Convergence of Quintic Spline Interpolation

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1 Inrnaonal Journal o ompur Applcaons Volum 7 No., Aprl onvrgnc o Qunc Spln Inrpolaon Y.P. Dub Dparmn O Mamacs, L.N..T. Jabalpur 8 Anl Sukla Dparmn O Mamacs Gan Ganga ollg O Tcnog, Jabalpur 8 ASTRAT In s papr, w av nvsga sanc, unqunss and rror bound o qunc spln nrpolaon. words : onvrgnc, Qunc spln, Inrpolaon, Error ound, Dcn.. INTRODUTION Pcws Quarc, Qunc and gr dgr spln ar popular or smoo and bs appromaon s Dboor [. Inrpolaon b lowr dgr spln ar wdl usd n mod o Pcws Polnomal appromaon o rprsn a non analc uncon. Inrpolaon b Qunc and gr dgr spln, mamum rror bwn a uncon and s nrpolan can b conrolld b ms spacng and suc uncon av no cornr a on o wo pcs and ror no mor daa ar rqurd an lowr ordr mod o g dsr accurac. Tror, qunc and gr dgr spln ar usul an lowr dgr spln or bs appromaon S [. In drcon o mor gr dgr spln Janzon and Hung [9 av oband opmal rror bounds or quarc and qunc nrpolaor splns s also Howll and Vrma [ Wong and Agrawal [, Gmllng Mlng [ av oband plc rror bounds or qunc and bqunc nrpolaor spln Dub [ as oband snc, unqunss and rror bound or gr dgr spln S also Rana and Dub [8. In prsn papr w sall oban prcs rror sma o dcn qunc spln nrpolaon macng gvn uncon valus a ms pons and s drvav a wo nror pons o nrval.. EXISTENE AND UNIQUENESS onsdr a ms P o [, gvn b = o <,..., < n = suc a + - =, =,,..., n -. For a posv ngr m, l [, dno s o all ral algbrac polnomals o dgr lss an or qual o m. For a uncon s dnd ovr P. w dno rsrcon o s ovr [, + b s. T class R,, P o dcn qunc splns dnd ovr P s gvn b R,,P={s:s [,, s, =,, n }, wrn R*,, P dnos class o all dcn qunc splns R,, P wc sas boundar condons s =, s n = n.. W nroduc ollowng nrpolaor condons: s =,, =,,..., n. s =, =,,..., n s =, =,,..., n. wr = - + / - = - + / - In ac, w sall prov ollowng: THEOREM.. L s, n r ss a unqu dcn qunc spln s n R*,, P wc sass nrpolaor condons. -.. Proo : onsdrng a qunc polnomal Pz on [,, can b asl vrd a Ez = E Q z + E. Q z + E/. Q z + E /. Q z + E O Q z + E. Q z. wr Q z z z z z Q z z z z z Q z [ 7z z 89z 8z Q z [ z 89z z 8z Q z [z 8z z z 7z Q z [ z z z 7z Now wrng = - /, o < <,. ma b prssd n rms o rsrcon s o s on [, + as ollows : s =. Q + +. Q + +. Q + +. Q + s. Q + s +. Q. wc clarl sass condons. -. and s s qunc n [, + or =,..., n-. Now obsrvng ac a s [,, ror, applng connu condon o scond drvav o s n., w g ollowng ssm o quaons :

2 Inrnaonal Journal o ompur Applcaons Volum 7 No., Aprl wr s s s F,,,..., n. F or Snc I ma b sn asl a cocn mar o ssm o quaons. s dagonall domnan and nc nvrbl. Ts compls proo o Torm... ERROR OUNDS In s scon o papr, w sall sma bounds o rror uncon b usng mod o Hall and Mr [ and o s drvavs,.. r = s r - r, r =, or spln nrpolan o Torm. wc ar bs possbl. L s b wc connuousl drnabl qunc spln uncon sasng condons o Torm.. Now consdrng [, and dnong unqu qunc polnomal b L [, wc agrs w gvn unconal valus and drvavs +, +, r and r +, or r=,, w av or [, +, s - s - < s - L [, + L [, -... W now procd o g ponws bounds o bo rms on rg and sd o.. T sma o scond rm can b oband b ollowng a wll known rmandr orm or polnomal nrpolaon o Davs [. L [, - < Wr - --F/9...! and F = = - / ma W n urn ovr anon o g a smlar bounds or s -L [,. From., w av s L[,. Q. Q Tus, s L [, Q. As Q Q 9 9 Tus, applng. n., w av. s L [, ma{, } k wr k = 9 9. L ma s or =, ror. ma b wrn as s L [, k < < n.7 Wr =ma, w now n procd o nd uppr bound or ndd n q..7. Rplacng s b n., w av F = sa.8 In vw o ac a s a lnar unconal wc s zro or polnomal o dgr or lss, w can appl Pano Torm Davs [ o oban [ d....9! Tus rom.9, w g, F [ d....! Furr, can b obsrvd rom.9 a or [ 7 Q..

3 Inrnaonal Journal o ompur Applcaons Volum 7 No., Aprl In ordr o sma ngral o r..s. o., w rwr abov prsson n ollowng smmrc orm abou o g. [ [ [ 9 From abov prsson, ollows a.7 [ d.. ombnng. w., w av F!..7.. Tus rom.8 and., ollows a 7!..7.. Now usng.,.7 along w. n., w av /9! F 9 9! F.. Wr / 9 9 F * sa.. THEOREM.. L s b qunc spln nrpolan o Torm. nrpolang a gvn uncon and [, n, ma /!... Wr, ma dnd b quaon. Also, w av, /!ma...7 Wr = / Furrmor, can b sn asl a n. b mprovd or an quall spacd paron. Inqual.7 s also bs possbl. Also, w av, ma /!...8 wr s posv consan. Equaon. provs. wras nqual.7 s a drc consqunc o..

4 Inrnaonal Journal o ompur Applcaons Volum 7 No., Aprl Now, w urn o s a nqual. s bs possbl n lm. onsdrng = /! and usng auc ormula Davs [, w av L [ /!, /! / / /!...9 Morovr, or uncon undr consdraon.8 ollowng rlaon olds or quall spacd knos,.7!! onsdrng or a momn 9.7! w av rom.,.7 s L[, Q Q!.7 [ 9 9 / }! Now combnng.9 and., w g or.7 9 s! / /... 9 From., s clarl obsrvd a. s bs possbl, provdd w could prov a.7!... In ac. s aand onl n lm. T dcul wll appar n cas o boundar condons,.., n. Howvr, can b sown a as w mov man subnrvals awa rom boundars,.7 /!. For a, w sall appl. nducvl o mov awa rom nd condons. n T rs sp n s drcon s o sabls a or som, =,..., n wc can b sown b conradcor rsul, l < or som =,...n-. Now makng us o.7, w g.7! 9 or som L.H.S. posv and r..s. s ngav. Tus w av a conradcaon. Tus, or =,..., n. Now rom. w can wr.7 9!.7 9! Snc, w g.7,...!9 Now agan usng. n., w av.7 9! 9 =,,..., n- Tus, s clar a a rpad us o. lads us o.7 9! [ /9 / Now, can b sn asl a rg and sd o..7! and nc n lmng cas ! wc vrs proo o.7..! Tus, corrspondng o uncon, and.7 mpl.7 n lm or! quall spacd knos. Ts compls proo o Torm.. REFERENES [ Dboor,., A Praccal Gud o splns, Appld Mamacal Scnc, Vol. 7 Sprngr - Vrlag, Nw York, 979. [ Hall,.A. and Mr, W.W., J. Appromaon Tor, 97, pp. -. [ Howl, G. and Vrma, A.., s Error ond o Quarc Spln Inrpolaon, J. Appron Tor 8 989, 8-7.

5 Inrnaonal Journal o ompur Applcaons Volum 7 No., Aprl [ Davs, P.J., Inrpolaon and appromaon. Nw Yrok, 9. [ Agrawal, R.P. and Wong, P.J.Y. Error Inquals n Polnomal Inrpolaon and r applcaon, umar Acadmc Publsrs 99. [ Gmllng, R.H.J. and Mlng, G. In Inrpolaon b vara Qunc splns o class...onsrucon n or o uncons 87, Ed Sndov.al. 987, -. [7 Rana, S.S., Dub, Y.P. and Gupa, P. onvrgnc o Dcn Qunc Spln Inrpolaon, Journal o Indan Ma Soc. Vol. No. 7 No.--, pp. -8. [8 Rana, S.S. and Dub, Y.P. s Error ound o Qunc Spln Inrpolaon, J. Pur and app. Ma 8 97-, 997. [9 Janzong, W. and Hung, D. Scna Snca 98, -. [ Dub, Y.P. s Error ound o Spln o Dgr s, In. Journal o Mamacs Analss, Vol., pp. -,.

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