcal Scences, vol. 8, 204, no. 46, HIKARI Ltd, Monotonc Interpolatng Curves by Usng Ratonal Cubc Ball Interpolaton Samsul Arffn Abdul Karm Department of Fundamental and Appled Scences Unverst Teknolog PETRONAS, Bandar Ser Iskandar 3750 Tronoh, Perak Darul Rdzuan Malaysa Copyrght 204 Samsul Arffn Abdul Karm. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Abstract Ths paper dscusses the monotoncty preservng of monotone data by usng ratonal cubc Ball nterpolant wth four parameters. The suffcent condton for the monotoncty of the ratonal nterpolant wll be derved on two of the parameters meanwhle the remanng two are free parameters that can be used to modfy the fnal shape of the monotonc nterpolatng curves. The degree smoothness acheved s C. The frst dervatve s estmated by usng arthmetc mean method (AMM) and geometrc mean method (GMM). Several numercal results wll be presented ncludng comparson wth exstng scheme. From the numercal results, ratonal cubc Ball gves vsually pleasng results. Keywords: Ratonal nterpolant, cubc Ball, monotone, suffcent, parameters, contnuty, numercal. Introducton In computer graphcs and scentfc vsualzaton, the dsplay curves and surfaces (or mages) are an mportant and man task for the desgner and expertse. Usually the gven data has ts own characterstcs such as postvty, monotoncty and convexty. For example f the data s monotone (the frst dervatve have ether postve value or negatve value), then the methods that have been used to

2 7260 Samsul Arffn Abdul Karm dsplay the curves or surfaces must be able to retan the orgnal shape of the data namely monotone. For examples the dose-response curves and surfaces n bochemstry and pharmacology are other examples n whch the monotoncty exsts n the data sets Belakov []. Ths paper s a contnuaton of our prevous research n Karm [2, 3, 4] and Karm and Kong [5]. There exst many methods that can be used for monotoncty preservng. Two early papers are Frtsch and Carlson [6] and Dougherty et al. [7]. Both methods requre the modfcaton(s) of the frst dervatve f the monotoncty of the data s not preserves. Karm and Kong [5], Sarfraz et al. [8, 9, 0], Sarfraz [, 2] and Hussan and Sarfraz [3] have dscussed the shape preservng nterpolaton by usng varous types of polynomal splne and ratonal cubc splne nterpolant. All the methods do not requre any modfcaton of the frst dervatve n order to mantan the monotoncty of the gven monotone data sets. Motvated by work of Karm [2, 3, 4], n ths paper the author wll extend the ratonal cubc Ball wth four parameters that has been ntated by Karm [2] for monotoncty preservng. The suffcent condton for the ratonal cubc Ball to be monotone on the entre gven nterval wll be derved on two parameters whle the remanng two parameters are free parameters that can be used to refne the fnal shape of the monotonc nterpolatng curves. The suffcent condtons provde vsually pleasng monotonc nterpolatng curves wth two free parameters, that can be further utlzed by the user. These papers have the followng contrbuton to the feld of scentfc vsualzaton and computer graphcs:. In ths paper the ratonal cubc Ball nterpolant (cubc/cubc) ntated by Karm [2] has been used for monotoncty preservng whle Hussan and Sarfraz [3] have used ratonal cubc splne (cubc/cubc) wth four parameters too. The suffcent condtons for monotoncty-preservng can be used to generate the monotonc nterpolatng curves. But by usng ratonal cubc Ball nterpolaton the computaton may be lower than the ratonal cubc splne nterpolaton. 2. Numercal comparson wth the work of Frstch and Carlson [6] for monotoncty preservng also has been done. Furthermore the frst dervatve need not to be modfed n whch the monotoncty s not preserves. Meanwhle the shape preservng by usng Frstch and Carlson [5] and Dougherty et al. [7] requre the modfcaton of the frst dervatve n whch the shape volaton s found. The ratonal cubc Ball nterpolaton do not requred any addtonal new knots compare to the work oflahtnen [8].

3 Monotonc nterpolatng curves The suffcent condton for monotoncty of the ratonal cubc Ball nterpolant s dfferent from the suffcent condton for monotoncty of the ratonal cubc splne (cubc/cubc) of the work by Hussan and Sarfraz [3]. Thus our ratonal cubc Ball nterpolant provde good alternatve to the exstng ratonal cubc splne scheme for monotoncty preservng nterpolaton. The remander of the paper s organzed as follows. Secton 2 gve the revew the ratonal cubc Ball nterpolant wth four parameters ntated Karm [2]. Secton 3 dscuss the methods to estmate the frst dervatves values meanwhle Secton 4 s devoted to the monotoncty preservng by usng ratonal cubc Ball nterpolant. The suffcent condton for the monotoncty of the ratonal cubc Ball nterpolant wll be derved. All numercal results are gven n Secton 5 ncludng comparson wth the work of Frstch and Carlson [6]. Secton 6 gves the summary and conclusons to the paper. 2. Ratonal Cubc Ball Interpolant In ths secton the ratonal cubc Ball wth four parameters proposed by Karm [2] wll be revewed n detals. Assumng that the scalar (or functonal) data s gven.e. { f f h / x, f,,..., n } where x x2... x n. Let h x x, and x x / h wth 0. For Where x x, x,,2,..., n, s x P, S () Q P f A A f and Q. The parameters,,, 0,,2,..., n. The ratonal cubc Ball nterpolant n () satsfes the followng C condtons:

4 7262 Samsul Arffn Abdul Karm s x f, s x f, s x d, s x d. (2) Smple algebrac manpulaton to () by usng C condtons n (2) wll gve the value of the unknown varable Aj, j,2. It s gven as follows: A f h d, A f h d. (3) 2 Some observatons can be made as follows: When, 2, the ratonal cubc Ball nterpolant defned by Eq. () s reduce to the followng standard cubc Ball polynomal n Hermte-lke form (Karm [2]). s x f 2 f h d 2 f h d f. (4) Furthermore where Sx can be rewrtten as follows: h E Q s x f f. (5) E d d. When,, or, 0, ratonal nterpolant n (5) converges to straght lne gven below: s x f f. (6) Thus the ratonal cubc Ball nterpolant of [2] has the capablty to reproduce the straght lne when the parameters satsfed:, 0, and,. 3. Determnaton of Dervatves For monotoncty preservng nterpolaton the frst dervatve values can be estmated ether by usng arthmetc mean method (AMM) or geometrc mean method (GMM). In ths paper both methods wll be used to estmate the values of the frst dervatves, d. Below the mathematcal formula of AMM:

5 Monotonc nterpolatng curves 7263 At the end ponts x and x n d d h h h2 2 h n hn hn2 n n n n2 (7) (8) At the nteror ponts, x, 2,..., n, the values of d are gven as d h h h h. (9) Meanwhle the GMM s defned as follows: At the end ponts x and x n d h h h2 h2 0, 0 or 0 3, 3, otherwse (0) d hn h n n hn2 hn2 n n, n2 0 0 or 0 n n, n2 otherwse () At nteror ponts, x, 2,..., n, the values of d are gven as h h h h h h d. (2) f3 f wth 3,, x x 3 fn fn2 and nn, 2. x x n n2 The AMM s smple and easy to used. Meanwhle the GMM wll gve the postve values for the frst dervatve, d f the data s monotone. Both methods have ther own advantages.

6 7264 Samsul Arffn Abdul Karm 4. Monotoncty-Preservng Usng Ratonal Cubc Ball Interpolant In ths secton the ratonal cubc Ball functon defned n Secton 2 wll be used for monotoncty preservng for strctly monotone data sets. We begn wth the followng assumpton: Let x, f,,..., n be a gven monotone data set, where x x2... x n. For a monotonc ncreasng (decreasng), the necessary condton should be: f f2... fn (or f f 2... fn for monotonc decreasng) (3) Equaton (3) s equvalent wth 0 (or 0 for monotonc decreasng data) (4) In ths secton, the necessary and suffcent condton for the C monotoncty of ratonal cubc Ball nterpolant wll be derved n detals. For monotonc (ncreasng or decreasng) the ratonal cubc Ball nterpolant s x n Eq. (), the frst dervatve must satsfy: d 0,,2,..., n. (for monotonc ncreasng) (5) Now, s x s monotonc ncreasng f and only f s x x x x n 0,. (6) Now after some smplfcaton the frst dervatve of the ratonal cubc Ball nterpolant s x s gven by: s x 4 4 j B j0 j 2 Q j (7)

7 Monotonc nterpolatng curves 7265 wth, 2, 2 B d B d 2 0 B 3 2 d, A 4 d and B 4 d d d d. 2 Furthermore, B 2 can be rewrtten as follows: B d d d d 2 4. Now, s x 0 f and only f B 0, j 0,,2,3,4. The necessary condtons for j monotoncty are d 0, d 0 and the suffcent condton for monotoncty can be obtaned from: Bj 0, j 0,, 2,3, 4. Clearly B 0 0, B 4 0. If the gven data s strctly monotone (.e. 0 ), then from B 0, B2 0and B3 0 wll gves us the followng condtons: B 0 f 2 d 0. (8) B2 0 f d d d d 4 0. and (9) B3 0 f 2 d 0. (20) Eq. (8), (9) and (20) provdes the followng nequaltes:

8 7266 Samsul Arffn Abdul Karm d. (2) d d 0. (22) d. (23) The suffcent condtons n (2), (22) and (23) can be stated as the followng theorem. Theorem : Gven a strctly monotonc ncreasng set of data satsfyng (3) or (4), there exst monotonc ratonal cubc Ball nterpolatng splne s x C x x nvolvng free parameters, that satsfy the followng, n suffcent condtons:, 0, d d d d,. Remark : The choces gven n Eq. (24) satsfed the condtons n Eq. (8), (9) and (20). Thus the monotoncty of the ratonal cubc Ball nterpolant s guaranteed. Remark 2: If the data are constant on certan nterval,.e. 0, then t s necessary to set d 0, hence sx f f s a constant on the nterval d x, x,,2,..., n. Ths shows that the ratonal nterpolant s monotone. (24) 5. Results and Dscusson In order to llustrate the monotoncty preservng nterpolaton by usng ratonal cubc Ball nterpolaton (cubc/cubc), two sets of monotone data taken from Akma [5] and Sarfraz et al. [9] were used. All the data sets are lsted n Table and Table 2 respectvely.

9 Monotonc nterpolatng curves 7267 Table. A monotone data from [5] x f d (GMM) d (AMM) Table 2. A monotone data from [9] x 0 f d (GMM) d (AMM) 0 (-2.833)

10 7268 Samsul Arffn Abdul Karm (a) (b) Fgure. Default Cubc Ball nterpolaton by usng (a) AMM and (b) GMM for data n Table. (a) (b) Fgure 2. Default Cubc Ball nterpolaton by usng (a) AMM and (b) GMM for data n Table 2.

11 Monotonc nterpolatng curves 7269 (a) (b) Fgure 3. Shape preservng by usng AMM wth (a) (b) 2 and (c) (c), 2 for data n Table 2.

12 7270 Samsul Arffn Abdul Karm (a) (b) Fgure 4. Shape preservng by usng GMM wth (a) (b) 2 (c) (c), 2 for data n Table.

13 Monotonc nterpolatng curves 727 (a) (b) Fgure 5. Shape preservng by usng AMM wth (a) (b), 2 for data n Table 2. (a) (b) Fgure 6. Shape preservng by usng GMM wth (a) (b), 2 for data n Table 2.

14 7272 Samsul Arffn Abdul Karm Fgure (a) and Fgure (b) and Fgure 2(a) and Fgure 2(b) shows the default cubc Ball nterpolaton.e. when, 2, for data lsted n Table and Table 2 respectvely. Fgure 3(a), 3(b) and 3(c) and Fgure 4(a), 4(b) and 4(c) shows the monotoncty preservng by usng AMM and GMM for data sets lsted n Table respectvely. Meanwhle Fgure 5(a) and 5(b) and Fgure 6(a) and 6(b) shows the monotoncty preservng by usng AMM and GMM for data sets lsted n Table respectvely. From Fgure 2 untl Fgure 6 t can be seen clearly that monotoncty preservng by usng ratonal cubc Ball nterpolant wth four parameters gve smooth and very vsual pleasng nterpolatng curves. Furthermore the GMM method gve more smooth monotonc nterpolatng curves as compare wth the monotonc nterpolatng curve by usng AMM method. In general, both methods are acceptable to estmate the frst dervatve values for monotoncty preservng by usng ratonal cubc Ball nterpolant. One queston stll remans to be answered: Whch method gve better results? Table 3, Table 4, Table 5 and Table 6 gves the value for all shape parameters for Fgure 3(a), Fgure 4(a), Fgure 5(b) and Fgure 6(b) respectvely. The other shape parameters can be calculated by usng Eq. (24). Fnally Fgure 7 and Fgure 8 show the shape preservng nterpolaton by usng Frstch and Carlson [6] methods for data n Table and Table 2 respectvely. Clearly the proposed ratonal cubc Ball nterpolaton wth four parameters gve better nterpolatng curves together wth more smooth results compare to the work of Frstch and Carlson [6]. Fgure 7. Shape preservng by usng Frstch and Carlson [6] for data n Table.

15 Monotonc nterpolatng curves 7273 Fgure 8. Shape preservng by usng Frstch and Carlson [6] for data n Table 2. Table 3.Shape parameters values for Fgure 3(a) Table 4.Shape parameters values for Fgure 4(a)

16 7274 Samsul Arffn Abdul Karm Table 5.Shape parameters values for Fgure 5(b) Table 6.Shape parameters values for Fgure 6(b) Fnal Remark: For the Akma data sets, the monotoncty preservng s appled only on the nterval [8, 5]. Ths s due to the fact that on the nterval [0, 8] the ratonal cubc Ball nterpolaton wll reproduce the straght lne snce on ths nterval snce 0,,...,5. 6. Conclusons These paper dscuss the use of ratonal cubc Ball nterpolant wth four parameters for monotoncty preservng. The suffcent condtons for the monotoncty of the ratonal cubc Ball nterpolant are derved on two parameters.e. and. Meanwhle the remanng two parameters and are free parameters that can be further utlzed to refne the fnal shape of the monotonc nterpolatng curves.

17 Monotonc nterpolatng curves 7275 The frst dervatve s estmated by usng AMM and GMM. Clearly the proposed ratonal cubc Ball nterpolant gve very smooth monotonc nterpolatng curves for both tested data sets. The free parameters and gve extra degree of freedom to the user n controllng the fnal shape of the monotonc nterpolatng 2 curves. The ratonal cubc Ball nterpolant also can used to generate the C monotonc nterpolatng curves. Works on bvarate nterpolatng also s underway by the author. Acknowledgment. The author would lke to acknowledge UnverstTeknolog PETRONAS (UTP) for the fnancal support receved n the form of a research grant: Short Term Internal Research Fundng (STIRF) No. 35/202and Mathematca Software. References [] G. Belakov. Monotoncty preservng approxmaton of multvarate scattered data, BIT, 45(4), , [2] Karm, S.A.A. Postvty Preservng by Usng Ratonal Cubc Ball Functon. ICOMEIA 204, May 204, The Gurney Resort, Penang. AIP Conf. Proc.##:##-##. [3] Karm, S.A.A. (203). Ratonal Cubc Ball Functons for Postvty Preservng. Far East Journal of Mathematcal Scences (FJMS). Vol. 82, No. 2, pp , 203. [4] Karm, S.A.A. GC Monotoncty Preservng usng Cubc Ball Interpolaton. Australan Journal of Basc and Appled Scences, 7(2): , 203. [5] Karm, S.A.A. and Kong, V.P. Monotoncty Preservng usng Quartc Splne. In AIP Conf. Proc. 482:26-3, 202. GC Ratonal [6] F.N. Frtsch and R.E. Carlson. Monotone pecewse cubc nterpolaton, SIAM J. Numer. Anal. 7: , 980. [7] R.L. Dougherty, A. Edelman, and J.M. Hyman, Nonnegatvty-, Monotoncty-, or Convexty-Preservng Cubc and Quntc Hermte Interpolaton, Mathematcs of Computaton, Volume 52(86) , 989.

18 7276 Samsul Arffn Abdul Karm [8] M. Sarfraz, M.Z. Hussan and F.S. Chaudary. Shape Preservng Cubc Splne for Data Vsualzaton, Computer Graphcs and CAD/CAM 0, 85-93, [9] M. Sarfraz, S. Butt and M.Z. Hussan. Vsualzaton of shaped data by a ratonal cubc splne nterpolaton, Computers & Graphcs 25: , 200. [0] Sarfraz, M.Z Hussan and M Hussan. Shape-preservng curve nterpolaton, Internatonal Journal of Computer Mathematcs, Vol. 89, No., 35-53, 202. [] M. Sarfraz. A ratonal cubc splne for vsualzaton of monotonc data. Computers & Graphcs 24(4):509-56, [2] M. Sarfraz. A ratonal cubc splne for the vsualzaton of monotonc data: an alternate approach, Computers & Graphcs 27: 07-2, [3] M.Z. Hussan and M. Sarfraz. Monotone pecewse ratonal cubc nterpolaton,, Internatonal Journal of Computer Mathematcs, Vol. 86, No. 3, March 2009, pp , [4] Lahtnen A. Monotone nterpolaton wth applcatons to estmaton of taper curves. Annals of Numercal Mathematcs 3:5-6, 996. [5] H. Akma. New method and smooth curve fttng based on local procedures. J. Assoc. Comput. Mech. 7: , 970. Receved: August, 204