Chapter Direct Method of Interpolation
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1 Chper 5. Direc Mehod of Inerpolion Afer reding his chper, you should be ble o:. pply he direc mehod of inerpolion,. sole problems using he direc mehod of inerpolion, nd. use he direc mehod inerpolns o find deriies nd inegrls of discree funcions. Wh is inerpolion? Mny imes, d is gien only discree poins such s (, y,, y, n yn, (, y n n. So, how hen does one find he lue of y ny oher lue of? Well, coninuous funcion f ( my be used o represen he n + d lues wih f ( pssing hrough he n + poins (Figure. Then one cn find he lue of y ny oher lue of. This is clled inerpolion. Of course, if flls ouside he rnge of for which he d is gien, i is no longer inerpolion bu insed is clled erpolion. So wh kind of funcion f ( should one choose? A polynomil is common choice for n inerpoling funcion becuse polynomils re esy o (A elue, (B differenie, nd (C inegre relie o oher choices such s rigonomeric nd eponenil series. Polynomil inerpolion inoles finding polynomil of order n h psses hrough he n + poins. One of he mehods of inerpolion is clled he direc mehod. Oher mehods include Newon s diided difference polynomil mehod nd he Lgrngin inerpolion mehod. We will discuss he direc mehod in his chper. (,..., ( 5..
2 5.. Chper 5. y (, y (, y (, y (, y Figure Inerpolion of discree d. f ( Direc Mehod The direc mehod of inerpolion is bsed on he following premise. Gien n + d poins, fi polynomil of order n s gien below n y n ( hrough he d, where,,..., n re n + rel consns. Since n + lues of y re gien n + lues of, one cn wrie n + equions. Then he n + consns,,,..., n cn be found by soling he n + simulneous liner equions. To find he lue of y gien lue of, simply subsiue he lue of in Equion. Bu, i is no necessry o use ll he d poins. How does one hen choose he order of he polynomil nd wh d poins o use? This concep nd he direc mehod of inerpolion re bes illusred using emples. Emple The upwrd elociy of rocke is gien s funcion of ime in Tble. Tble Velociy s funcion of ime. (s ( (m/s
3 Direc Mehod of Inerpolion 5.. Figure Grph of elociy s. ime d for he rocke emple. Deermine he lue of he elociy 6 seconds using he direc mehod of inerpolion nd firs order polynomil. Soluion For firs order polynomil inerpolion (lso clled liner inerpolion, he elociy gien by y ( + (, y f ( (, y Figure Liner inerpolion.
4 5..4 Chper 5. Since we wn o find he elociy 6, nd we re using firs order polynomil, we need o choose he wo d poins h re closes o 6 h lso brcke 6 o elue i. The wo poins re 5 nd. Then 5, ( 6. 78, ( gies ( 5 + ( ( + ( Wriing he equions in mri form, we he Soling he boe wo equions gies.9.94 Hence ( , 5 A 6, ( m/s Emple The upwrd elociy of rocke is gien s funcion of ime in Tble. Tble Velociy s funcion of ime. (s ( (m/s Deermine he lue of he elociy 6 seconds using he direc mehod of inerpolion nd second order polynomil. Soluion For second order polynomil inerpolion (lso clled qudric inerpolion, he elociy is gien by + + (
5 Direc Mehod of Inerpolion 5..5 y (, y (, y f ( (, y Figure 4 Qudric inerpolion. Since we wn o find he elociy 6, nd we re using second order polynomil, we need o choose he hree d poins h re closes o 6 h lso brcke 6 o elue i. The hree poins re, 5, nd. Then, ( , ( 6. 78, ( gies ( ( ( 4 ( 5 + ( 5 + ( ( + ( + ( Wriing he hree equions in mri form, we he Soling he boe hree equions gies Hence , A 6, ( ( ( m/s (
6 5..6 Chper 5. The bsolue relie pproime error second order polynomil is % obined beween he resuls from he firs nd Emple The upwrd elociy of rocke is gien s funcion of ime in Tble. Tble Velociy s funcion of ime. (s ( (m/s Deermine he lue of he elociy 6 seconds using he direc mehod of inerpolion nd hird order polynomil. b Find he bsolue relie pproime error for he hird order polynomil pproimion. c Using he hird order polynomil inerpoln for elociy from pr (, find he disnce coered by he rocke from s o 6s. d Using he hird order polynomil inerpoln for elociy from pr (, find he ccelerion of he rocke 6s. Soluion For hird order polynomil inerpolion (lso clled cubic inerpolion, we choose he elociy gien by (
7 Direc Mehod of Inerpolion 5..7 y (, y (, y (, y (, y Figure 5 Cubic inerpolion. f ( Since we wn o find he elociy 6, nd we re using hird order polynomil, we need o choose he four d poins closes o 6 h lso brcke 6 o elue i. The four poins re, 5, nd. 5. Then, ( , ( 6. 78, ( , ( gies ( ( ( ( 4 ( 5 + ( 5 + ( 5 + ( ( + ( + ( + ( (.5 + (.5 + (.5 + ( Wriing he four equions in mri form, we he Soling he boe four equions gies
8 5..8 Chper Hence ( ,.5 ( ( 6 +.4( ( m/s b The bsolue percenge relie pproime error for he lue obined for (6 beween second nd hird order polynomil is % c The disnce coered by he rocke beween s nd 6s cn be clculed from he inerpoling polynomil ( ,. 5 Noe h he polynomil is lid beween nd. 5 nd hence includes he limis of inegrion of nd 6. So 6 ( s( ( s 6 d 6 ( d m d The ccelerion 6 is gien by d ( 6 ( d 6 Gien h ( ,. 5 d ( ( d d ( d ,.5 ( ( ( m/s 6
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