Chaper 5. Newon s Diided Dierence Inerpolaion Aer reading his chaper you should e ale o:. derie Newon s diided dierence mehod o inerpolaion. apply Newon s diided dierence mehod o inerpolaion and. apply Newon s diided dierence mehod inerpolans o ind deriaies and inegrals. Wha is inerpolaion? Many imes daa is gien only a discree poins such as y y... n yn y n n. So how hen does one ind he alue o y a any oher alue o? Well a coninuous uncion may e used o represen he n daa alues wih passing hrough he n poins Figure. Then one can ind he alue o y a any oher alue o. This is called inerpolaion. O course i alls ouside he range o or which he daa is gien i is no longer inerpolaion u insead is called erapolaion. So wha kind o uncion should one choose? A polynomial is a common choice or an inerpolaing uncion ecause polynomials are easy o A ealuae B diereniae and C inegrae relaie o oher choices such as a rigonomeric and eponenial series. Polynomial inerpolaion inoles inding a polynomial o order n ha passes hrough he n poins. One o he mehods o inerpolaion is called Newon s diided dierence polynomial mehod. Oher mehods include he direc mehod and he Lagrangian inerpolaion mehod. We will discuss Newon s diided dierence polynomial mehod in his chaper. Newon s Diided Dierence Polynomial Mehod To illusrae his mehod linear and quadraic inerpolaion is presened irs. Then he general orm o Newon s diided dierence polynomial mehod is presened. To illusrae he general orm cuic inerpolaion is shown in Figure. 5..
5.. Chaper 5. y y y y Figure Inerpolaion o discree daa. y Linear Inerpolaion Gien and i a linear inerpolan hrough he daa. Noing y and y y y assume he linear inerpolan is gien y Figure Since a and a giing So giing he linear inerpolan as
Newon s Diided Dierence Inerpolaion 5.. y y y Figure Linear inerpolaion. Eample The upward elociy o a rocke is gien as a uncion o ime in Tale Figure. Tale Velociy as a uncion o ime. s m/s 7.4 5 6.78 57.5.5 6.97 9.67 Deermine he alue o he elociy a 6 seconds using irs order polynomial inerpolaion y Newon s diided dierence polynomial mehod. Soluion For linear inerpolaion he elociy is gien y Since we wan o ind he elociy a 6 and we are using a irs order polynomial we need o choose he wo daa poins ha are closes o 6 ha also racke 6 o ealuae i. The wo poins are 5 and. Then 5 6. 78 57. 5 gies
5..4 Chaper 5. 6.78 57.5 6.78 5.94 Figure Graph o elociy s. ime daa or he rocke eample. Hence 6.78.94 5 5 A 6 6 6.78.946 5 9.69 m/s I we epand 6.78.94 5 5 we ge.9.94 5 and his is he same epression as oained in he direc mehod. Quadraic Inerpolaion Gien and i a quadraic inerpolan hrough he daa. Noing y y y y y y and y assume he quadraic inerpolan is gien y
Newon s Diided Dierence Inerpolaion 5..5 A A giing A Giing Hence he quadraic inerpolan is gien y Figure 4 Quadraic inerpolaion. y y y y
5..6 Chaper 5. Eample The upward elociy o a rocke is gien as a uncion o ime in Tale. Tale Velociy as a uncion o ime. s m/s 7.4 5 6.78 57.5.5 6.97 9.67 Deermine he alue o he elociy a 6 seconds using second order polynomial inerpolaion using Newon s diided dierence polynomial mehod. Soluion For quadraic inerpolaion he elociy is gien y Since we wan o ind he elociy a 6 and we are using a second order polynomial we need o choose he hree daa poins ha are closes o 6 ha also racke 6 o ealuae i. The hree poins are 5 and. Then 7. 4 5 6. 78 57. 5 gies 7.4 6.78 7.4 5 7.48 57.5 6.78 6.78 7.4 5 5.94 7.48
Newon s Diided Dierence Inerpolaion 5..7.766 Hence 7.4 7.48.766 5 A 6 6 7.4 7.486.7666 6 5 9.9 m/s I we epand 7.4 7.48.766 5 we ge.5 7.7.766 This is he same epression oained y he direc mehod. General Form o Newon s Diided Dierence Polynomial In he wo preious cases we ound linear and quadraic inerpolans or Newon s diided dierence mehod. Le us reisi he quadraic polynomial inerpolan ormula where Noe ha and are inie diided dierences. and are he irs second and hird inie diided dierences respeciely. We denoe he irs diided dierence y he second diided dierence y and he hird diided dierence y where and are called rackeed uncions o heir ariales enclosed in square rackes. Rewriing
5..8 Chaper 5. This leads us o wriing he general orm o he Newon s diided dierence polynomial or n daa poins y y... n yn n yn as n... n... n where n n n... n n n... h where he deiniion o he m diided dierence is m m... m... m... m From he aoe deiniion i can e seen ha he diided dierences are calculaed recursiely. For an eample o a hird order polynomial gien y y y and y Figure 5 Tale o diided dierences or a cuic polynomial. Eample The upward elociy o a rocke is gien as a uncion o ime in Tale.
Newon s Diided Dierence Inerpolaion 5..9 Tale Velociy as a uncion o ime. s m/s 7.4 5 6.78 57.5.5 6.97 9.67 a Deermine he alue o he elociy a 6 seconds wih hird order polynomial inerpolaion using Newon s diided dierence polynomial mehod. Using he hird order polynomial inerpolan or elociy ind he disance coered y he rocke rom s o 6 s. c Using he hird order polynomial inerpolan or elociy ind he acceleraion o he rocke a 6 s. Soluion a For a hird order polynomial he elociy is gien y Since we wan o ind he elociy a 6 and we are using a hird order polynomial we need o choose he our daa poins ha are closes o 6 ha also racke 6 o ealuae i. The our daa poins are 5 and 5.. Then 7. 4 5 6. 78 57. 5.5 6. 97 gies 7.4 6.78 7.4 5 7.48
5.. Chaper 5. 5 6.78 57.5.94 7.48 7.48.94.766.5 57.5 6.97 4.48 5 6.78 57.5.94 5.5.94 4.48.4445.766.5.766.4445 5.447 Hence
Newon s Diided Dierence Inerpolaion 5.. 7.4 7.48.766 5 5.547 5 A 6 6 7.4 7.486.7666 6 5 5.547 6 6 56 9.6 m/s The disance coered y he rocke eween s and 6 s can e calculaed rom he inerpolaing polynomial 7.4 7.48.766 5 5.547 5 4.54.65.4.5447. 5 Noe ha he polynomial is alid eween and.5 and hence includes he limis o and 6. So 6 s s 6 d 6 4.54.65.4.5447 d 4 6 4.54.65.4.5447 65 m c The acceleraion a 6 is gien y d a 6 6 d d a d d 4.54.65.4.5447 d.65.648.64 a 6.65.6486.646 9.664 m/s INTERPOLATION Topic Newon s Diided Dierence Inerpolaion Summary Teook noes on Newon s diided dierence inerpolaion. Major General Engineering Auhors Auar Kaw Michael Keelas Las Reised Decemer 9 We Sie hp://numericalmehods.eng.us.edu 4