Convexity preserving interpolation by splines of arbitrary degree

Transcription

1 Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete set of data s gven usng splnes of arbtrary degree, whch preserves the convexty of gven set of data. Mathematcs Subject Classfcaton 2000: 65D05, 65D07, 41A05, 41A15. Keywords phrases: splne, nterpolaton, convexty preservng nterpolaton. 1 Introducton It s well known that problems concernng nonnegatvty, monotoncty, or convexty preservng nterpolaton have receved consderable attenton, because of ther nterest n computer aded desgn n other practcal applcatons [1]. Problem of constructon of nterpolatng curve whch preserves the convexty of the ntal dscrete set of data stll remans n the focus of nvestgators [2]. In what follows an algorthm preservng the convexty of gven set of data s presented. 2 Interpolatng splnes of arbtrary degree Let us assume that the mesh : a = x 0 < x 1 <... < x n = b s gven on the nterval [a, b] f = f(x ), = 0(1)n, are the correspondng data ponts. Problem of constructon of an nterpolaton functon S, such that nterpolaton condtons S(x ) = f, = 0(1)n, are held S C 2 [a, b], s consdered. c 2010 by I. Verlan 54

2 Convexty preservng nterpolaton by splnes of... Let us ntroduce splnes as follows: on [x, x +1 ] S(x) = f + (f +1 f )t + + h2 M (1 t)((1 t) α 1) where the followng notatons are used: t = (x x )/h, h = x +1 x, S (x ) = M. + h2 M +1t(t α 1), (1) The α s a free parameter of the splnes (1) has to satsfy the condton α > 1. From (1) for the frst dervatve of splne we get: where S (x) = δ (1) h (M ((α + 1)(1 t) α 1) M +1 ((α + 1)t α 1)),(2) δ (1) = (f +1 f )/h, for the second dervatve, respectvely: S (x) = M (1 t) α 1 + M +1 t α 1. (3) Obvously, at the knots of the mesh the second dervatve s the contnuous one. For the frst dervatve at the knots of the mesh we have S (x ) = δ (1) 1 + h 1M 1 α 1 (α 1 + 1) + h 1M α S (x +) = δ (1) h M α + 1 h M +1. Requrng the contnuty of the frst dervatve of the splne at the knots of the mesh we obtan the followng system of lnear algebrac equatons: 55

3 I. Verlan where c M 1 + a M + b M +1 =, = 1(1)n 1, (4) c = h 1 α 1 (α 1 + 1), a = h 1 α h α + 1, b = h, = δ (1) δ (1) 1. The system (4), presented above, s the undetermned one. Snce the system (4) provdes only n 1 lnear equatons n n + 1 parameters M, t follows that two addtonal lnearly ndependent condtons are needed n order to have a determned system of equatons. In what follows we ll consder that the end condtons of the type M 0 = f 0 M n = f n are used as addtonal condtons. In fact, t s easy to prove that the system of equatons (4) has the dagonally domnant matrx of coeffcents, therefore the soluton of ths system exsts t s the unque one for fxed parameters of the splne. 3 Convexty preservng algorthm In ths secton t s consdered that the ntal set of data s the convex one, namely, 0, = 1(1)n 1. From the formulae (3) t mmedately follows, that n order to preserve the convexty of ntal data the soluton of the system (4) has to be the nonnegatve one. So, let s choose the value of free parameter as t follows: 56

4 Convexty preservng nterpolaton by splnes of... ( (2) 2δ α max +1, 2δ(2) +1 ). (5) There s no problem to prove that n ths case the soluton of the system (4) s the nonnegatve one. Ths concluson s based on the fact that for the coeffcents of the matrx of lnear algebrac equatons (4) the followng relatons are vald: c a 1 < 1 2 In ths case we get that b a +1 < 1 2. c 1 b a 1 a +1 As a result, takng nto account [3] t can be concluded that the soluton of the system (4) s the nonnegatve one. As a result the splne, constructed usng condton (5), preserves the convexty of the ntal set of data. 4 Conclusons In fact not only the problem of convexty preservng nterpolaton represents the nterest, but also the problem of constructon of nterpolants whch have the same number of nflecton ponts as the ntal set of data preserve the convexty or concavty of data. References [1] B. I. Kvasov. Methods of shape-preservng splne approxmaton, World Scentfc, Sngapore,

5 I. Verlan [2] V. V. Bogdanov, Yu. S. Volkov, Selecton of parameters of generalzed cubc splnes wth convexty preservng nterpolaton, Sb. Zh. Vychsl. Mat., 9:1 (2006), pp [3] I. I. Verlan. Postve solutons of systems of lnear algebrac equatons wth Jacoban matrces of coeffcents. Mat. Issled. No 104, Program. Obespech. Vychsl. Komlpeks. (1988), pp (n Russan) I.Verlan, Receved July 5, 2010 Department of Appled Mathematcs, Moldova State Unversty 60 Mateevch str. Chşnău, MD2009, Moldova Insttute of Mathematcs Computer Scence 5 Academe str., Chşnău, MD2028, Moldova E mal: verlan@yahoo.com 58