Chapter 12-b Integral Calculus - Extra

Transcription

1 C - Itegrl Clulus Cpter - Itegrl Clulus - Etr Is Newto Toms Smpso BONUS Itroduto to Numerl Itegrto

2 C - Itegrl Clulus Numerl Itegrto Ide s to do tegrl smll prts, lke te wy we preseted tegrto: summto. Numerl metods just try to mke t ster d more urte Bs Numerl Itegrto Ide: Wegted sum o uto vlues to ppromte tegrl d LL - Tsk: Fd pproprte s wegts d te s odes.

3 C - Itegrl Clulus Bs Numerl Itegrto We wt to d tegrto o utos o vrous orms o te equto kow s te Newto-Cotes tegrto ormuls. Newto-Cotes ormul Assume te vlue o deed o [,] s kow t eqully sped pots,,...,, were, d. Te, d were, wt te step sze equl to / /. Te 's re lled wegts. Te 's re lled odes. Te preso o te ppromto depeds o. Te odes e eqully sped or ueqully sped. Bs Numerl Itegrto Error lyss Te error o te ppromto s te deree etwee te vlue o te tegrl d te umerl result: error ε d - Σ Te errors re requetly ppromted usg Tylor seres or. Te error lyss gves strt upper oud o te error, te dervtves o re vlle. 6

4 C - Itegrl Clulus Newto-Cotes Formul Te wegts re derved rom te Lgrge polyomls L. Tey deped oly o te 's; ot o te uto ƒ. L Deret metods use deret polyomls to get te 's. Newto-Cotes Closed Formule -- Use ot ed pots Trpezodl Rule : Ler Smpso s /-Rule : Qudrt Smpso s /8-Rule : Cu Boole s Rule : Fourt-order Newto-Cotes Ope Formule -- Use oly teror pots mdpot rule 7 Trpezod Rule Strgt-le ppromto Te trpezod rule ppromtes te rego uder te grp o te uto s trpezod d lultg ts re. Appromte tegrl y te trpezod s re. d [ ] L 8

5 C - Itegrl Clulus Trpezod Rule - Dervto We use te Lgrge ppromto or uto over te tervl - Lgrge terpolto, gve y L L... L For te se,, wt te tervl - : L d let,,, d ; L Te, we tegrte to ot te trpezod rule 9 Trpezod Rule - Dervto Itegrtg: d L d L d d d > Te wegts deped oly o! [ ] Ts ppromto my e poor. Te ppromto error s: ε d - Σ - / η, η Є [, ]. Tus, te tegrd s ove.e., postve seod dervtve-, te error s egtve. Tt s, te trpezodl rule overestmtes te true vlue o te tegrl.

6 C - Itegrl Clulus Trpezod Rule - Emple Evlute te tegrl Et soluto e d e Grp d e e e Trpezod Rule - Emple Evlute te tegrl e d Trpezodl Rule: Appromto Error ε - / η η s umer etwee d. Let s tke η. Te ε - /*[*ep**ep***ep*] -9.8 Trpezodl Rule I e 8 d [ ] e ε 7.% 6.96

7 C - Itegrl Clulus Smpso s /-Rule Kepler s Rule Appromte te uto y prol [ ] d L d d,,,, let L L Smpso s /-Rule - Dervto Use qudrt Lgrge terpolto:

8 C - Itegrl Clulus Smpso s /-Rule - Dervto Itegrte te Lgrge terpolto d L d d d d d [ ] Ag, te wegts deped oly o! Toms Smpso 7 76, Egld Smpso s /8-Rule Appromte y u polyoml d 8 [ ] L 6

9 C - Itegrl Clulus 7 L [ ] 8 - ; Ld d Lgrge terpolto Itegrte to ot te rule Smpso s /8-Rule 8 Emple: Smpso s Rules Evlute te tegrl Smpso s /-Rule: Appromto Error ε - /88 η η s umer etwee d. Se η >, te error s egtve oversootg. Let s tke η.. Te, ε - /88 [ ep*.*[6*.] Smpso s /-Rule d e [ ] % I e d e e ε

10 C - Itegrl Clulus Emple: Smpso s Rules Evlute te tegrl Smpso s /8-Rule e 8 I e d 8 / [ ] ε.7% 6.96 d Smpso s /8-Rule: Appromto Error ε - /68 η η s umer etwee d. 9 Mdpot Rule Newto-Cotes Ope Formul d were η Є [,]. m η Te mdpot rule mouts to ompute te re o te retgle Note: Ts rule does ot mke y use o te ed pots. m

11 C - Itegrl Clulus Two-pot Newto-Cotes Ope Formul Appromte y strgt le d 8 [ ] η Tree-pot Newto-Cotes Ope Formul Appromte y prol d 7 [ ] η

12 C - Itegrl Clulus Better Numerl Itegrto Composte tegrto Composte Trpezodl Rule Composte Smpso s Rule Rrdso Etrpolto Romerg tegrto Composte Trpezod Rule To mprove te Trpezod Rule, rst splts te tervl o tegrto [,] to N smller, uorm sutervls, d te pples te trpezodl rule o e o tem. 7 Two segmets 7 Tree segmets Four segmets 7 My segmets

13 C - Itegrl Clulus Composte Trpezod Rule Use te Trpezod Rule tervls. Te, dd tem togeter. d d d [ ] [ ] L [ ] [ L L ] LL d Composte Trpezod Rule Evlute te tegrl I e d, I [ ] ε 7.%, I [ ]. ε.7%, I [ ] ε 9.7% 8,. I [.... ] ε.% 6,. I [.. L..7 ].9 ε.66% 6

14 C - Itegrl Clulus Composte Trpezod Rule wt Uequl Segmets Evlute te tegrl I e d Use te ollowg s: {,,.,.} I. d d d [ ] [ ]. [.] [. ] d [ e ] [ e e ] [ e.e ] 7 8 [.e e ] 97.8 ε.%. 7 Composte Smpso s Rule Peewse Qudrt ppromtos

15 C - Itegrl Clulus Composte Smpso s Rule Evlute te tegrl Usg, I Usg, I [ ] e d 8 [ e e ] 8. ε 7.96% I [ ] 6 8 [ e e e e ] ε 8.7% 9 Composte Smpso s Rule wt Uequl Segmets Evlute te tegrl Usg.,. I e d I d d [. ].. ε.76% [. ] [.e e ] [ e.e e ]

16 C - Itegrl Clulus Guss Qudrtures Newto-Cotes Formule - Nodes: Use evely-sped utol vlues - Wegts: Derved rom ppromto requred to e equl or polyoml o order lower or equl to te degree o te polyomls used to ppromte te uto. Gve odes, est! - Prolem: C eplode or lrge Ruge s peomeo Q: C we use more eet wegts d odes? Yes! Guss Qudrtures - Guss qudrture sets te odes d te wegts su wy tt te ppromto s et we. s low order polyoml. Best oe or ot, odes d wegts! Guss Qudrtures I t, Guss qudrture s more geerl t smple tegrto, t omputes ppromto to te wegted tegrl: d Guss Qudrtures - Selet utol vlues t o-uormly dstruted pots to eve ger ury. Te vlues re ot predetermed, ut ukows to e determed. - Cge o vrles > te tervl o tegrto s [-,]. - Guss-Legedre ormule or odes d wegts - Wt odes, delvers et swer s - t -order polyoml, s lose to - t -order.

17 C - Itegrl Clulus Guss Qudrtures Te Guss-Legedre qudrture rule s geerl stted s: d te 's re lled te wegts, te 's re lled te qudrture odes. Te ppromto error term, ε, s lled te truto error o tegrto. Te gol s to get et swer s - t -order polyoml. Wt, we get et swer s 9 t -order polyoml. For Guss-Legedre qudrture, te odes re ose to e zeros o ert Legedre polyomls. Tey re ot trvl to ompute. Guss Qudrtures Nodes d Wegts To get rgt swer or To get rgt swer or To get rgt swer or d d d ψ d / ψ To get rgt swer or j or j,..., - j d ψ j >A system o equtos ukows.

18 C - Itegrl Clulus By ostruto we get rgt swer or j, j,..., j j-, > eoug to get te rgt swer or y polyoml o order -. Emple: > system o equtos d ukows: Guss Qudrtures Nodes d Wegts d d d d 6 Cge o Itervl or Guss Qudrture Coordte trsormto rom [,] to [-,] Ts e doe y e trsormto o t d ge o vrles. t t d dt t t t d dt t

19 C - Itegrl Clulus 7 Guss Qudrture o [-, ] For, we ve our ukows,,,. Tese re oud y ssumg tt te ormul gves et results or tegrtg geerl rd order polyoml. It lso e doe y oosg,,, su tt t yelds et tegrl or,,,. d L d : - Guss Qudrture Geerl ormulto: 8 Guss Qudrture o [-, ] Et tegrl or,,, Four equtos or our ukows d Cse d d d d d I

20 C - Itegrl Clulus 9 Guss Qudrture o [-, ] Now, oose,,,,, su tt te metod yelds et tegrl or,,,,,. Ag,,,,,, re lulted y ssumg te ormul gves et epressos or tegrtg t order polyoml. : Cse d - Guss Qudrture o [-, ] d d d d d d / / 9 / 8 / 9 / 9

21 C - Itegrl Clulus Guss Qudrture o [-, ] Appromto ormul or 8 I d Emple : Guss Qudrture Evlute I te t dt Coordte trsormto t t I te dt ; dt d e d d Two-pot ormul I d e e ε.%

22 C - Itegrl Clulus Emple : Guss Qudrture Tree-pot ormul 8 I d e e.6 e ε.79% Four-pot ormul I d.78.6 [ ] [ ] 97.7 ε.7% Emple : Guss Qudrture. 6 Evlute I. 997 e d π Coordte trsormto t I π.6 e ; dt. 8d t [.8 ].8.8 dt e d π π d

23 C - Itegrl Clulus Emple: Guss Qudrture Two-pot ormul I d e π π π.767* [.8 ] e ε.6% [.8 ] Tree-pot ormul I.8 π.8 [.8.6 ] 8 [.8] [.8 e e e π * d.8 π 9 ε.7% ] Multdmesol Itegrls So r, we oetrted o oe-dmesol tegrls. To ompute tegrls multple dmesos, oe ppro s to prse te multple tegrl s repeted oe-dmesol tegrls. But, evetully, we ru to te so-lled urse o dmesolty. Four or more dmesos re omplted d, ote, mprese. Tere re two metods tt work well:.mote Crlo: Bsed o repeted uto evlutos, ot repeted tegrtos usg oe-dmesol metods. Populr lgortm: Mrkov Mote Crlo MCMC, w lude te Metropols-Hstgs lgortm d Gs smplg.. Sprse grds: Bsed o oe dmesol qudrture rule, ut uses reursve omto o uvrte results. 6

24 C - Itegrl Clulus Q: Wt's te tegrl o /d? A: A turl log! 7