Shape preserving third and fifth degrees polynomial splines

Transcription

1 Amercan Journal of Appled Mathematcs 014; (5): Publshed onlne September 30, 014 ( do: /j.ajam ISSN: (Prnt); ISSN: X (Onlne) Shape preservng thrd and ffth degrees polynomal splnes Vladmr Ivanovch Pnchukov Sberan dvson of Russan Academy of Sc., In-te of Computatonal Technologes, Novosbrsk, , Russa Emal address: To cte ths artcle: Vladmr Ivanovch Pnchukov. Shape Preservng Thrd and Ffth Degrees Polynomal Splnes. Amercan Journal of Appled Mathematcs. Vol., No. 5, 014, pp do: /j.ajam Abstract: Ths paper s devoted to the development of postvty and monotonsty preservng lnear splne technques, namely, technques whch are based on deas appled n the feld of hgh order TVD (Total Varaton Dmnshng) methods for numercal solvng compressble flow equatons. Thrd and ffth degrees polynomal splnes are constructed. Thrd degree splnes nclude two varants, namely, monotonsty preservng and postvty preservng splnes. These splnes may be consdered as modfcatons of classcal cubc splne and may be dentcal to ths splne for good data. These splnes get shape preservaton at the cost of reducng smoothness tll C^1. To restore C^smoothness ffth degree polynomal splnes are consdered, whch are constructed as a sum of base cubc shape preservng splnes and ffth degree terms, whch are chosen to provde contnuty of the splne second dervatve. These C^ffth degree polynomal splnes are observed to preserve monotonsty or postvty for all consdered data wth these propertes. Keywords: Classcal Cubc Splne, Interpolaton, Shape Preservng, Postvty, Monotonsty, Polynomal Functons 1. Introducton It s known that hgh accuracy nterpolatons are connected wth possble producng of undesrable oscllatons near ponts of nterpolated functon dscontnutes or ponts of dscontnutes of the fncton frst dervatve. The problem of nterpolatons constructng, whch do not produce oscllatons, may be consdered as a part of the more total problem of nterpolatons constructng, whch preserve the data shape. Data may be postve, constraned, monotone, convex accordng to ther shapes. For example, some substances n chemstry, physcs or bology [1,] are always postve. There are processes n these scences whch are descrbed by monotoncally ncreasng or decreasng curves. Constraned data are avalable n stock markets, where stock prces often move n ncreasng or decreasng channels. Convex nterpolatons look more natural and pleasant for convex data. Pesewse nterpolaton schemes may be classfed as local, whch may be calculated n any subnterval ndependently of other subntervals, and global, whch calculaton nvolves solvng the system of I or I*k equatons, I number of nterpolaton knots. Frst class contans, for example, monotonsty preservng pecewse cubc Hermte nterpolatons [1,3], trgonometrc splnes [4,5]. Second class contans, for example, B-splnes [6,7], ratonal splnes [8-10], dscrete hyperbolc tenson splnes [11,1], monotoncty preservng cubc splne [13], based on lmtaton of some terms n equatons for the splne coeffcents defnton. Ths lmtaton uses deas dampng undesrable oscllatons appled n calculatons of gasdynamcs flows by hgh order methods [14]. Recent paper s devoted to the development of ths approach, whch allows to receve lnear equatons, provdng shape preservaton. Below classcal cubc splne s descrbed n a form, convenent to the term lmtaton applcaton. Next secton s devoted to representaton of monotonsty preservng C 1 cubc splne [13]. Ffth degree polynomal C splne s descrbed n secton 4. Secton 5 s concerned wth postvty preservng cubc splne.. Splnes Let {(x,u ), =0,1,,3,...,I} be the gven set of data ponts defned over the nterval [0,a], where 0 = =x 0 <x 1 <x <,...,< x I = a. Condtons s(x )=u, and, at nner knots 0<<I, s'(x +0)=s'(x -0), s''(x +0)=s''(x -0), are used. We deal here wth splne dervatve values s (x ) = v, 0 I. Snce the frst dervatve s a pecewce second degree polynomal, ths functon may be wrtten n the subnterval x, x ] as follows [ 1

2 Amercan Journal of Appled Mathematcs 014; (5): s (x)=v 1 (1-ξ ) + v ξ + c(1-ξ ) ξ, ξ =(x-x 1 h 1/ = x - x 1, )/h 1/, c any constant, whch may be defned after ntegratng of ths formulae. If to take nto account nterpolaton condtons s(x 1 )=u 1, s(x )=u, next expresson may be receved: 3 3 ( ) = 1 ( ξ + ξ ) + ( ξ ξ ) s x u 1 3 u 3 ( ) 1 + ξ(1 ξ) v 1 1 ξ v ξ h.. (1) Ths expresson garantues contnuty of the splne frst dervatve. The splne second dervatve may be wrtten as follows ( ) s x = (1ξ 6)(u u ) h 1 1 ( ξ ) ( ξ ) + v v 6 h 1 1 If to use ths formulae and the smlar formulae for the neghbourng subnterval [х,x + 1 ], the second dervatve dscontnuty jump s''(x ) may be receved: x Table 1. Data [11]. u We observe ntervals where splne ncreases, and ntervals where splne decreases, whle data ncrease everywhere. The cubc splne scheme, preservng monotoncty of monotonc data, s descrbed below. ( x ) ( x 0) ( x 0) s = s + s = 6(u u ) h + 6(u u ) h (v + v ) h (v + v ) h () Next desgnatons are used below h = h 1/ h +1/ /( h 1/ +h +1/ ), δ +1/ = = (u + 1 -u )/h +1/, Z +1/ = δ +1/ / h +1/. The contnuty requrement for the splne second dervatve s''(x ) =0 leads to the equaton, relatng three consecutve values of the splne frst dervatve v 1, v, v + 1 v 1 /h 1/ +4v /h +v + 1 /h +1/ =3(Z +1/ : +Z 1/ (3) ). (4) If to assume zero values of the splne second dervatve at nterval ends, next relatons may be receved v 0 +v 1 =3δ 1 /, v I +v I 1 =3δ I 1/. (5) So, a closed lnearly ndependent system of equatons for calculatons of splne dervatve values v, 0 I, s derved for nonunform knot spacng. It s known that cubc splne may produce undesrable oscllatons near ponts of nterpolated functon dscontnutes or dscontnutes of the frst dervatve. For example, the classcal cubc splne nterpolant to data [11], reported n table 1, s shown n fgure 1. Fg. 1. Classcal cubc splne, data [11]. 3. Monotonsty Preservng Cubc Splne The problem of undesrable oscllatons producng exsts both n the feld of nterpolaton technques and n the feld of hgh order methods for compressble flows calculatons. There s a popular approach, started n [14], to overcome ths producng of undesrable oscllatons, based on lmtaton of some terms n fnte dfference equatons for solvng gasdynamc problems. A smlar approach was suggested for the splne constructng n [13]. To defne monotonsty preservng splne we need n the functon-delmtator: Delm(b,y)=max[-b, mn(b, y)], b>0. (6) Here y an argument, whch should be lmted, b a parameter delmtator. Let us defne the dscrete functon Z not only n ponts x +1/ =(x +x + 1 )/ (see form. (3)), but also n ponts x : Z = Delm[ Z +1/, Z 1/ ]. (7)

3 164 Vladmr Ivanovch Pnchukov: Shape Preservng Thrd and Ffth Degrees Polynomal Splnes Next resultng equatons for splne frst dervatve values were suggested n [13]: v 1 p /h 1/ +(3-p )v / h +v + 1 p /h +1/ = R, (8) R =3Delm[p ( Z +1/ + Z 1/ ), (Z +1/ +Z 1/ )], (9) p = mn[1, ( Z )/( Z +1/ + Z 1/ )], (10) where the dscrete functon Z s calculated by formulas (3), (7). If to take nto account end condtons (5) and the defnton (3) for parameters h, t s easy to establsh, that the coeffcent matrx of the system (5),(8) s dagonally domnant and thus nvertble. Therefore a ungue soluton of ths system exsts. It may be shown [13], that these equatons produce monotonsty preservng splne. Namely, next theorem s proved n [13]: Theorem 1. If data u, 0 I, are decreasng, δ +1/ 0, (ncreasng, δ +1/ 0 ), then cubc splne, defned on the bass of formulas (5),(8)-(10), s a decreasng, s (x) 0, (ncreasng, s (x) 0) contnues functon wth the contnues frst dervatve. To llustrate monotonsty preservaton by ths splne, we consder agan data [11], reported n table 1. Fgure shows splne (5), (8)-(10), whch s ncreasng n agreement wth theorem 1. Fg. 3. The composte functon. Fg. 4 shows splne (8)-(10) appled to the composte functon. It may be observed that extrems of data and splne dffer and ntervals of splne negatve values exst. The splne scheme preservng the data postvty s consdered n secton 5. Fg. 4. Monotonsty preservng splne, the composte functon. 4. Ffth Degree Polynomal С² Splne Fg.. Monotonsty preservng splne, data [11]. To study the monotonc splne applcaton to data wthout sngle drecton of ncreasng or decreasng the composte functon u=y 0 (x), 0 x 1, defned by formulas (11)-(13), s consdered. Ths functon ncludes trangular, rectangular and parabolc regons (see fg.3). y 0 =mn[(x-0.1)/0.1, (0.3-x)/0.1], 0.1 x 0.3, (11) y 0 = 1, 0.4 x 0.6, (1) y 0 =[1-(x-0.8) /0.01] 1 /, 0.7 x 9. (13) It should be noted, that f p 1, then equatons (4) and (8) are dentcal, snce t s easy to see, that Delm[b, x] x when x b, consequently, the rght sde of the equaton (8) may be transformed to the form R =3(Z +1/ +Z 1/ ) the equaton (4) rght sde. Parameters p become less unt near ponts of dscontnutes of the nterpolated functon or t s frst dervatve. In ths case the equaton (4), whch s equvalent to the contnuty condton for the splne second dervatve, s not satsfed and we have С¹ nterpolaton. We have C² nterpolaton n ntervals, where the relaton p =1 s true, whch s equvalent to the nequalty Z /( Z +1/ + Z 1/ ) 1. If to take nto account formulas (3) and (7), ths nequalty may be rewrtten as follows mn[ δ +1/, δ 1/ ] ( h 1/ +h +1/ ( δ +1/ / h +1/ + δ 1/ / h 1/ ) h 1/ ) h +1/.

4 Amercan Journal of Appled Mathematcs 014; (5): If to consder the case δ +1/ δ 1/, h 1/ =h +1/, the last nequalty may be transformed to the form δ +1/ δ 1/ ( -1) 1.8 δ 1/ Ths nequalty s true for good nterpolated functons nearly everywhere (namely, outsde vsntes of extrema ponts u =0), thus splne provdes С² nterpolaton nearly everywhere. In a total case, when the nterpolated functon has dscontnutes or dscontnutes of the frst dervatve, splne s C¹ contnues. To ncrease the splne smoothness and to provde contnuty of the splne second dervatve, a ffth degree addtonal term s used here. Let next formulae s used nstead of the formulae (1): S (x)=s(x)+ξ (1-ξ ) [ q ξ - q 1 (1-ξ )]r 1/, (14) sgnfcant oscllatons appearng. The wrtten above choce of parameters r +1/ leads to formulas q =0.5 s''(x ) /( Z +1/ + Z 1/ ), 0<<I, (16) q 0 =0, q I =0. (17) So, we have receved splne, whch has the С² smoothness smlarly to classcal cubc splne. But recent splne does not produce undesrable oscllatons. It s emprcal fact and monotonsty preservng s not proved. Fg. 4 shows thrd and ffth degree splnes appled to step functon data: u(0.0)=0.0. u(1.0)=0.0, u(3.0)=1.0, u(4.0)=1.0. Both splnes are monotonc, C²-splne looks more pleasant, to our opnon. where s(x) s splne (8)-(10). Ths expresson contans two famles of parameters q, 0 I, and r 1/, 0< I. Parameters q are chosen to provde contnuty of the splne second dervatve, parameters r 1/ are chosen to mnmze undesrable oscllatons, whch may be resulted from ffth degree term addton. The formulae (14) leads to the expresson S''(x -0)=s''(x -0)+q r 1/ /h 1/. Smlarly, f to consder the subnterval [x,x + 1 ], next expresson may be derved S''(x +0)=s''(x +0)- q r +1/ h +1/ If to subtract the prevous relaton from the last one, the second dervatve contnuty condton leads to the formulae S''(x +0)-S''(x -0)=0 q =0.5 s''(x ) /( r +1/ /h +1/ + r 1/ /h 1/ ), where s''(x ) s gven by the expresson (). Ths formulae may be dealt only at nner knots, 0<<I. Zero values of parameters q are used at end knots =0 and =I. Tral calculatons show, that the choce r +1/ = Z +1/ h +1/ = u + 1 -u (15) provdes absence of undesrable oscllatons. Other choces, for example, the choce r +1/ = h +1/, may provde Fg. 5. Monotonsty preservng cubc splne and C² ffth degree splne, step functon nterpolaton. C² ffth degree splne s appled also to data [11] (see table 1) and to Rado chemcal data (see table ). Both data are monotonc. Monotonc curves are observed n both cases. Thrd degree and ffth degree splnes are smlar, dfferences between these splnes are located near break ponts of nterpolated functons (see, for example, fg. 5). 5. Postvty Preservng Cubc Splne To present ths splne, we need n the more complcated defnton of parameters p, then the defnton (10): If Z +1/ Z 1/ 0 then p =0 else (18) p =mn[1, ( Z )/( Z +1/ + Z 1/ )], These parameters are used n equatons (8)-(9). To study property of splne (8)-(9),(18) we should note that f Z +1/ Z 1/ 0, then p =0 and hence the equaton (8) yelds the relaton v =0, whch may be consdered as a new end condton. So, to study postvty preservaton of new splne (8),(9),(18) we should nvestgate autonomously splne propertes for each nterval of monotonsty of dscrete data. Let the left end of the monotonsty nterval s placed at the knot x j, the rght end at the knot x J. End condtons of three

5 166 Vladmr Ivanovch Pnchukov: Shape Preservng Thrd and Ffth Degrees Polynomal Splnes type are possble for ths nterval, due to the locaton of ths nterval n the global nterval [0,a]: v j +v j + 1 = 3δ j +1/, v J =0, f j=0 and J<I, (19) v j =0, v J +v =3δ /, f j>0 and J=I, (0) v j =0, v J =0, f j>0 and J<I. (1) Тheorem. Splnes, defned by formulars (8),(9),(18),(19), (8),(9),(18),(0) or (8),(9),(18),(1) are contnues functons wth contnues frst dervatves, and f dscrete data are decreasng, δ +1/ 0, (ncreasng, δ +1/ 0) throughout the nterval [x j,x J ], 0=x 0 x j, x J x I =a, then splnes are decreasng, s (x) 0, (ncreasng, s (x) 0) throughout ths nterval. The contnuty of splnes and ts frst dervatve s followed from the formulae (1). To nvestgate monotonsty preservaton by new splnes we should consder two cases. In the frst case extremas are at neghbourng knots, x and x J. If to dfferentate the formulae (1) and to substtute to the resultng formulae relatons v =v J =0, v 1 receve s (ξ )=6ξ (1-ξ )(u J -u )/h / =v =0, we may, 0 ξ 1, for the mentoned subnterval. Snce ξ (1- ξ ) 0, expressons s (ξ ) and (u J -u )/h / have the same sgn, thus splne s ncreasng (decreasng) f u u J (u u J ) through the subnterval [x, x J ]. Wnen the monotonsty nterval contans three or more knots, we should nvestgate the equaton (8), added by relatons (9), (18) and end condtons (19), or (0), or (1). Proof of monotonsty preservaton by splne s nearly the same for all end condtons. The proof plan s next. We establsh some lmtatons on values of the splne frst dervatve at nterpolaton knots. These lmtatons allow to prove, that the splne frst dervatve has necessary sgn at every pont of the monotonsty nterval. Whch lmtatons should be done? Of course, a frst lmtaton s postvty of splne frst dervatve values at nterpolaton knots for ncreasng data and ther negatvty for decreasng data. A second lmtaton s a lmtaton on modul of these values. A convergng teraton s defned for solvng equatons (8)-(9). We prove that f necessary lmtatons are defned by startng frst dervatve values, these lmtatons are satsfed for all teraton numbers. For example, let be the case of end condtons (0) and ncreasng data u. We suppose strcly ncreasng data, u -u 1 >0 becouse f for some knot u -u 1 =0, then t s possble to consder a less monotonsty nterval wth strcly ncreasng data. It s easy to see, that f data u are strcly ncreasng, that s to say Z 1/ >0, Z +1/ >0, then Z +1/ =Z +1/, Z 1/ =Z 1/, besde of t, the functon Delm n formulas (7), (9), (18) has postve arguments and thus may be transformed, Delm(a,b) =mn(a,b). As a result, the formulae (18) may be wrtten as follows p = mn[1, mn(z +1/,Z 1/ )/(Z +1/ +Z 1/ )]. If to substtute ths expresson to the formulae (9) and to omt agan modulus sgns, next expresson may be receved R =3mn{mn[ (Z +1/ + Z 1/ ), mn(z +1/,Z 1/ ) ],( Z +1/ +Z 1/ )}, Snce a consecutve applcaton of the functon mn s equvalent to a sngle applcaton of ths functon to a lst of all arguments, next equaton may be derved nstead of the equaton (8) v 1 p /h 1/ +(3-p )v / h +v + 1 p /h +1/ = =3mn( Z 1/, Z 1/ + Z +1/, Z +1/ ). If to substtute parameters Z to ths equaton from the defnton (3) and f to place teraton ndexes k and k-1 to ths equaton and to end relatons (0), the Yacoby teraton may be defned by equatons v k (3-p )/h = 3mn( δ 1/ /h, δ 1/ /h 1/ +δ +1/ /h +1/ -v 1 p /h 1/ -v + 1 p /h +1/, δ +1/ /h )-, v k j = v J =0, v k J =3δ / -v 1 () I. (3) We have mentoned above that ths teraton should preserve some lmtatons. Let next relatons are true for the teraton number k-1: 0 v 1 k 1.5 mn(δ 1/ 0 v 1, δ +1/ k J 1.5 δ 1/ ), j+1 J-1, (4) J. (5) Zero startng values of the frst dervatve may be used n =0, j J, whch satsfes to relatons ths teraton v 0 (4)-(5). We should prove that these relatons are true for the teraton number k too: 0 v k 1.5 mn(δ 1/, δ +1/ ), j+1 J-1, (6) 0 v k J 1.5 δ /. (7) To prove these relatons we need n next auxlary nequaltes

6 Amercan Journal of Appled Mathematcs 014; (5): k v δ +1/ k v δ 1/, (8). (9) Really, f the ndex relaton +1 J s true, then nequalty (8) follows from the relaton (4) and obvous property of the functon mn, mn(a,b) a, mn(a,b) b, else ths nequalty s a part of the double nequalty (5). Smlarly, f the ndex relaton -1 j s true, then the nequalty (9) follows from the relaton (4) and mentoned above property of the functon mn, else ths nequalty follows from the frst relaton (3), v 1 =0. Inequaltes (8)-(9) are proved. k J If to replace dscrete parameters δ 1/, δ +1/ n the relaton () by ther least value mn(δ 1/, δ +1/ ) and f to take nto account the defnton h = =h 1/ h +1/ receved /( h 1/ +h +1/ ), next relaton may be v k (3-p )/h mn(δ 1/, δ +1/ ) 3mn( /h,/h, /h )- -v 1 p /h 1/ -v + 1 p /h +1/, If to use the obvous relaton mn( /h, /h, /h )=/h and f to replace subtrahends by ther maxmum values, whch are gven by nequaltes (8),(9), next relaton may be derved v k (3-p )/h mn(δ 1/, δ +1/ )6/h -1.5 (δ 1/ /h 1/ + δ +1/ /h +1/ )p. The defnton (18) yealds the relaton p mn(δ +1/, δ 1/ ) z /(δ +1/ +δ 1/ /h 1/ /h +1/ + ). If to ncrease the subtrahend n the last formulae by usage of ths relaton, we have δ +1/ v k (3-p )/h mn(δ 1/, )6/h -3 mn(δ +1/, δ 1/ )/h =0. Thus, left parts of relatons (6), namely, postvty of parameters v s proved for nner knots. We should prove t s postvty for the end knot J, that s to say, we should prove left part of the relaton (7). If to ncrease the subtrahend n the second formulae (3) by usage of the relaton (5), we have v k J 3 δ / δ 1/ J. Snce the rght sde of ths relaton s postve, new value of the splne dervatve v s postve at the end knot =J. We have the steady condton v =0 at the end knot =j. Thus, new values of the splne dervatve v are postve at all knots. To prove rght sdes of nequaltes (6)-(7), we should take away postve subtrahend and take away the second argument of the functon mn n the relaton (). Snce both operatons may only ncrease ths expresson, we have v k (3-p )/h 3 mn( δ 1/ /h, δ +1/ /h ). If to replace the parameter p by the most possble value 1 at the left sde of ths relaton, we may only decrease ths sde. As a result we have v k 1.5 mn(δ 1/, δ +1/ ). So, the rght nequalty (6) s true at nner knots. If to take away the postve subtrahend at the rght sde of the second equalty (3), we have v k J 1.5 δ /. Hence the rght nequalty (7), contanng the coeffcent 1.5 nstead of the coeffcent 1.5 here, s true too. So, the consdered teraton preserves necessary evaluatons. Ths teraton may be wrtten by the formulae V k =AV +B, where V s a vector-column wth components v, A={ a l }, j J, J l J, s a matrx, whch has only two dagonals, where elements dffer from zero. They are placed near the man dagonal. The dagonal above the man dagonal contans elements 0, q j + 1,...,q, where q =-p h /[h +1/ (3-p )]. The dagonal under the man dagonal contans elements r j + 1,,r, -1/, where r = -p h /[h 1/ (3-p )]. It s easy to establsh, that the matrx norm may be wrtten as A = max J j l = J l j = a l =max (0,w j + 1,...,w,1/), where w = r + q =h [1/ h 1/ +1/h +1/ ]/[ (3-p )]=p /(3-p ). Snce the most possble value of w s ½ (ths value s acheved when p =1), then A ½<1, hence ths teraton converges. The proof of convergence and neseccary lmtatons preservaton s nearly the same for splnes wth end condtons (19) and (1), but the matrx norm may be wrtten as A = max (1/,w j + 1,...,w,0) 1/ and A = max (0,w j + 1,...,w,0) 1/, correspondngly. Snce evaluatons (6)-(7) for splne dervatve values are

7 168 Vladmr Ivanovch Pnchukov: Shape Preservng Thrd and Ffth Degrees Polynomal Splnes true for each teraton number k, these evaluatons are true for the teraton consequence lmt. Now we should prove that nequaltes (6)-(7) provde necessary sgn of the splne dervatve at any pont. Agan we consder the case of ncreasng data, δ 1/ 0, δ +1/ 0. The dfferentaton of the formulae (1) gves the result that s =6ξ (1-ξ ) δ 1/ +v 1 (1-ξ )(1-3ξ )+ v ξ (3ξ -), 0 ξ 1. After some algebra ths relaton may be rewrtten as follows s = ξ (1-ξ )[6 δ 1/ -.5 v 1 ξ -.5 v (1-ξ )]+v 1 (1-ξ )(1-3ξ +.5ξ )+ +v ξ (.5ξ -ξ +.5). All three terms n ths expresson are postve. Necessary sgn of last two terms follows from postvty of polynomals, whch are factors of these terms. To show postvty of the frst term, most dangerous values of parameters v 1, v should be consdered. These values are gven by next nequaltes, whch are resulted from relatons (4)-(5) (the proof of these nequaltes s smlar to the proof of nequaltes (8)-(9) and so s omtted): fg.3) and to Rado chemcal data, reported n table. Splnes are monotonc n each nterval of data monotonsty, thus data postvty s preserved n agreement wth the proved above theorem. Rado chemcal data nclude lttle scale regon near the start of coordnate system, contanng four knots and shown n the addtonal fragment of fg. 7. x Table. Rado chemcal data u e e v δ 1/, v 1.5 δ 1/. It should be noted, that f -1=j, then v 1 =0 (see form.(0)), but ths case s not consdered here snce more dangerous case, whch s gven by nequaltes, wrtten above, s nvestgated. If to multply the frst nequalty by the factor.5 ξ, the second nequalty by the factor.5(1-ξ ) and to sum up, the resultng nequalty may be wrtten as follows.5 v 1 ξ +.5 v (1-ξ ) δ 1/ 6δ 1/. The frst term s thus postve too. So, the splne dervatve s postve at each pont n the ncreasng data nterval. Negatvty of the splne dervatve n the decreasng data nterval may be proved smlarly. Theorem s proved. Fg. 7. Postvty preservng C¹ splne, rado chemcal data. modfcaton of postvty preservng splne s also consdered wth usage of formulas (14)-(16). It s observed, that ffth degree splne scheme produces postve curves when ths scheme s appled to postve rado chemcal data and to the composte functon (see form. (11)-(13)). These curves are closed to those produced by postvty preservng C¹ splne (fgs. 6,7) and so are omtted here. C 6. Concluson Fg. 6. Postvty preservng C¹ splne, the composte functon (see fg. 3) Fgs. 6,7 shows results of a postvty preservng splne scheme applcaton to the composte functon (form. (11)-(13), We consder postvty and monotonsty preservng splnes, whch are lnear and thus do not requre teraton to calculate splne coeffcents. For comparson, ratonal splnes [8-10] requre solvng systems of nonlnear equatons or lnear equatons contanng shape parameters dependng globally on unknowns, thus nonlnear systems are solved n both cases. Cubc splnes, consdered here, are proved to preserve monotonsty or postvty of monotonc or postve data. These splnes may be consdered as classcal cubc splne C¹ modfcatons. Ffth degree polynomal splnes are developed to provde contnuty of the second dervatve. These splnes are constructed as a sum of base cubc C¹ splnes mentoned

8 Amercan Journal of Appled Mathematcs 014; (5): above and ffth degree terms, whch are chosen to get C smoothness. These splnes and base cubc splnes appled to monotonc or postve data are observed to have the same shape preservaton propertes. References [1] M.Z. Hussan, M. Sarfaz, M. Hussan, Scentfc Data Vsualzaton wth Shape Preservng C¹ Ratonal Cubc Interpolaton, European J. of Pure and Appled Mathematcs, Vol. 3, N., 010, [] S. Butt, and K. W. Brodle, Preservng postvty usng pecewse cubc nterpolaton, Computers and Graphcs, 17(1), 1993, [3] D. Kahaner, C. Moler, and S. Nash, Numercal Methods and Software, Prentce Hall, Englewood Clffs, NJ, 1989, 7. [4] M. Abbas, J. M. Al, A. A. Majd, Postvty Preservng Interpolaton of Postve Data by Cubc Trgonometrc Splne. MATEMATIKA, Vol. 7, N. 1, 011, [5] X-An. Han, Y M. Chen, and X. Huang, The cubc trgonometrc Bezer curve wth two shape parameters. Appled Math. Letters : [6] L. L. Schumaker, Constructve aspects of dscrete polynomal splne functons. Approxmaton Theory. G.G. Lorentz (ed.). Akademc Press. New York. 1973, [7] S. de Boor, Splnes as lnear combnatons of B-splnes: A survey. Approxmaton Theory II. G.G. Lorentz, C.K. Chu, and L.L. Schumaker (Eds.) Academc Press. New York. 1976, [8] Q. Sun, F. Bao, Q. Duan, Shape-preservng Weghted Ratonal Cubc Interpolaton, J. of Computatonal Informaton Systems. 8:18, 01, [9] M. Shrvastava, J. Joseph, C ratonal cubc splne nvolvng tensor parameters. Proc. Indan Acad. Sc. (Math. Sc.), Vol. 110, N. 3, 000, [10] R. Delbourgo, J. A. Gregory, C ratonal quadratc splne nterpolaton to monotonc data. IMA J. Numer. Anal. Vol. 5, 1983, [11] H. Akma, A new method of nterpolaton and smooth curve fttng based on local procedures, J Assoc. Comput. Machnery. Vol. 17, 1970, [1] N. S. Sapds, P. D. Kakls, An algorthm for constructng convexty and monotoncty pre- servng splnes n tensons // Comput. Aded Geometrc Desgn. Vol. 5, 1988, [13] V. I. Pnchukov, Monotonc global cubc splne, J. of Comput. & Mathem. Phys., Vol. 41, N., 001, (Russan) [14] A. A. Harten, A Hgh resoluton scheme for the computaton of weak solutons of hyperbolc conservaton laws // J. Comput. Phys. Vol. 49, N. 3, 1983,