Interpolation. 1. What is interpolation?

Transcription

1 Iterpoltio. Wht is iterpoltio? A uctio is ote give ol t discrete poits such s:.... How does oe id the vlue o t other vlue o? Well cotiuous uctio m e used to represet the + dt vlues with pssig through the + poit. The we c id the vlue o t other vlue o. This is clled iterpoltio. O course i lls outside the rge o or which the dt is give it is o loger iterpoltio ut isted is clled etrpoltio. So wht kid o uctio should we choose? A polomil is commo choice or iterpoltig uctio ecuse polomils re es to - Evlute - Dieretite d - Itegrte s opposed to other choices such s sie or epoetil series. Polomil iterpoltio ivolves idig polomil o order tht psses through the + poits. Oe o the methods is clled the direct method o iterpoltio. Other methods iclude Newto s divided dierece polomil method d Lgrgi iterpoltio method.. Direct Method The direct method o iterpoltio is sed o the ollowig priciple. I we hve '+' dt poits it polomil o order '' s give elow through the dt where... re + rel costts. Sice + vlues o re give t + vlues o oe c write + equtios.

2 The the '+' costts... c e oud solvig the + simulteous lier equtios. To id the vlue o t give vlue o simpl sustitute the vlue o i the polomil orm. Emples :. Assume the ollowig dt is give: Determie the vlue o the uctio correspodig to usig the direct method o iterpoltio d third order polomil. Solutio: For third order polomil iterpoltio cuic iterpoltio we choose the uctio give. The gives Writig the our equtios i mtri orm we hve

3 Solvig the ove our equtios gives Hece Use direct method to compute the third order polomil iterpoltio ssocited with the ollowig dt: Solutio: We choose the uctio give The gives Writig the our equtios i mtri orm we hve

4 Solvig the ove our equtios gives Hece Newto s divided dierece iterpoltio To illustrte this method we will strt with lier d qudrtic iterpoltio the the geerl orm o the Newto s Divided Dierece Polomil method will e preseted... Lier iterpoltio Give it lier iterpolt through the dt. Note tht d ssumig lier iterpolt mes: Sice t : d t : The so Ad the lier iterpolt Becomes:.. Qudrtic iterpoltio

5 Give d it qudrtic iterpolt through the dt. Note tht d ssume the qudrtic iterpolt give At At the At the Hece the qudrtic iterpolt is give Figure 5.4. Qudrtic iterpoltio

6 .. Geerl Form o Newto s Divided Dierece Polomil I the two previous cses we oud how lier d qudrtic iterpoltio is derived Newto s Divided Dierece polomil method. Let us lze the qudrtic polomil iterpolt ormul where Note tht d re iite divided diereces. d re irst secod d third iite divided diereces respectivel. Deotig irst divided dierece the secod divided dierece d the third divided dierece where d re clled rcketed uctios o their vriles eclosed i squre rckets. We c write: This leds to the geerl orm o the Newto s divided dierece polomil or dt poits... s where where the deiitio o the th m divided dierece is... m m m m m

7 From the ove deiitio it c e see tht the divided diereces re clculted recursivel. For emple o third order polomil give d EXAMPLE : Assume the ollowig dt is give: = =.7 = 7.9 Determie the vlue o the uctio correspodig to usig secod order polomil iterpoltio usig Newto s Divided Dierece polomil. Solutio: Let us lze the qudrtic polomil iterpolt ormul:.7 DIVIDED DIFFERENCE TABLE: i i i = =.7.4 = 7.9 The clculte

8 Emple : Assume the ollowig dt is give: = = 4 = Determie the vlue o the uctio correspodig to usig secod order polomil iterpoltio usig Newto s Divided Dierece polomil. Solutio: Let us lze the qudrtic polomil iterpolt ormul: 4 DIVIDED DIFFERENCE TABLE: i i i = = = 6.79 The clculte Emple : Use Newto s Divided Dierece polomil to compute the irst secod third d ourth order polomil iterpoltio ssocited with the ollowig dt = = = = 4 4 = Solutio: DIVIDED DIFFERENCE TABLE

9 i i i i k i k l 4 = = = = =5 6. Lier iterpoltio irst order: sed o d Qudrtic iterpoltio secod order: sed o d 6 8. c. Third order polomil iterpoltio: sed o d d. Fourth order polomil iterpoltio: sed o d Error Estimtio i Newto s Iterpoltig Polomils. Structure o iterpoltig polomils is similr to the Tlor series epsio i.e. iite divided diereces re dded sequetill to cpture the higher order derivtives.. For th-order iterpoltig polomil logous reltioship or the error is: R R. I dditiol poit + + is ville the R This result is sed o the ssumptio tht the series is strogl coverget. i.e. + th -order predictio is closer to the true vlue th the th -order predictio.

10 4. Lgrgi Iterpoltio Polomil iterpoltio ivolves idig polomil o order tht psses through the + poits. Oe o the methods to id this polomil is clled Lgrgi Iterpoltio. Lgrgi iterpoltig polomil is give : i where i stds or the give t L i i th order polomil tht pproimtes the uctio dt poits s... L i i i d L i is weightig uctio tht icludes product o terms with terms o i omitted. Emple : Assume the ollowig dt is give: = 8 = =.9.79 Estimte the vlue o t usig the Lgrgi method d irst order polomil? Solutio For irst order Lgrge polomil iterpoltio lso clled lier iterpoltio we choose the vlue o s give Hece Li i i L L L L

11 You c see tht L. d L re like weightges give to the vlues o t. d 4. 5 to clculte the vlue o t 4.. Emple : Assume the ollowig dt is give: =8 =9 = = Estimte the vlue o t usig the Lgrgi method d secod order polomil? Solutio For secod order Lgrge polomil iterpoltio lso clled qudrtic iterpoltio we choose the vlue o give L L L Li i i L L L Hece Emple : Assume the ollowig dt is give: =8 =9 = = = Estimte the vlue o t usig the Lgrgi method d third-order polomil? Solutio For third-order Lgrge polomil iterpoltio we choose the vlue o give L.99 L.9544 L.49 L.798

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