CS321. Introduction to Numerical Methods

Transcription

1 CS Itroducto to Numercl Metods Lecture Revew Proessor Ju Zg Deprtmet o Computer Scece Uversty o Ketucky Legto, KY 6 6 Mrc 7,

2 Number Coverso A geerl umber sould be coverted teger prt d rctol prt seprtely Covert teger umber rom bse β < to bsed Covert rctol umber rom bse β < to bsed

3 Repeted Dvso To covert bse teger umber to bse β < umber, we c use repeted dvso lgortm 576 / / / 5 / 5 remder remder remder remder 5 5 So we ve It c be urter coverted to bry umber usg te bry octl tble Note tt te rst dgt you obt s te rst dgt et to te rd pot

4 Iteger Frcto Splt Te teger rcto splt process s used to covert rctol bse umber to bse β umber. Multply te rctol umber by β;. Tke te teger prt s te rst et dgt;. Repet te process o te remg rctol prt Hece, we ve.7.

5 Normlzed Scetc Notto We wrte umber te ormlzed scetc otto orm s Oly mce umbers c be represeted ectly computer To store lotg pot umber usg bts Sg o q eeds bt Iteger m eeds bts Number q eeds bts m q Sgle precso IEEE stdrd lotg pot c7. 5

6 Represetto Errors 6 Te reltve error correct roudg s Te ut roudo error s Flotg pot represetto Note tt Avod loss o sgct dgts rtolzto Tylor epsos or commoly used uctos l l y y t

7 Root Fdg Algortms Bsecto metod, Newto s metod, d sect metod Bsecto metod s slow, but s gurteed to d root Coose two tl pots wt deret sgs, compute te mddle pot d determe two ew pots bsed o te sg o te mddle pot Number o bsecto steps or some tolerce b log b log log 7

8 Newto s Metod ' Newto s metod coverges qudrtclly, wle te bsecto does lerly Newto s metod eeds te rst dervtve vlue t ec terto Itl pot must be sucetly close to te soluto pot Newto s metod c be used to solve system o ler equtos d oler equtos [ ' Were k s te Jcob mtr k k k k ]

9 Sect Metod 9 Sect metod tres to tke dvtge o Newto s metod, but does ot compute te dervtve vlues Te dervtve vlue s ppromted by Te sect metod geertes tertes Covergece rte s betwee tt o Newto s metod d bsecto metod, oly oe uctol evluto s eeded t ec terto Need two tl pots to strt wt, my use bsecto metod to geerte te rst terte '

10 Iterpolto Gve set o ccurte dt, d polyoml ucto o degree ot ger t tt represets tese dt d possble te mssg dt For Lgrge orm, d te crdl uctos Te Lgrge polyoml s Te crdl ucto s costructed s j j l j l p, l j j j j

11 Newto Form Costruct te polyoml step by step wt Note tt terpolto polyomls costructed by Lgrge or Newto orm re te sme. I ct, tere s oly oe uque polyoml o degree ot ger t to terpolte dt set o + Tere re my ger order polyomls tt my stsy te terpolto codtos, too y p or k k k c p p k k k k k k k p y c

12 Dvded Derece ], [ ],, [ ],,, [ ],, [ ], [ ] [ p

13 Dervtve Appromto Sded O ppromto o rst dervtve Hger order ppromtos Tylor epsos Subtrctg tese two equtos, we ve '! '''! "! '! '''! "! ' 5! '''! ' 5 5

14 d Order Appromto Ater droppg te ger order terms, we ve secod order ppromto ormul s Te ledg tructed terms o ppromto sceme s Hece te ppromto s o O. Te ppromto error goes to s st s. Te ect tructo error s ''' 6 ''' ''' 6 ' 5! '''! ' 5

15 5 Rcrdso Etrpolto Rcrdso etrpolto estmtes te vlue o ψ rom some computed vlues o ψ er Multply te d equto by d subtrct t rom te st equto Hece c be computed s ccurte s O ' ' ' '

16 Iterpolto Polyoml We c ppromte te ucto by polyoml p o order, suc tt p To compute, we use te ppromto p Hger order polyomls re voded becuse o oscllto Let p terpolte t two pots, d Te rst dervtve o p s p, p', ' 6

17 7 Trpezod Rule Te tervl [,b] s rst dvded to subtervls [, + ],. Te bsc trpezod rule or te subtervl [, + ] s Te totl re uder te curve s For uorm spcg = b /, te composte trpezod rule s d ; b A P T d ; P T

18 Smpso s Rule Smpso s rule s tree pot umercl tegrto rule usg te mddle pot o te tervl wt deret wegts or ec pot d Te error term o ts ppromto s 9 For some pot ξ, +. Ts sould be compred to te error term o te smple trpezod rule O Smpso s metod c be used dptvely to ree te spcg te subtervl tt yelds lrge error 5 Strtegy d crtero or dptto: usg Smpso s metod o te wole tervl d te o te two l tervls

19 Guss Elmto How to perorm Guss elmto o gve ler system, wy te ïve Guss elmto s ot robust Wt c we do to mke Guss elmto more robust to solve rbtrry ler system Wt s te order o te log operto cout Guss elmto / Wt s trdgol system, wt codto s sucet to gurtee tt tere s o pvotg eeded or solvg trdgol system Wt s te dvtges d dsdvtges o tertve metods versus Guss elmto drect metod, wt re te tertve metod we studed, wt type o ler systems tt tertve metods my be dvtgeous 9

20 Sple Appromtos Wy we eed sple ppromtos Wt re te codtos or vrous degrees o sple, eed to be ble to very tese codtos How c we sy tt sple ppromto s better t polyoml terpolto Wt s turl sple, ow my rbtrry codtos re speced geerl cubc sple order to obt turl sple S t = S t = Gve set o polyomls deed o seprte tervls, gure out cert costts to mke up cubc or gve sple

21 Lest Squres Dt re ccurte, polyoml terpolto does ot mke sese Try to mmze te totl error lest squres sese Tke te prtl dervtves d set tem to zero to obt orml equtos. Guss elmto my be used to solve te orml equtos Lest squres curve ttg my ot pss troug y o te gve dt pots, but t s supposed to represet te true tred o te dt How to coose bss uctos to mke computto eser Wt re te specl polyomls we studed tt c be used s bss ucto or good curve ttg