Numerical Integration. Lecture Objectives. Numerical Integration. Ch. 21

Transcription

1 Numerl tegrto Ch. Leture Ojetves To solve vrous types o egeerg prolems usg umerl tegrto To e le to determe whh type o tegrto tehque to use or spe ppltos ost eet Numerl tegrto Very ommo operto egeerg, Emples? Futos tht re dult or mpossle to lytlly tegrte ote e umerlly tegrted Dsrete dt tegrto.e, epermetl, mye uevely sped dt We wll osder two umerl tegrto tehques: Newto Cotes Guss Qudrture

2 Newto Cotes tegrto Formul Most ommo umerl tehque Reple omplted uto or tulted dt wth wth ppromte uto tht we esly tegrte d d Nth order polyoml strght le prol Newto Cotes tegrto Formul Apply peewse to over the rge < < OPEN & CLOSED orms o Newto-Cotes Ope orm tegrto lmts eted eyod the rge o dt lke etrpolto; ot usully used or dete tegrto Closed orm dt pots re loted t the egg d ed o tegrto lmts re kow Fous Newto Cotes tegrto Formul Trpezodl Rule Use rst order polyoml, strght le to ppromte our uto d d

3 Newto Cotes tegrto Formul Trpezodl Rule Ths s the orm wdth verge heght Error or sgle pplto Truto Error ξ wdth E t Averge heght ξ '' - Et or ler uto - utos wth d d hgher order dervtves wll hve some error Trpezodl Rule Multple Appltos Dvde the tervl to segmets wth eqully sped se pots d d h segmet wdth h d h o h h h o Trpezodl Rule Multple Appltos Put the orm wdth verge heght o wdth Averge heght Totl Trpezodl error sum o dvdul errors Et '' ξ Appromte E t y estmtg me d dervtve over the etre tervl '' '' ξ E ''

4 Trpezodl Rule Notes. For ely ehved utos sgle pplto o the trpezod rule wll gve suet ury or my egeerg purpose. For hgh ury lrge, omputtol eort s hgher. Roud O Error wth lrge wll lmt the ury o the trpezod rule Smpso s / Rule - Use hgher order polyoml to ppromte our uto d order Lgrge polyoml uque polyoml tht psses through dt pots d Smpso s / Rule - Ater tegrtg & smplyg: h 4 [ ] h h segmet wdth h wdth 6 4 Averge heght 4

5 Smpso s / Rule Error 5 4 E ξ 88 t Et or rd order polyomls Error goes lke - 5 ompred to rd power o trpezodl rule Smpso s / Rule Multple Appltos Requres Eve umer o segmets o 4,,5,4,6 segmets pots wdth Averge heght Error, thrd order urte eve though we oly use pots 5 4 E 4 8 Smpso s /8 Rule - A odd umer o segmets wth eve umer o pots ormul use rd order polyoml to ppromte C e used wth Smpso s / rule to evlute eve or odd umer o segmet prolems. d d h h[ ] 8 8 wdth Averge heght 5

6 Smpso s /8 Rule Error 5 4 E ξ 648 t Et or rd order polyomls, slghtly more urte th / rule Smpso s / rule s preerred se the sme ury s heved wth ew pots. Guss Qudrture Newto-Cotes e., trpezodl rule & Smpso s the tegrl ws determed y lultg the re uder the urve oetg pots d where we evlute the uto t the ed pots. Guss Qudrture Cosder pots log strght le etwee d where postve d egtve errors le to redue totl error d gve mproved estmte o the tegrl. Uses uequl o-uorm spg est or utos ot tulr dt. Guss Qudrture Method o Udetermed Coeets Trpezodl Rule: Should gve et results ostt or strght le y y --/ -/ --/ -/ 6

7 7 Guss Qudrture Method o Udetermed Coeets d d y --/ -/ Evlute Et tegrl Evlute ppromto Set equl to eh other: Guss Qudrture Method o Udetermed Coeets y d d / / --/ -/ Guss Qudrture Method o Udetermed Coeets Equtos & ukows solve or o d : Susttute o d k to the orgl equto: Equvlet to the Trpezodl Rule!

8 Two Pot Guss Legedre Formul Eted the method o udetermed oeets: o & ukow ostts o & ukow lotos etwee & We ow hve 4 ukows eed 4 equtos! Two Pot Guss Legedre Formul Need to ssume utos g: d Prol & u utos wll gve us totl o 4 equtos We wll get pt ler tegrto ormul ormul tht wll e et or us! To smply the mth & provde geerl ormul selet lmts o tegrto to e & Normlzed oordtes Two Pot Guss Legedre Formul Evlute the tegrls or our 4 equtos: d d d d d 8

9 Two Pot Guss Legedre Formul Rewrte the equtos: Solve or o,, o d : / / Two Pot Guss Legedre Formul Pot Guss Legedre Formul or tegrto lmts to : / / rd order urte Need to hge vrles to trslte to other tegrto lmts Two Pot Guss Legedre Formul Chgg the lmts o tegrto: trodue ew vrle d tht represets our geerlzed ormul where we use to Assume d s lerly relted to d d d d 9

10 Two Pot Guss Legedre Formul Susttute o d k to our orgl ler ormul d Derette wth respet to d : d d d Susttute these vlues o d d the orgl tegrl to eetvely hge the lmts o tegrto wthout hgg the vlue o the tegrl. trs trs [ ] d o d Guss-Legedre Qudrture uses roots o Legedre Polyomls to lote the pot t whh the tegrd s evluted d - Guss-Legedre Qudrture uses roots o Legedre Polyomls to lote the pot t whh the tegrd s evluted d ukow weghtg oeets Guss Pots: spe vlue where the tegrd s evluted -

11 Guss-Legedre Qudrture uses roots o Legedre Polyomls to lote the pot t whh the tegrd s evluted d ukow weghtg oeets The vlues o w d re hose so tht the ormul wll e et up to & ludg polyoml o degree m-, where m s the umer o pots. E: Pot Et rd order polyoml Guss Pots: spe vlue where the tegrd s evluted Guss-Legedre Qudrture Geerl orm Pot Applto rom our dervto, we oud o d or the tegrto lmts to - d dd d - o d Guss-Legedre Qudrture Pt Applto Trsormto proedure d trs o trs d d d - o trs trs [ ] d o trs o trs d

12 . 4 Guss-Legedre Qudrture Smple pot Emple tegrte the ollowg uto rom. to.8: d d 4 d d dd / / Step : Trsorm lmts d Guss pots o & rom geerl orm trs o trs Step : perorm summto trs trs [ ] o [ ] [.675 ] Error Estmte -pot Guss-Legedre Formul E t 4 [! ] [!] ξ < ξ < Where s the umer o pots the ormul rememer, -pot ormul tegrtes polyoml o - etly! ξ s the th dervtve ter the hge o vrle the mgtude o the hgher order dervtves derese or oly rese slowly wth resg, Guss ormuls re Sgtly more urte th Newto-Cotes ormuls. Mtl Multpot Emple Bult Mtl tegrto methods trpzy*d le ^ qud,, tegrto lmts