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Research Inventy: International Journal Of Engineering And ScienceVol.2, Issue 9 (April 2013), Pp 24-30Issn(e): 2278-4721, Issn(p):2319-6483, Www.Researchinventy.Com24The Deficient C1Quartic Spline Interpolation1,Y.P.Dubey , 2,Anil Shukla1,Department Of Mathematics, L.N.C.T. Jabalpur 482 0032,Department Of Mathematics, Gyan Ganga College Of Technogy, Jabalpur.ABSTRACT: In this paper we have obtained the best error bounds for deficient Quartic Spline interpolationmatching the given functional value at mid points and its derivative with two interior points with boundarycondition.Key Words : Error Bounds, Deficient, Quartic Spline, Interpolation, best error.1. INTRODUCTIONThe most popular choice of reasonably efficient approximating functions still continues to be in favourof Quartic and higher degree (See deBoor [1]). In the study of Piecewise linear functions it is a disadvantagethat we get corners at joints of two linear pieces and therefore to achieve prescribed accuracy more data thanhigher order method are needed. Various aspects of cubic interpolatory splines have been extended number ofauthors, Dubean [3], Rana and Dubey [10]. Morken and Reimers [6]. Kopotum [5] has shown the equivalenceof moduli of smoothness and application of univariate splines.Rana and Dubey [8] generalized result of Garryand Howell [4] for quartic spline interpolation. Rana, Dubey and Gupta [9] have obtained a precise errorestimate concerning quartic spline interpolation matching the given functional value at intermediate pointsbetween successive mesh points and some boundary conditions.II. EXISTENCE AND UNIQUENESS.Let a mesh on [0, 1] be given by }10{ 10 nxxxP with 1 iii xxh for i=1, 2,.....n.Let Lk denote the set of all algebraic polynomials of degree not greater than K and si is the restrictions of s on],[ 1 ii xx the class S (4,2, P) of deficient quartic spline of deficiency 2 is defined by }.,...,2,1],1,0[:),2,4( 1niforLsCssPS ki where in S* (4,2, P) denotes the class of all deficient quartic spline S (4,2, P) which satisfies the boundaryconditions.)()(),()( 00 nn xfxsxfxs .........(2.1)We introduced following problem.2.1 PROBLEMSuppose f exist over p, there under what restriction on ih there exist a unique spline interpolation),2,4(* PSs , of f which satisfies the interpolating condition.nifsfsfsiiiiii......2,1)()()()()()(Where iiiiiiiii hxhxhx32,21,31 In order to investigate Problem 2.1 we consider a quartic Polynomial q(z) on [0, 1] given by)7()3/2()(31)(21)( 321 PqzPqzPqzq )()1()()0( 54 zPqzPq (2.5)(2.2),(2.3)(2.4)
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The Deficient C1Quartic Spline…25Where 321 144288208647)( zzzzzP 322 5412999247)( zzzzzP 323 54873637)( zzzzzP 7727186741971681)(4324zzzzzP 325 721024147)( zzzzzP We are now set to answer problem (2.1) in following theorem.THEOREM 2.1 :If 1ih ih there exists unique deficient quartic spline S in S*(4,2,P) which satisfies theinterpolatory conditions (2.2) - (2.4) and boundary condition (2.1).2.2 PROOF OF THE THEOREMLet 10,1 thxxtiithen in view of condition (2.1) - (2.4).We now express (2.5) in terms of restriction si of s to ],[ 1 ii xx as follows :-)()()()()()()( 321 tPfhtPfhtPfxs iiiiii )()()()( 541 tPxstPxs ii (2.6)which clearly satisfies (2.1) - (2.4) and si(x) is a quartic in ],[ 1 ii xx for i=1.....nSince S is first time continuously differentiable on [0, 1]. Therefore applying continuity condition of firstderivative, we get-176 1111 4)16860( iiiiiii hsshhhs iF say) (2.7)Where ))()((64 11 iiiii fhfhF ])(3)(24[ 11 iiii ffhh )(24)(3[ 11 iiii ffhh In order to prove theorem 2.1 we shall show that system of equation (2.7) has unique set of solution.Since 0116172 1 ii hh if ii hh 1 .Therefore, the coefficient matrix of the system of equations (2.7) is diagonally dominant and hence invertible,this complete the proof of theorem 2.1.III. ERROR BOUNDSFollowing method of Hall and Meyer [2], in this sections, we shall estimate the bounds of errorfunction )()()( xsxfxe for the spline interpolant of theorem 2.1 which are best possible. Let s(x) be thefirst time continuous differentiable quartic spline function satisfying the conditions of theorem 2.1. Nonconsidering ]1,0[5Cf and writing Mi [f, x] for the unique quartic which agree with)(),(),(),( 1iiii xffff and )( 1ixf , we see that for ii xxx ,1 and we have)(],[],[)()()()( xsxfMxfMxfxsxfxe ii (3.1)
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The Deficient C1Quartic Spline…26In order to obtain the bounds of e(x), are proceed to get pointwise bounds of the both the terms on the righthand side of (3.1). The estimate of the first term can be obtained by following a well known theorem of Cauchy[ 7 ] i.e. FttttthxfxfM ii )32)(31)(21)(1(!5)(,5Where11iihxxt , and )(max )5(10xfUx (3.2)To get the bounds of ],[)( xfMxs il , we have from (2.6).)]()()()([],[)( 541 tPxetPxexfMxS iiil (3.3)Thus )()()()(],[)( 541 tPxetPxexfMxs iiii (3.4)Let the )(max0inixeexist for i=jThen (3.4) may be written as |)(||)(||)(|],[)( 54 tPtPxexfMxs jii (3.5)Now 254 361477)1(21)()( tttttPtP 236334 ttt =k(t) Say (3.6))()(],[)( tkxexfMxs jii (3.7)Now we proceed to obtain bound of )( jxeReplacing )()( ii xebyxS in (2.7),We have1111 4)16860(176 iiiiiii heehheh iiiiii fhhfhF )16860(176 11 )(4 11 fEhf ii Say where iF define in (2.7)(3.8)In view of that E(f) is a linear functional which is zero for Polynomial of degree 4 or less, we can apply Peanotheorem [7] to obtain 1145)(!4)()(jjxxt dyyxEyffE (3.9)Thus from (3.9) we have11])(|!41)( 4jjxxt dyyxEFfE (3.10)Further it can be observed from (3.9) that for 11 ii xyx4)[( tyxE
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The Deficient C1Quartic Spline…27])(12)(96[)()([643314141 yyhhyhyh iiiiiiii 413131 )()16860(])(48)(12[ yxhhyyhh iiiiiii 411 )(4 yxh ii (3.11)Rewriting the above expression in the following symmetric form about jx , we get41 ][4 yhxh iii 1 ii xy 434135)()(4 iiiii hyxhyxhii y 341 )(26)(15[4 yxhyxh iiii ]38)(8)(24 4322iiiii hhyxhyx ii y ])(7)(15[4 341 yxhyxh iiii ii yx 314)(7)(15[4 yxhyxh iiii ii xy1212314)(24)(26)(15[4 iiiiii hyxyxhyxh]38)(8 4131 iii hhyx11 ii y ]35)()[(4 41314 iiiii hyxhyxh11 ii y 41][(4 iii hyxh11 ii yx (3.12)Thus it is clear from above expression that ])( 4 yxE is non-negative for11 ii xyxTherefore it follows that
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The Deficient C1Quartic Spline…28 1144114587.18)(iixxiiii hhhhdyyxE (3.13) )!5(87.18)(4411 iiii hhhhfE (3.14)Thus from (3.8) and (3.14), it follow that - Fhhhhhhxeiiiiiii)116172(!587.18)(14411(3.15)Now using (3.2), (3.7) along with (3.15) in (3.1), we have FtkhFttttthxe )()!5(5687.18)32()31()21()1()!5(18)(55 (3.16))(!5)(5tcFhxe (3.17)Where18|32||31|)1([21)(ttttttc )36334()361477()1(5687.18 22tttttt ]= k*(t) (say)Thus we prove the following theorem.3.1 THEOREM : Let s(x) be the quartic spline interpolation of theorem 2.1 interpolatinga given function and ]1,0[)5(Cf then!5)(*)(|5htkxe 10maxn|)(| 5xf (3.18)Also, we have)(max!5|)(| 510511xfhKxeni (3.19)Where5687.181 KFurther more, it can be seen easily that k* (t) in (3.18 ) can not be improved for an equally spacedpartition. In equality (3.19) is also best possible. Also equation (3.16) proves (3.18) whereas (3.19) is directconsequence of (3.15).Now, we turn to see that inequality (3.19) is best possible in limit case.Considering!5)(5xxf and using Cauchy formula [7], we have)21()32()31()1(!5!5,!5555ttttthhxtMi (3.20)
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The Deficient C1Quartic Spline…29Moreover, for the function under consideration (3.8) the following relations holds for equally spaced knots.11554228176!587.18)!5( iii eeehxE (3.21)Consider for a momentiii ehee 151)!5(5687.18(3.22)we have from (3.5).],[)( xfMxs i (3.23)))()((!55687.18545tQtQh)1441587()21(!55687.18 25ttth )1441587)(21(!55687.18)()( 25ttthxfxs )}31)(32)(21)(1(!55 ttttth)1441587)(21)(5687.18[(!525ttthi )]31()32()21()1( ttttt (3.24)from (3.24 ), it is clear that (3.16) is best possible. Provided we could prove that)!5(5687.18 511heee iii (3.25)In fact (3.25) is attained only in the limit. The difficulty will appear in case of boundary condition i.e.)()( 0 nxexe = 0. However it can be shown that as we move many subinterval away from the boundaries.!55687.18)(51hxe For that we shall apply (3.21) inductively to move away from the end conditions 0)()( 0 nxexe .The first step in this direction is to establish that 0)( ixe for some i, 1=1, 2,....n which can be shown bycontradictory result. Let 0)( ixe for some i=1,2,.....n-1.Now we rewrite equation (3.8)!587.184228176511heee iii (3.26 ) 115422817621)(max!55687.18 iiii eeexehSince R.H.S is negative!587.185h561 (3.27)This is a contradiction.
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The Deficient C1Quartic Spline…30Hence 1,....,2,1,0)( niforxe iNow from (3.26), we can write1154176!587.18228 iii eehe!522887.18 5hei Again using (3.27) in (3.21), we get2281721!587.18228 5heiRepeated use of (3.21), we get .......2281722281721)!5(22887.1825hei (3.28)Now it can easily see that the right hand side of (3.28) tending to5)!5(5687.18h!55687.18)(5hxe i (3.29 )which verifies the proof of (3.19).Thus, corresponding to the function!5)(5xxf , (3.28) and (3.29) imply!55687.18)(5hxe i in the limit forequally spaced knots this completes the proof of theorem 3.1.REFERENCES[1] Carl Deborr, A Practical Guide to Springer Applied Mathematical Sciences, Vol. 27, Spoinger - Verlag, New York, 1979.[2] C.A. Hall and W.W. Meyer, J. Approx.theory 16(1976), 105-22.[3] Dubean and J. Savoier. Explicit Error bounds for spline interpolation on a uniform partition, J. approx. theorem 82 (1995), 1-14.[4] Gary Howell and A.K. Verma, Best error bounds for quartic spline interpolation, J. Approx. theory 58 (1989), 58-67.[5] K.A. Kopotun, Univairate Splines equivalence of moduli of smoothness and application, Mathematics of Computation, 76(2007), 930-946.[6] K. Marken and M. Raimers. An unconditionally convergents method for compecting zeros of splines and polynomials,Mathematics of Computation 76 (2007), 845-866.[7] P.J. Davis, Interpolation and Approximation Blaisedell New York, 1961.[8] 8 S.S. Rana and Y.P. Dubey, Best error bounds of deficient quartic spline interpolation. Indian, J. Pur.Appl.Maths; 30(1999), 385-393.[9] S.S. Rana, Y.P. Dubey and P. Gupta. Best error bounds of Deficient c quartic spline interpolation International Journal of App.Mathematical Analysis and Application Jan.-Jun 2011 Vol. No. 1, pp. 169-177.[10] S.S. Rana and Y.P. Dubey, Local Behaviour of the deficient discrete cubic spline Interpolation, J. Approx. theory 86(1996), 120-127.
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