INTEGER INTERVAL VALUE OF NEWTON DIVIDED DIFFERENCE AND FORWARD AND BACKWARD INTERPOLATION FORMULA

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1 Volume 8 No. 8, ISSN: (o-lie versio) url: hp:// ijpam.eu INTEGER INTERVAL VALUE OF NEWTON DIVIDED DIFFERENCE AND FORWARD AND BACKWARD INTERPOLATION FORMULA A.Arul dass M.Dhaapal Deparme of mahemaics, M.Kumarasamy college of egieerig, Karur, Idia aruldass34@gmail.com Deparme of mahemaics, M.Kumarasamy college of egieerig, Karur, Idia mdhaapal46@gmail.com ABSTRACT I his paper A ew approaches o solve he approximae soluio of he ewos ierpolaio formula for uequal iervals ad also equal iervals.the soluio ca be used o compue y umerically specified he value of [ x x] ear o x ] i he ierval aalysis mehod.the [ x mehod provide highly precise approximae soluios.umerical illusraio demosrae more effecive ad effcie. Key words: ewo divided differece, forward ad backward ierpolaio formula.. Iroducio The erm ierpolaio ivolve ierpolaig some improbabiliy iformaio from a give se of experieced iformaio. The echique of ierpolaio is geerally used as a high ool i sciece ad egieerig. he mos rouiely ecouered mahemaical models i egieerig ad sciece are i he cosiue of differeial equaios.there are all umerical mehods for differeiaio, iegraio, ad he soluio of ordiary ad parial differeial equaios. These mehods are based o he work of fiie differeces. Therefore, he moive of he Fiie Differece Mehods ad Ierpolaio is o develop he idy ermiology used i he calculus of fiie differeces ad o make he relaioships bewee ie differeces ad differeial operaors, which are eeded i he umerical soluio of ordiary ad parial differeial equaios. early here are umber of ierval mehods for approximaig he iiial value problem which cosiss i a ordiary differeial equaio ad a iiial value of he fucio ha should be foud.i his paper we make a beauiful way for he ieger ierval value of ewos divided differeces ad ewos ierpolaio formulas. 45

2 Le s [ s, s ], [, ]. Prelimiaries (I). Addiio s [ s, (II). Subracio s [ s, ] (III). Muliplicaio ] s. [mi( s, s,, ),max( s, s,, (IV). Divisio [ i, j] [ i, j].[, ] If [ k, l] [ k, l] l k (V). s [ s, s ] for (VI). Iverse [ s, s ] for [ s, ] [, ], for [ s, ] s s (VII). s, s ] [ s, s ], ifs [ [ s, s ], ifs [,max{ s, }], oherwise 3. Proposed Mehod 3. Newos divided differece ierpolaio formula. Le f ( s ), f ( s ), f ( s ) be he values of f (s ) correspodig o he argumes s s,.., s o ecessarily equally spaced. From he defiiio of divided differeces, f ( s ) f ( s) We have (, f s s) s s )] 46

3 f s ) f ( s ) ( s s ) f ( s, )..() ( s f ( s, s ) f ( s, s ) f ( s, s, s ) s s f s, s ) f ( s, s ) ( s s ) f ( s, s, ) () ( s Usig his i () f s ) f ( s ) ( s s ) f ( s, s ) ( s s )( s s ) f ( s, s, ).....(3) ( s Also f ( s, s, s, s ) f ( s, s, s ) f ( s, s, s ) s From(3) we have f s, s, s ) f ( s, s, s ) ( s s ) f ( s, s, s, ) ( f ( s ) f ( s ) ( ) (, ) ( )( ) (,, ) ( )( )( ) (,,, s s f s s s s s s f s s s s s s s f s s s ) Proceedig i his maer, we ge f ( s ) f ( s ( (, ( )( ) (,, ) ( )( )( ) (,,, )... ( )( ) s s) f s ) )( )( s s s ) (, s s,,... f s s s s s s s f s s s3 s s s s s s s f s s s ) Bu,,,... h f s s s ) sice ( ) divided differece will be zero if f (s) is a polyomial of ( s degree.hece f ( s ) f ( s ( (, ( )( ) (,, ) ( )( )( ) (,,, )... ( )( ) s s ) f s ) )( )( s s s ) (, s s,,... f s s s s s s s f s s s3 s s s s s s s s f s s s ) s This formula is called Newos divided differece ierpolaio formula. 3. Newos forward differece ierpolaio formula Le f ( s ) be a fucio which akes he values,,... correspodig o he values s s,.., s where he values of are equally spaced. Ie, s i s ih, i,,,.. Suppose we wa o fid he values of whe s s ph where p isay real umber. Le ( s p ) p h 47

4 E p ( ) s ( ) p sice E ( ( p p ) p p )( p ) [ p! 3! 3...] ( ( p p ) )( p p p ) 3 p p! 3! s s Where p h This formula is called Newos forward ierpolaio formula. 3.3 Newos backward differece ierpolaio formula This formula is used for ierpolaig a value of for a give s ear he ed of a able of values. Le,,... be he values of f ( s ) for s s,.., s where s i s ih, i,,,.. Suppose we wa o fid he values of whe s egaive i his case). Le ( s p ) p p E ( s ) ) p h ( E sice ( ) s ( ) p usig E... ( ( p p ) )( p p p ) p...! 3! 3 p s s Where p h This formula is called Newos backward ierpolaio formula. ph where p isay real umber( p is 48

5 4. Numerical illusraio Problems. Usig ewos divided differece formula,fid he value of f [7 9] give he followig daa. x : [ 3 5] [ 4 6] [ 6 8] [ 9 ] [ ] [ 4] f (x ) : [ 47 49] [ 99 ] [ 93 95] [ 899 9] [ 9 ] [ 7 9]. Soluio: The divided differece able is give below s f (s ) (s ) [ 3 [ 4 [ 6 [ 9 [ [ 5] 6] 8] ] ] 4] [ 47 [ 99 [ 93 [ 899 [ 9 [ 7 49] ] 95] 9] ] 9] [ 5 [ 96 [ [ 38 f [ 48 54] 98] 4] 3] 4] f (s ) [ 4 [ [ 6 [ 3 6] ] 8] 34] 3 f (s ) [ [ [ ] ] ] 4 f (s ) [ [ ] ] From he give daa, x [3 5], x [ 4 6 ], x [6 8], 3 x [9 ], 4 x [ ], 5 x [ 4] By Newos divided differece ierpolaio formula f ( s ) f ( s ( (, ( )( ) (,, ) ( )( )( ) (,,, )... ( )( ) s s ) f s ) )( )( s s s ) (, s s,,... f s s s s s s s f s s s3 s s s s s s s s f s s s ) s f ([7 9]) [47 49] [[7 9] [3 5]][5 54] [[7 9] [3 5]][[7 9] [4 6]][4 6] [[7 9] [3 5]][[7 9] [4 6]][[7 9] [6 8]][ ] [ 47 49] [ 6][5 54] [ 6][ 5][4 6] [ 6][ 5][ [ ] 544 3][ ] 49

6 Problems. Usig ewos forward ierpolaio formula,fid f [..3] from he followig daa x [.9.] [.] [..3] [..4] [.3.5] f (x ) [.45.65] [ ] [ ] [ ] [ ] Soluio: Forward differece able x [ ], x [..3], h [..], p [..3] x y f ( x) y y [.9.] [.45.65] [ [. [. [.3.].3].4].5] [ [.466 [.48 [ ].398].446].4775] [.86 [.875 [.96 [ ].955].98].9955] [.7865 [.8685 [ ].855].95] 3 y [ [ ] 3.78] ( ( p p ) )( p p p ) 3 p p! 3!....3][-.9 [ ] [. y [.45.65] [..3][ ] -.7] [ ] [ [.45.65] [ ] [ [.45.65] [-o ] [ ] ].4] [ ].8573 Problems 3. Usig ewos backward ierpolaio formula,fid whe s [6 8] from he followig daa 5

7 x [ 9 ] [ 4 6] [ 9 ] [ 4 6] [ 9 3] f (x ) [ ] [ ] [ ] [ 3 39] [ ] Soluio: backward differece able is give below s s 3 4 [ 9 ] [ ] [ 4 [ 9 [ 4 [ 9 6] ] 6] 3] [ 6. [4.55 [ 3 [.55 39] 48.3] 43.65] 34.65] [.37 [ [ 3.65 [ ] 7.55] 4.45].65] [64.35 [ 58. [ ] 58.] 5.3] [.75.55] [ 3.65 [..5] 33.5] ( ( p p ) )( p p p ) p...! 3! 3 p s [9 3], [ ], h [5 5] s s h p [6 8][9 [5 5] 3] [.] [ ] [ [.][.8][ [6 6].][ 7.45 [.][.8].65] [ 5.9 [ ].8] [.][.8][.8][ [..5] [4 4] 5.3].8] [ ] [ ] [ ] [-.9.76] [ ] [ ] [ ]

8 5. Coclusio The proposed ierval value of ewos divided differeces ad ewos ierpolaio formula is simple o lear. We Evaluae he ewos ierpolaio wih more accuracy ad much more less amou of error`s o obai he soluios of real life siuaios. Newo ierpolaio is simply aoher echique for obaiig he same ierpolaig polyomial as was obaied usig he Lagrage formulae. Noe ha o compue higher order differeces i he ables, we ake forward differeces of previous order differeces isead of usig expaded formulae. The order of he differeces ha ca be compued depeds o how may oal daa pois are available. I his mehod is eiher o slow i case of h beig small or oo iaccurae, i case of h is o small for real life siuaios. Refereces. E. Hasa ad G. W. Walser, Global opimizaio usig Ierval Aalysis, Marcel Dekker, New York, 3.. Karl Nickel, O he Newo mehod i Ierval Aalysis. Techical repor 36, Mahemaical Research Ceer, Uiversiy of Wiscosio, Dec97 3. Hase E. R (988), A overview of Global Opimizaio usig ierval aalysis i Moore(988) pp E. R. Hase, Global Opimizaio Usig Ierval Aalysis, Marcel Dekker, Ic., New York, Helmu Raschek ad Jo G. Rlke, New Compuer Mehods for Global Opimizaio, Wiley, New York, K. Gaesa ad P. Veeramai, O Arihmeic Operaios of Ierval Numbers, Ieraioal Joural of Uceraiy, Fuzziess ad Kowledge - Based Sysems, 3 (6) (5), G.Veeramalai ad R.J.Sudararaj, Sigle Variable Ucosraied Opimizaio Techiques Usig Ierval Aalysis IOSR Joural of Mahemaics (IOSR-JM), ISSN: Volume 3,Issue 3, (Sep-Oc. ), PP G.Veeramalai, Ucosraied Opimizaio Techiques Usig Fuzzy No Liear Equaios Asia Academic Research Joural of Muli-Discipliary,ISSN: 39-8.Volume, Issue 9, (May3), PP Eldo Hase, Global opimizaio usig ierval aalysis- Marcel Dekker, 99. Hase E. R (979), Global opimizaio usig ierval aalysis-he oe dimesioal case, J.Opim, Theory Applicaio, 9, G.Veeramalai ad P.Gajedra, A New Approaches o Solvig Fuzzy Liear Sysem wih Ierval Valued Triagular Fuzzy Number Idia Scholar A Ieraioal Mulidiscipliary Research e-joural, ISSN: 35-9X,.Volume, Issue 3, (Mar6), PP G.Veeramalai, Eige Values of a Ierval Marix CLEAR IJRMST, Vol--No-3, Ja-Jue, ISSN: Louis B. Rall, A Theory of ierval ieraio, proceedig of he America Mahemaics Sociey, 86z:65-63, 98. 5

9 4. Louis B. Rall, Applicaio of ierval iegraio o he soluio of iegral equaios. Joural of Iegral equaios 6: 7-4, Ramo E. Moore, R. Baker Kearfoh, Michael J. Cloud, Iroducio o ierval aalysis, SIAM, 5-7, Philadelphia, Hase E.R (978a), Ierval forms of Newo s mehod, Compuig, K. Gaesa, O Some Properies of Ierval Marices, Ieraioal Joural of Compuaioal ad Mahemaical Scieces, () (7), E. R. Hase ad R. R. Smih, Ierval arihmeic i marix compuaios, Par, SIAM. joural of Numerical Aalysis, 4 (967), E. R. Hase, O he soluio of liear algebraic equaios wih ierval coefficies, Liear Algebra Appl, (969), E. R. Hase, Boudig he soluio of ierval liear Equaios, SIAM Joural of Numerical Aalysis, 9 (5) (99), P. Kahl, V. Kreiovich, A. Lakeyev ad J. Roh, Compuaioal complexiy ad feasibiliy of daa processig ad ierval compuaios Kluwer Academic Publishers, Dordrech (998). S. Nig ad R. B. Kearfo, A compariso of some mehods for solvig liear ierval Equaios, SIAM Joural of Numerical Aalysis, 34 (997), R. E. Moore, Mehods ad Applicaios of Ierval Aalysis, SIAM, Philadelphia, E. R. Hase ad R. R. Smih, Ierval arihmeic i marix compuaios, Par, SI AM. Joural of Numerical Aalysis, vol. 4, pp. 9, A. Neumaier, Ierval Mehods for Sysems of Equaios, Cambridge Uiversiy Press, Cambridge, Omar A. AL-Sammarraie,Geeralizaio of Newos Forward Ierpolaio Formula, IJSRP, Volume 5, Issue 3, March 5,ISSN Biswaji Das ad Dhriikesh Chakrabary, Newos backward ierpolaio: Represeaio of umerical daa by a polyo-mial curve, IJAR 6; (): Nasri Aker Ripa, Aalysis of Newos Forward Ierpolaio Formula,IJCSET, (E-ISSN: 44-64) Volume, Issue 4, December 53

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