The Deficient Discrete Quartic Spline Over Uniform Mesh

Discrete splines have been introduced by Mangsarian and Schumaker [7] in connection with certain studied of minimization problem involving differences. Discrete cubic splines which interpolate given functional values at one points lying in each mesh interval of a uniform mesh have been studied in [2]. The case of these points coincide with the mesh points of a non uniform mesh was studied earlier by Lyche [5], [6]. To compute non-linear splines interactively Malecolm [3] used discrete splines. Mangasarian and Schumaker [8] used discrete splines for best summation formula. For some different constructive aspects of discrete splines, we refer to Schumaker [10], Astor and Duris [1], Jia [4] and Rana and Dubey [9]. In this paper we have obtained existence, uniqueness and convergence properties of dificient quartic spline interpolation over uniform mesh which matches the given functional values at mesh points and mid points with boundary condition of second difference. Let us consider a mesh P on [a, b] which is defined by bxxxPn......0:10 For i=1,2,...n. iP shall denote the length of the mesh interval ],[1iixx, P is said to be a uniform mesh if iP is constant for all i. Throughout, h will represent a given positive real number. Consider a real function s(x, h) defined over [0, 1] which is such that its restriction is on ],[1iixxis polynomial of degree 4 or less i=1, 2.....n. Then s(x, h) defines a deficient discrete quartic splines with deficiency 1 if ),()(1}{}{hxSDhxSDiijniijn 2,1,0j (1.1) Where the difference operator jnDfor a function f is defined by

hxfhxfxfDxfxfDnn2

)()()(),()(}1{}0{

hxfxfhxfxfDn and 0,),(}{}{)(nmxfDDfDnnmnnmn The class of all deficient discrete quartic splines with deficiency 1 satisfying the boundary condition. ),(),(0}2{0}2{hxfDhxfDnn

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),(),(}2{}2{hxfDhxfDnnnn (1.2) is denoted by R (4, 1, P, h). Now writing 12iiixx, we introduced the following interpolating condition for given function f. ),(),(hxfhxsii 1 = 0, 1,.....n (1.3) ),(),(hfhsii i = 1,2,.....n and pose the following. PROBLEM : Given h > 0, for what restriction on P does there exist a unique s(x, h)  R (4, 1, P, h) which satisfies the condition (1.2) and (1.3)?

Let P(z) be a discrete quartic spline Polynomial on [0, 1], then we can show that

)(2

1)()1()()0()(321zREzREzREzE

 

)()1()()0(524}2{zREDzREDnn (2.1) Where

432221489648)7872(66

1)(zzzhzhAAzR

432222489648)1824(6

1)(zzzhzhAzR

4322239619296)1(966

1)(zzzhzhAzR

43222246)417()25(3)42(6

1)(zzhzhzhAzR

43222256)74(6)12(6

1)(zzhzhzhAzR

Where 





54

Now we are set to answer problem A in the following. Theorem 2.1. For h > 0, there exist a unique deficient discrete quartic spline s(x, h)  R (4,1,P,h) which satisfies conditions (1.2) and (1.3). Proof the Theorem 2.1 : Denoting )(ixx by t, 0< t< 1. We can write (2.1) in the form of restriction ),(hxsiof the quartic spline ),(hxs on ],[1iixx as follows :- )()()()(),(211zRxfzRxfhxsiiii )(),()()(4}2{23zRhxsDPzRfini ),()(1}2{52hxsDzRPin where

22232441)(48))(7872(6[6

1)(PxxhPxxhAPAPzRii

4)(48)(96iixxPxx]

3242))(1824[(6

1)(PxxhAPzRi

])(48)(96)(4843222xxPxxPxxhii

2223243)(96)()1(96[6

1)(PxxhPxxhAPzRii

])(96)(19243iixxPxx

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)25())(412[6

1)(23244hPxxhAPzRi

])(6))(417()(43222iiixxPxxhPxx

2223245)(6))(12[(6

1)(PxxhPxxhAPzRii

])(6))(74(432iixxPxxh Clearly ),(1hxsis a quartic on 1,[iixxfor i=0,1,...n-1 and satisfies (1.2) and (1.3). Now applying continuity of first difference of ),(hxsiat ix, given by (1,1), we get the following system of equation - ),(()74()12(1}2{2222hxsDhhPhiih 2222}2{)417()42(2),(hhPhhxsDiin 22221}2{)74()12(),(hhPhhSDin iF i = 1,2,...n Say Where

))]()(}(192)1(96{)()96)7872{(2122222iiiffhhxfhPh Write iiinmhmhxD)(),(}2{ for all i. Say We can easily see that excess of the absolute value of the coefficient of mi over the sum of the absolute value of the coefficient of 1imand 1imin (2.3) under the condition of theorem 2.1 is given by )8(6)(22hPhdi which is clearly positive. Therefore the coefficient matrix of the system of equation (2.3) is diagonally dominant and hence invertible. Thus the system of equation has unique solution. This complete the proof of theorem 2.1.

For a given h > 0, we introduce the set egeranisjjhRnint: and define a discrete interval as follows : hhR1,01,0 For a function f and three disjoint points 321,,xxx in its dominant, the first and second decided difference are defined by



21

xfxffxx 

and



)(

,,,,

13

fxxfxxfxxx 

respectively. For convenience, the write 2ffor fDh2and )(}2{}2{ihixfDforf and w(f,p) for the modulus of continuity of f. The discrete norm of a function f over the interval [0,1]h is defined by |)(|max||'||1,0xffx without assuming any smoothness condition on the data f, we shall obtain in the following the bounds for the error function over the discrete interval [0,1]h. Theorem 3.1 : Suppose s(x, h) is the discrete quartic spline interpolant of theorem 2.1, then ),(),()(||||1}2{pfwhpkhCei (3.1) ),(),()(|||12}1{pfwhpkhCei (3.2)

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and ),(),(*||)(||2pfwhPkpxe (3.3) where ),(*&),(),,(1hpkhpkhpk are positive constant of p and h Proof of theorem 3.1. To obtain the error estimate (3.1) first we replace )(1hmby iiiniLfhxsDxe}2{}2{}2{),()( Say (3.4) To estimate row max norm of the matrix iLin (3.4), we shall need the following lemma due to Lyche [5]. Lemma 3.1. Let miia1}{ and njjb1}{ be a given sequence of non-negative real numbers such that jiba, then for any real valued function f defined on a discrete interval [0,1]h, we have

where hjkkiyx]1,0[, for relevant values of i, j and k. It may be observed that the right hand side (3.4) is written as |],[],[||)(|1010fjjjfiiiiyybxxaL (3.5)

1222215548)9121bhPhpaba

266aba= 22222)47(1)12(bhhPh 32222377)417()42(ahhPhaba

42224)3024(1bPhpa

and 4010110xyxi 1114111,iixyxxx hxxi120 ixy10 ixx21 hxyi111 ixx30 hxyi30 hxx131 ixx31 iy40 ixy41 iiiixyyxxx515015150,,, hxyxyxxhxxiiii61606160,,, hxyhxxxxiii170171170,, 71y Clearly in (3.5) }{ia and }{jb are sequences of non-negative real numbers such that

),(

7 1 7

1hPNbajjii (Say) Thus applying Lemma 3.1 in (3.5) for i=7=j and k=1, we get ||,(),(||)(||}1{PfwhPNLi (3.6) Now using the equation (2.2) and (3.6) in (3.3) we get ),(),()(||)(||}1{11}2{PfwhPKhCxe (3.7) where K(P,h) is some positive function of P and h.

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We next to proceed to obtain a upper bound for )(xe, replacing )(hmi by }2{iein equation (2.2), we obtain )()()()()([),(1}2{151}2{42fMxetQxetQPhxeiiiii (3.8) Now we write M, (f) in term of divided difference as following : fyyvxxufMjjjiiii1010,[],[)( (3.9) Where

Pu

24322222

Pu

34322222

Pu

Pu and 111101110,,,iiiixyyxxx hxyhnxxxiii202120,, 21y hxyhxxxxiii130131130,, 31y xyxyxxxiii41404140,,, Clearly observing that



4 1

2224

ptt

),(*hpN Say We again applying Lemma 3.1 in (3.9) for i=j=4 and k=1 to see that ),(),(*|)(|}1{PfwhPNfMi (3.10) Thus using (3.7) and (3.11) in (3.8) we get the following : ),(),(*||)(||}1{PfwhPKPxe (3.11) Where K* (P,h) is a positive constant of P and h, this is the inequality (3.3) of Theorem (3.1) . We now proceed to obtain an upper bound of }1{ie, from equation (2.4) we get )()()(),(}1{3}1{21}1{}1{tQftQftQfhxsiiiii )(),()(),(}1{5}2{12}1{4}2{2tQhxsPtQhxsPii (3.12) Thus )(])()([),(6}1{}2{1}1{4}2{2}1{fUtQetQePhxAeiiiii (3.13) Where )()()()(}1{3}1{21}1{1tQftQftQffUiiii ),(6)()(}1{}1{5}2{1}1{4}2{2hxfAtQftQfPiii By using Lemma 3.1 and first and second divided difference in Ui (f) as follows:-



4 1 4 1

(3.14)

)(96)1,3(4848)39,36(222hZZgZhgp

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)(48)3(2430322222hZZhZZp Where 1222221)(96)3(4848)9,12(bhZZhZZgpa 22)30,24(bgpa

3222223)(24)3)(17,4()30,12()4,2(bhZZhZgZggpa

42222224)(243)(7,4(12)1,2(bhZZhZgZhgpa

and 11211131302010,yxxyxxxxii ,,204140110xyyxxyi hxyhxxhxyii303121,, hxyhxxii140141, From equation (3.7) put value of }2{iein (3.13) we

get upper bound of }1{ie. This is inequality (3.2) of theorem 3.1.

1. P.H. Astor and C.S. Daris. Discrete L Splines Number. Math 22 (1974), 393-402. 2. H.P. Dikshit and P.L. Powar. Discrete cubic spline interpolation number Math. 40 (1982) 71-78. 3. M.A. Mallom. NOn linear spline function, Report Stan CS-73-372 Stanford University, 1973. 4. Rong Qingjia, Totall positivity of the discrete spline Collcation matrix J. Approx. Theory 39 (1983) 11-23. 5. T. Lyche. Discrete Cubic spline interpolation Report RRI5, University of Oslo 1975. 6. T. Lyche. Discrete Cubic Spline Interpolation BIT 16 (1976) 281-290. 7. O.L. Mangasarian and L.L. Schumaker. Discrete Spline via Mathamtical Programming SIAMJ. Control 9(1971), 174-183. 8. ------------------Best summation formula and discrete splines. SIAM J. Numeral 10 (1973), 448-459. 9. S.S. Rana and Y.P. Dubey. Local Behaviour of the deficient cubic spline interpolation J. Approx. theory 86 (1996), 120-127. 10. L.L. Schumaker, Constructive aspect of discrete polynomial spline function in approximation theory (G.G. Lorent ed); Academic Press, New York 1973.