Interpolasi dan Turunan Numerik (Rabu, 2 Maret 2016) Hidayatul Mayyani G

Transkripsi

1 Iterpolsi d Turu Numeri (Rbu Mret 6) Hidytul Myyi G55535 Outlie: Iterpolsi Lier - Poliomil Lgrge - Poliomil Newto - Vdermode Mtris - Ivers Iterpolsi - Iterpolsi Neville Glt Iterpolsi Turu Numeri Estrpolsi Ricrdso

2 . Iterpolsi Lier. Poliomil Lgrge Sutu poliomil P(t) derjt - yg ui dpt diostrusi yg mellui titi t yg berbed. Iterpolsi Poliomil digu utu mecri titi-titi tr dri bu titi ( ) ( ) ( )... ( ) deg meggu pedet ugsi polyomil pgt -; = Sutu ugsi dpt di iterpolsi e dlm betu iterpolsi polyomil lgrge. Betu umum poliom Lgrge derjt = = Dlm l ii d = =... = ( ) Iterpolsi dri du bu titi ( )d ( ) megsil polyomil berderjt (iterpolsi lier) yitu: = ( ) ( ) + ( ) ( ) = ( ) + ( ) ( ) = + ( ) ( ) ( ) Iterpolsi dri tig titi ( ) d ( ) d ( ) megsil poliomil berderjt yitu: Deg = = + + = =

3 Coto : Dri ugsi = diberi tig bu titi dt dlm betu tbel: 4 6 y Tetu (35)deg poliom Lgrge derjt. Gu lim g be. Peyelesi: Poliom derjt = (perlu tig bu titi) = + + = (3.5)= (. )(. ) = = (3.5)= (. )(. ) =.47 = (3.5)= (. )(. ) =.5 Jdi (3.5)= (.83333)(.579)+(.47)(.577)+(.5)(.575) =.573 b. Poliomil Newto Iterpolsi poliom Newto dpt ditulis: i i p ( j ) i j p Koeisie diostrusi meggu bed bgi:... Deg otsi yg bru iterpolsi poliom betu Newto dpt ditulis dlm betu: i p... i ( j ) i j Ji diberi sebrg ugsi () d titi d. Bed bgi orde pertm dri ugsi dl: ( ) Bed bgi orde du utu 3 titi yg berbed D seterusy i i i j ( ) j

4 Jdi oeisie... diperole dri: ) ( ) ( Secr umum ) ( ) ( ) ( j j i j j i i ) ( ) (... ) (... j j i j j i i Coto: Dietui: ( ) ( ) ( ) ( ) (dri ugsi l ) Dity: Perir l deg iterpolsi Newto orde e-3 Peyelesi: 3 3 p

5 (.5) i i ( i ) First Secod Tird p 3 3 p 3 () = +.465(-)-.53(-)(-4)+.8(-)(-4)(-6) p 3 () =.69 Gmbr gri perbdig tr ugsi l d mpiry meggu iterpolsi Newto c. Vdermode M tris Betu umum dri poliomil yg megiterpolsi + titi dl =

6 Sistem persm lier: = = = = = = = = Betu mtris dri persm lier di ts. = : Mtris V disebut sebgi mtris Vdermode. Mtris tersebut merup mtris osigulr seigg memilii solusi tuggl. Utu mecri peyelesiy dpt digu elimisi Guss. Coto: Cri iterpolsi poliomil pd dt (-)()()(6) meggu Sistem Vdermode. Peyelesi: Sistem Vdermode dri titi-titi (-)()()(6) dl = = = = = = = = Mtris Vdermode: Utu medpt solusiy digu Gussi Elimitio

7 M diperole persm lier dri mtris di ts dl + = = + = = = = Seigg poliomily berbetu: = = + d. Ivers Iterpolsi Sebu proses iverse iterpoltio serig digu utu megmpiri sutu ivers ugsi. Sediy ili = tel diitug pd. Meggu tble beriut y Betu poliomily dl: = memilii ivers pd odisi tertetu. Ivers tersebut dimpiri ole =.. Coto gri ugsi = b. Coto gri ugsi =

8 e. Iterpolsi Neville Misl dl ili di titi dri persm polyomil orde ol (ost) yg mellui titi ( ) seigg =. Demii jug yg mellui ( )( )...( ). Seljuty misl dl ili di dri persm orde stu yg mellui titi ( ) ( ). Deg cr yg sm dl ili di yg mellui titi ( )( )...( ) seluru s titi. Dimuli deg poliomil ost =. Deg memili d > + dideiisi ugsi reursi: = +... Diperole utu = 4 = ( )... ( ) Seljuty dilu peyeder otsi =... Dim utu i meuju poliomil iterpolsi derjt pd j+ ode.... = + = ( ) ( ) Diperole = = = = = = = = = = = = = = =

9 Coto Diberi ugsi: = Kit megprosimsi ili (3) (3) (3)= =.5 (3) (3)= =.4 (3) (3)= =.5 Deg meggu rumus Neville it meetu ili d (3 ) (3) (3 ) (3) (3) (3)= = ( ) (3) (3)= = ( ) = 3 (3 ) (3) (3 ) (3) = 35 Seljuty sil yg it perole it msu e dlm tbel beriut: Seljuty it dpt megitug deg meggu (3 ) (3) (3 ) (3) (3) (3)= = =.35

10 Diperole : Deg demii ili mpir 35 (3)meggu rumus Neville derjt dl. Glt Iterpolsi Teorem I Ji p dl poliomil berderjt tertiggi yg megiterpolsi pd + titi yitu d ji otium =! ( ) Glt Rt-Rt Iterpolsi Kre ili c tid dietui m mpiri deg c = = seigg rumus mejdi = ( + )! ( ) Lemm I (Lemm Bts Ats) Deiisi = + utu i= m utu sutu deg =(b-)/ dl jr trtiti Teorem II 4! Misl ugsi yg memeui otiu pd b d. Misl p dl poliomil berderjt yg megiterpolsi pd + titi di termsu titi iry m di 4( + ) deg =(b-)/ dl jr trtiti.

11 Keterg : = m Coto Diberi empt bu titi beriut mpiri ili cos(.5) deg metode Lgrge d Newto smpi derjt 3 sert idetiisi glty Nili sejti cos(.5)= Dri metode Lgrge diperole cos(.5) Dri metode Poliomil Newto diperole cos(.5).69 Bts ts esl yitu.. = Glt bsolut Lgrge = < Glt bsolut Newto = < Seigg poliomil derjt 3 sud cuup teliti utu megmpiri ili cos(.5) Teorem III Ji p dl poliomil berderjt yg megiterpolsi pd titi m utu sutu bu titi iterpolsi = ( ) Deg disebut selisi terbgi yg digu dlm iterpolsi poliom Newto seigg teorem III dpt digu utu mecri glt poliom Newto

12 Teorem IV Ji otiu pd b d dl + titi yg berd di itervl b m utu di (b) =! Corollry I (Selisi Terbgi) Ji dl poliomil berderjt m semu selisi terbgi dl ol utu + Glt Li dlm Iterpolsi Poliomil Fugsi Ruge: = ( + ) pd itervl -55 Misl poliomil yg megiterpolsi ugsi ii pd + titi pd itervl - 55 m lim m = Aibty semi by titi iterpolsi m semi besr glty. Pili li yg lebi bi yitu deg meggu titi Cebysev deg itervl stdr - + = cos ( ) + Utu titi yg berd di sutu itervl b mejdi = ( + )+ ( ) + + ( )

13 Iterpolsi poliomil deg titi Cebysev megmpiri ugsi Ruge lebi bi dibdig deg iterpolsi poliomil deg titi dt. 3. Turu Numeri Proses mecri mpir ili umeri sutu turu dri ugsi yg diberi pd sutu titi. Bisy digu eti : Fugsi tid dietui secr esplisit y dietui dt empirisy sj yitu {( ); = } Fugsi dietui secr esplisit tetpi betuy rumit.. Pedet turu umeri Hmpir selisi mju (bed mju) Hmpir selisi mudur (bed mudur) Hmpir selisi pust (bed pust) Rumus utu etig mpir dits diperole dri espsi deret Tylor di seitr = : = + ( )+ Misl = + m didpt : ( ) +!

14 ( + ) = + + Rumus selisi mju utu turu pertm :! + Deg = ( + ) ( + ) () dl glt dri mpir ugsi.! y y y = () Rumus selisi mju utu turu edu : Deg ( + ) ( + ) ( + ) ( + ) ( + ) ( + )+ () dl glt dri mpir ugsi. Misl = m didpt : ( ) = + Rumus selisi mudur utu turu pertm :! Deg ( ) = + ( ) () dl glt dri mpir ugsi.!

15 y y- y = () - Rumus selisi mudur utu turu edu : Deg ( ) ( ) ( ) ( ) ( )+ ( ) () dl glt dri mpir ugsi. Bil edu persm beriut slig diurg : ( + ) = + + ( ) = + Diperole rumus selisi pust utu turu pertm:!! + Deg ( + ) ( ) = ( + ) ( ) dl glt dri mpir ugsi. 3!

16 y y y - y = () - Bil edu persm beriut slig ditmb : ( + ) = + + ( ) = + Diperole rumus selisi pust utu turu edu :!! + = ( + ) + ( ) ( + ) + ( ) Deg dl glt dri mpir ugsi. Coto: Dietui ugsi =. Tetu mpir () d () utu =. d =.5 deg meggu mpir selisi mju selisi mudur selisi pust d itug glty. Jwb : = = = 4

17 4. Estrpolsi Ricrdso Metode utu memperole rumus mpir turu deg orde yg lebi tiggi dri mpir deg orde yg lebi red disebut deg estrpolsi. Metode tersebut diembg ole Lewis Fry Ricrdso di wl bd seigg metode tersebut emudi diel deg estrpolsi Ricrdso. Diterp pd turu umeri utu memperole solusi yg lebi teliti. Hmpir turu bed pust deg orde ()= ( )+ = + +. () = () ()= 3 () () = 3 Deg () () = ()+ 3 ( )+ (() ) () = + () +. = = () + ; dl orde glt yg dipi

18 Subtitusi: () () () = 3 () () ()= 3 Setip perlus estrpolsi Ricrdso mei orde glt dri mejdi