Chapter Lagrangian Interpolation

Transcription

1 Chaper 5.4 agrangan Inerpolaon Afer readng hs chaper you should be able o:. dere agrangan mehod of nerpolaon. sole problems usng agrangan mehod of nerpolaon and. use agrangan nerpolans o fnd deraes and negrals of dscree funcons. Wha s nerpolaon? Many mes daa s gen only a dscree pons such as x y x y... x n yn x y n n. So how hen does one fnd he alue of y a any oher alue of x? Well a connuous funcon f x may be used o represen he n daa alues wh f x passng hrough he n pons Fgure. Then one can fnd he alue of y a any oher alue of x. Ths s called nerpolaon. Of course f x falls ousde he range of x for whch he daa s gen s no longer nerpolaon bu nsead s called exrapolaon. So wha knd of funcon f x should one choose? A polynomal s a common choce for an nerpolang funcon because polynomals are easy o A ealuae B dfferenae and C negrae relae o oher choces such as a rgonomerc and exponenal seres. Polynomal nerpolaon noles fndng a polynomal of order n ha passes hrough he n daa pons. One of he mehods used o fnd hs polynomal s called he agrangan mehod of nerpolaon. Oher mehods nclude Newon s dded dfference polynomal mehod and he drec mehod. We dscuss he agrangan mehod n hs chaper. 5.5.

2 5.5. Chaper 5.5 y x y x y x y Fgure Inerpolaon of dscree daa. x y f x x The agrangan nerpolang polynomal s gen by f x n n x f x where n n f n x sands for he n h order polynomal ha approxmaes he funcon y f x gen a x y x y... x y x y and x n n n n daa pons as x x x x x s a weghng funcon ha ncludes a produc of n erms wh erms of omed. The applcaon of agrangan nerpolaon wll be clarfed usng an example. Example The upward elocy of a rocke s gen as a funcon of me n Table. Table Velocy as a funcon of me. s m/s n n

3 agrangan Inerpolaon 5.5. Fgure Graph of elocy s. me daa for he rocke example. Deermne he alue of he elocy a 6 seconds usng a frs order agrange polynomal. Soluon For frs order polynomal nerpolaon also called lnear nerpolaon he elocy s gen by

4 5.5.4 Chaper 5.5 y x y f x x y x Fgure near nerpolaon. Snce we wan o fnd he elocy a 6 and we are usng a frs order polynomal we need o choose he wo daa pons ha are closes o 6 ha also bracke 6 o ealuae. The wo pons are 5 and. Then ges Hence

5 agrangan Inerpolaon m/s You can see ha. 8 and. are lke weghages gen o he eloces a 5 and o calculae he elocy a 6. Quadrac Inerpolaon y x y x y f x x y x Fgure 4 Quadrac nerpolaon. Example The upward elocy of a rocke s gen as a funcon of me n Table. Table Velocy as a funcon of me. s m/s a Deermne he alue of he elocy a 6 seconds wh second order polynomal nerpolaon usng agrangan polynomal nerpolaon. b Fnd he absolue relae approxmae error for he second order polynomal approxmaon.

6 5.5.6 Chaper 5.5 Soluon a For second order polynomal nerpolaon also called quadrac nerpolaon he elocy s gen by Snce we wan o fnd he elocy a 6 and we are usng a second order polynomal we need o choose he hree daa pons ha are closes o 6 ha also bracke 6 o ealuae. The hree pons are and 5. Then ges Hence m/s

7 agrangan Inerpolaon b The absolue relae approxmae error a for he second order polynomal s calculaed by consderng he resul of he frs order polynomal Example as he preous approxmaon a % Example The upward elocy of a rocke s gen as a funcon of me n Table. Table Velocy as a funcon of me s m/s a Deermne he alue of he elocy a 6 seconds usng hrd order agrangan polynomal nerpolaon. b Fnd he absolue relae approxmae error for he hrd order polynomal approxmaon. c Usng he hrd order polynomal nerpolan for elocy fnd he dsance coered by he rocke from s o 6 s. d Usng he hrd order polynomal nerpolan for elocy fnd he acceleraon of he rocke a 6 s. Soluon a For hrd order polynomal nerpolaon also called cubc nerpolaon he elocy s gen by

8 5.5.8 Chaper 5.5 Fgure 5 Cubc nerpolaon. Snce we wan o fnd he elocy a 6 and we are usng a hrd order polynomal we need o choose he four daa pons closes o 6 ha also bracke 6 o ealuae. The four pons are 5 and 5.. Then ges x y x y x y x y x f x y

9 agrangan Inerpolaon Hence m/s b The absolue percenage relae approxmae error a for he alue obaned for 6 can be obaned by comparng he resul wh ha obaned usng he second order polynomal Example a.69% c The dsance coered by he rocke beween s o s 6 can be calculaed from he nerpolang polynomal as

10 5.5. Chaper Noe ha he polynomal s ald beween and. 5 and hence ncludes he lms of and 6. So 6 s 6 s d d m d The acceleraon a 6 s gen by 6 d a d 6 Gen ha d a d d d a m/s Noe: There s no need o ge he smplfed hrd order polynomal expresson o conduc he dfferenaon. An expresson of he form ges he derae whou expanson as d d

11 agrangan Inerpolaon 5.5. INTERPOATION Topc agrange Inerpolaon Summary Texbook noes on he agrangan mehod of nerpolaon Maor General Engneerng Auhors Auar Kaw Mchael Keelas as Resed December 9 Web Se hp://numercalmehods.eng.usf.edu