Numerical Methods for Eng [ENGR 391] [Lyes KADEM 2007] Direct Method; Newton s Divided Difference; Lagrangian Interpolation; Spline Interpolation.

Transcription

1 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 CHAPTER V Itepolto d Regesso Topcs Itepolto Regesso Dect Method; Newto s Dvded Deece; Lgg Itepolto; ple Itepolto Le d o-le Wht s tepolto? A ucto s ote gve ol t dscete pots such s o? How does oe d the vlue o t othe vlue Well cotuous ucto e used to epeset the dt vlues wth pssg though the pot The we c d the vlue o t othe vlue o Ths s clled tepolto O couse lls outsde the ge o o whch the dt s gve t s o loge tepolto ut sted s clled etpolto should we choose? A polol s coo choce o tepoltg ucto ecuse polols e es to o wht kd o ucto - Evlute - Deette d - Itegte s opposed to othe choces such s se o epoetl sees Polol tepolto volves dg polol o ode tht psses though the pots Oe o the ethods s clled the dect ethod o tepolto Othe ethods clude Newto s dvded deece polol ethod d Lgg tepolto ethod 3 3 Itepolto & Regesso 6

2 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 Dect Method The dect ethod o tepolto s sed o the ollowg pcple I we hve '' dt pots t polol o ode '' s gve elow though the dt whee e el costts ce vlues o e gve t vlues o oe c wte equtos The the '' costts c e oud solvg the sulteous le equtos Ah!!! do ou eee pevous couse!!! To d the vlue o t gve vlue o spl susttute the vlue o the polol o But t s ot ecess to use ll the dt pots How does oe the choose the ode o the polol d wht dt pots to use? Ths cocept d the dect ethod o tepolto e est llustted usg eple Eple The upwd veloct o ocket s gve s ucto o te Tle Tle Veloct s ucto o te t [s vt [/s Detee the vlue o the veloct t t6 s usg the dect ethod d st ode polol Detee the vlue o the veloct t t6 s usg dect ethod d thd ode polol tepolto usg dect ethod Itepolto & Regesso 6

3 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 75 v t [s t [s Fgue 5 Veloct vs te dt o the ocket eple 3 Newto s dvded deece tepolto To llustte ths ethod we wll stt wth le d qudtc tepolto the the geel o o the Newto s Dvded Deece Polol ethod wll e peseted 3 Le tepolto Gve t le tepolt though the dt Note tht d ssug le tepolt es: ce t : d t : The so Ad the le tepolt Itepolto & Regesso 6

4 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 Itepolto & Regesso 63 Becoes: 3 Qudtc tepolto Gve d t qudtc tepolt though the dt Note tht d ssue the qudtc tepolt gve At At the At the Hece the qudtc tepolt s gve

5 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 Itepolto & Regesso 6 Fgue 5 Qudtc tepolto 33 Geel Fo o Newto s Dvded Deece Polol I the two pevous cses we oud how le d qudtc tepolto s deved Newto s Dvded Deece polol ethod Let us lze the qudtc polol tepolt oul whee Note tht d e te dvded deeces d e st secod d thd te dvded deeces espectvel Deotg st dvded deece [ the secod dvded deece [ d the thd dvded deece [ [ [

6 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 whee d e clled cketed uctos o the vles [ [ eclosed sque ckets [ We c wte: [ [ [ Ths leds to the geel o o the Newto s dvded deece polol o dt pots s whee [ [ [ M [ [ th whee the deto o the dvded deece s [ [ [ Fo the ove deto t c e see tht the dvded deeces e clculted ecusvel Fo eple o thd ode polol gve d 3 [ [ [ 3 [ [ [ 3 [ [ 3 [ 3 [ 3 3 Itepolto & Regesso 65

7 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 Eple Use the se pevous dt o the upwd veloct o ocket to detee the vlue o the veloct t t6 s usg thd ode polol tepolto usg Newto s Dvded Deece polol Lgg Itepolto Polol tepolto volves dg polol o ode tht psses though the pots Oe o the ethods to d ths polol s clled Lgg Itepolto Lgg tepoltg polol s gve L whee stds o the th ode polol tht ppotes the ucto gve t dt pots s L j j j j d L s weghtg ucto tht cludes poduct o tes wth tes o j otted Eple Use the se pevous dt o the upwd veloct o ocket to detee the vlue o the veloct t t6 s usg thd ode polol tepolto usg thd ode polol tepolto usg Lgg polol tepolto 5 ple Method o Itepolto ple ethod ws toduced to solve oe o the dwcks o the polol tepolto I ct whe the ode ecoes lge cses osclltos ppe the esultg polol Ths ws show Ruge whe he tepolted dt sed o sple ucto o 5 o tevl o [- Fo eple tke s equdsttl spced pots [- d d t these pots s gve Tle Itepolto & Regesso 66

8 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 Tle : equdsttl spced pots [ Fgue55 5 th ode polol vs ect ucto Now though these s pots we c pss th ode polol though the s dt pots Whe plottg the th ode polol d the ogl ucto ou c otce tht the two do ot tch well o e ou wll cosde choosg oe pots the tevl [- to get ette tch ut t dveges eve oe see gue elow I ct Ruge oud tht s the ode o the polol ecoes te the polol dveges the tevl o < < 76 d 76 < < Fgue56 Hghe ode polol tepolto s d de Itepolto & Regesso 67

9 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 5 Le sple tepolto Gve t le sples to the dt Ths spl volves og the cosecutve dt though stght les o the ove dt s gve scedg ode the le sples e gve Fgue57 Le sples Note the tes o the ove ucto e spl slopes etwee d 5 Qudtc ples I these sples qudtc polol ppotes the dt etwee two cosecutve dt pots The sples e gve Itepolto & Regesso 68

10 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 c c c Now how to d the coecets o these qudtc sples? Thee e 3 such coecets c To d 3 ukows we eed 3 equtos d the sulteousl solve the These 3 equtos e oud the ollowg Ech qudtc sple goes though two cosecutve dt pots c c c c c c Ths codto gves equtos s thee e qudtc sples gog though two cosecutve dt pots The st devtves o two qudtc sples e cotuous t the teo pots Fo eple the devtve o the st sple s c The devtve o the secod sple s c Itepolto & Regesso 69

11 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 d the two e equl t gvg ll t the othe teo pots 3 3 ce thee e - teo pots we hve - such equtos Now the totl ue o equtos s 3 equtos We stll the eed oe oe equto We c ssue tht the st sple s le tht s: Ths gves us 3 equtos d 3 ukows These c e solved ue o techques used to solve sulteous le equtos Itepolto & Regesso 7

12 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 Regesso Wht s egesso? Regesso lss gves oto o the eltoshp etwee espose vle d oe o oe depedet vles to the etet tht oto s coted the dt The gol o egesso lss s to epess the espose vle s ucto o the pedcto vles Dult o t d ccuc o cocluso deped o the dt used Hece o-epesettve o popel copled dt esult to poo ts d coclusos Thus o eectve use o egesso lss oe ust Ivestgte the dt collecto pocess Dscove lttos dt collected Restct coclusos ccodgl Oce egesso lss eltoshp s oted t c e used to pedct vlues o the espose vle det vles tht ost ect espose o ve hpotheszed csul odels o the espose The vlue o ech pedcto vle c e ssessed though sttstcl tests o the estted coecets ultples o the pedcto vles 3 Le egesso Le egesso s the ost popul egesso odel I ths odel we wsh to pedct espose to dt pots dt egesso odel gve whee d e the costts o the egesso odel A esue o goodess o t tht s how pedcts the espose vle s the gtude o the esdul ε t ech o the dt pots ε Idell ll the esduls ε e zeo oe hve oud equto whch ll the pots le o the odel Thus zto o the esdul s ojectve o otg egesso coecets The ost popul ethod to ze the esdul s the lest sques ethod whee the esttes o the costts o the odels e chose such tht the su o the squed esduls s zed tht s ze ε Wh ze the su o the sque o the esduls? Wh ot o stce ze the su o the esdul eos o the su o the solute vlues o the esduls? Altetvel costts o the odel c e chose such tht the vege esdul s zeo wthout kg dvdul esduls sll Fo eple let us lze the ollowg tle To epl ths dt stght le egesso odel Itepolto & Regesso 7

13 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 d usg zg ε s cte to d o d we d tht o Fgue 58 Y Fgue58 Regesso cuve o vs dt The su o the esduls ε s show the tle elow pedcted ε - pedcted ε o does ths gve us the sllest eo? It does s ε But t does ot gve uque vlues o the petes o the odel A stght-le o the odel: Y 6 Itepolto & Regesso 7

14 Nuecl Methods o Eg [ENGR 39 [Les KADEM Fgue59 Regesso cuve 6 o vs dt lso kes ε s show the tle elow pedcted ε - pedcted ε ce ths cteo does ot gve uque egesso odel t cot e used o dg the egesso coecets Wh? Becuse we wt to ze ε Deettg ths equto wth espect to d we get ε ε _ Itepolto & Regesso 73

15 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 Puttg these equtos to zeo gve ut ths s possle Theeoe uque vlues o d do ot est You thk tht the eso the zto cteo esduls ccel wth postve esduls o s zg look t the dt gve elow o equto ollowg tle It kes ε ε does ot wok s tht egtve cteo e ette? Let us ε s show the pedcted ε - pedcted ε The vlue o ε lso ests o the stght le odel 6 No othe stght le o ths dt hs ε < Ag we d the egesso coecets e ot uque d hece ths cteo lso cot e used o dg the egesso odel Let us use the lest sques cteo whee we ze ε s clled the su o the sque o the esduls Itepolto & Regesso 7

16 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 Itepolto & Regesso 75 Fgue5 Le egesso o vs dt showg esduls t tpcl pot To d d we ze wth espect to d : gvg Notg tht olvg the ove equtos gves: 3 3 ε

17 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 Itepolto & Regesso 76 Redeg _ we c ewte Nole odels usg lest sques Epoetl odel Gve we c t e to the dt The vles d e the costts o the epoetl odel The esdul t ech dt pot s e E The su o the sque o the esduls s E e To d the costts d o the epoetl odel we ze deettg wth espect to d d equtg the esultg equtos to zeo

18 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 o e e e e e e e e These equtos e ole d d thus ot closed o to e solved s ws the cse o the le egesso I geel tetve ethods ust e used to d vlues o d Howeve ths cse c e wtte eplctl tes o s e e usttutg gves e e e e Ths equto s stll ole equto d c e solved uecl ethods such s secto ethod o sect ethod Gowth odel Gowth odels coo scetc elds hve ee developed d used successull o specc stutos The gowth odels e used to desce how soethg gows wth chges egesso vle ote the te Eples ths ctego clude gowth o populto wth te Gowth odels clude c e whee d c e the costts o the odel At d s The esduls t ech dt pot e E c e The su o the sque o the esduls s Itepolto & Regesso 77

19 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 Itepolto & Regesso 78 E c e To d the costts d c we ze deettg wth espect to d c d equtg the esultg equtos to zeo [ c c c c e e e e [ 3 c c c e e e [ 3 c c c e e e c The t s possle to use the Newto-Rphso ethod to solve the ove set o sulteous ole equtos o d c 3 Polol Models Gve dt pots use lest sques ethod to egess the dt to th ode polol < L L The esdul t ech dt pot s gve E The su o the sque o the esduls s gve E To d the costts o the polol egesso odel we put the devtves wth espect to to zeo tht s

20 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 Itepolto & Regesso 79 Wtg these equtos t o gves The ove sste s solved o Logthc Fuctos The o o the log egesso odels s l β β Ths s le ucto etwee d l d the usul lest sques ethod pples whch s the espose vle d l s the egesso 5 Powe Fuctos The powe ucto equto desces scetc d egeeg pheoe: The ethod o lest sques s ppled to the powe ucto st lezg the dt ssupto s tht s ot kow I the ol ukow s the le elto ests etwee d The lezto o the dt s s ollows: l l l The esultg equto shows le elto etwee l d l

21 Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 Itepolto & Regesso 8 We c put l l w z l the o e we get w z w z w w z w w z ce d c e oud the ogl costts o the odel e e