CS Numercl Alyss Lecture 4 Numercl Itegrto Proessor Ju Zg Deprtmet o Computer Scece Uversty o Ketucky Legto, KY 456 6 Octoer 6, 5
Dete Itegrl A dete tegrl s tervl or tegrto. For ed tegrto tervl, te result s umer A dete tegrl does ot ve tegrto tervl. Te result o dete tegrl tdervtve s clss o uctos Numercl tegrto s or computg dete tegrls Fudmetl Teorem o Clculus: s d s d cos C F' d F F' t dt F F F
Numercl Itegrto
Prtto 4 Te dete tegrl o ucto c e vewed s te re uder curve. Ts pot o vew leds us mes to compute dete tegrl Let P e prtto o te tervl o [,] s We ve sutervls s [, + ]. Let m e te gretest lower oud o oegtve ucto o [, + ] s d M s te lest upper oud o te sme sutervl P : m : sup M
Prtto
Lower d Upper Sums 6 Te lower sums d upper sums o correspodg to te gve prtto P s I we cosder te dete tegrl o oegtve s te re uder te curve, we ve or ll prttos P I s cotuous o [,], te te ove equlty dees te dete tegrl. Te tegrl lso ests s mootoe eter cresg or decresg o [,] ; ; M P U m P L P U d P L ; ;
Upper d Lower Bouds
Rem Itegrle Fuctos I te gretest lower oud equls te lest upper oud or ll prttos o [,],.e., U ; P sup L, P Te s sd to e Rem tegrle Every cotuous ucto deed o closed d ouded tervl o te rel le s Rem tegrle We ve lm were P, P,.., P, re sequece o prttos suc tt te legt o te lrgest sutervl P coverges to s We c costruct ested reed prttos P L ; P P d lm U ; P 8
Computto. eed procedure to evlute. determe prtto ow my sutervls o te tervl [,]. compute m d M o ec sutervl 4. compute te sums L; P d U; P 5. ppromte vlue s oted ; P L P d U ; 6. te error o ts ppromto s ouded y ; P L P U ; 9
Trpezod Rule A strtegy tt s etter t estmtg ot te upper d te lower ouds o te re eet te curve s to use trpezods Te tervl [,] s rst dvded to sutervls [, + ],. A typcl trpezod s te sutervl [, + ] s ts se, d te two vertcl sdes re d +. Te re s gve y te se tmes te verge egt. Te sc trpezod rule or te sutervl [, + ] s Te totl re uder te curve s d d T ; P A [ ]
Trpezod Rule
Uorm Spcg Te legts o te sutervls prtto c e deret. For st computto, uorm prtto o te tervl my e dvtgeous Let e te umer o sutervls, te = / s te uorm tervl spcg. Te odl pots re = +, =,,,. Hece te composte trpezod rule s T ; P [ ] Note tt te ed pot o tervl s te strtg pot o te et tervl. Ts ct c sve lmost l o te computto, d [ ]
Trpezod Rule wt Uorm Spcg
Error Alyss I ests d s cotuous o [,], te error o te composte trpezod rule T s or some ξ, d T " O Proo. We rst prove te result or =, = d =. Tt s " d Ts smpled ormul wll e proved wt te elp o polyoml terpolto Dee polyoml o degree oe tt terpoltes t d p [ ] 4
Error Alyss It ollows tt p d [ [ ] ] Usg te error ormul or te polyoml terpolto, we ve Itegrte t o ot sdes, p "[ ] d p d "[ ] d 5
Error Alyss Usg te Me Vlue Teorem or Itegrls So we ve "[ ] d We te do cge o vrle, d let g t d g" t d [ "[ [ t dt, g' t t "[ s] ], 6 ] ] " '[ d t " t ] 6
Error Alyss 4 7 Te usg te result or te specl cse, Ts s te error ormul or te trpezod rule wt oly oe sutervl. Let [,] e dvded to equl sutervls y pots " ] [ " ] [ ] [ g g g dt t g dt t d ], wt sutervl[,,...,,
Error Alyss 5 8 Let e te tervl legt Sum over ll sutervls to get te composte trpezod rule Note tt =-/, we use Itermedte-Vlue Teorem o Cotuous Fuctos, " ] [ d " " " " ] [ d
Emple Sow tt d " Need to dee F t t We c epd F+ usg Tylor seres d F F F' F" F"'! Te y te Fudmetl Teorem o Clculus, we ve F t = t. Note tt F =, F t= t, F t= t, d so o. 9
We ve Emple We c lso pply te Tylor seres drectly o t s ' " "'! Addg o ot sdes o d multplyg t y /, we ot [ d ' "! ] ' " 4 Sutrctg rom, we lly get d [ ] "
Estmte Grd Spcg Emple. I te composte trpezod rule s used to compute e wt error o t most.5-4, wt s te uorm grd spcg? From te grp o te secod dervtve decresg ucto " 4 d e We d tt " " We eed " 6.5 4 It ollows tt.7. Te umer o sutervls s [/] = 58
Recursve Trpezod Ide Wt c we do te tl prtto o tervl s ot e eoug?
Recursve Trpezod Formul Gve prmeter, dvdg [,] to eqully spced sutervls, we ve Note tt = d = / Notce tt R, c e vewed s dvdg ec sutervl o R,to two equl su sutervls. I we lredy computed R,, ow c we compute R, ceply? ; P T, R
4 Recursve Formul I R,s vlle, R, c e computed s For usg = /. Itl strtg vlue s Te trck s to oly sum te ucto vlues t every oter grd pots Proo. Note tt wt C = [ + ]/ d ] [,, k k R R ] [, R, C R, j C j R
Recursve Formul Hece, we ve R, R, j j Ec term te rst sum tt correspods to eve vlue o de s ccelled y term te secod term. Te l result s oly te odd vlues o de We c use te recursve trpezod ormul to compute sequece o ppromtos to dete tegrl usg te trpezod rule, wtout recomputg te vlues t pots tt ve lredy ee computed te prevous step k [ k ] 5
6 Two Dmesol Itegrto For oe dmesol umercl tegrto o [,], usg uorm spce = / For two dmesol tegrto o ut squre A d ] [ j j j j j A A j A A dy y A dy y A dy d y,,,,,
Romerg Algortm Recursve composte trpezod metod For = / d R, R, [ ] R, Usg Rcrdso etrpolto, we c ve R, j R, j 4 For j d j. Ts s te Romerg lgortm, wc my yeld etter ppromte vlues or lrger j k [ k [ R, j R, j ] j ] 7
Dervg Romerg Algortm Composte trpezod rule o - sutervls wt = / - d te coecets deped o ut ot o Ater oe reemet d replcg wt d wt /, we ve Sutrctg te st equto rom 4 tmes te d equto were or d R, d R, 4 6 4 4 4 4 64 4 4 6 6 6 6 6 4 5 d R, 6 R, R, [ R, R,] 6 8
More Romerg Algortm We could pply te etrpolto de repetedly to get were 6 d R, 4 6 4 5 8 8 R, R, 5 [ R, R,] Ts tme, te tructo error s o st order A ew steps o etrpolto my geerte very ccurte ppromtos Too my etrpoltos my mke te computto tedous 9
Geerl Etrpolto Etrpolto processes c e ppled more geerl cses were te error term c e represeted s wt < α < β < γ, we sow ow te rst term o te error epso s lted. Let Replcg y / yelds Multplyg y α c E c L c L c L
Geerl Etrpolto Cot. Sutrctg te prevous two equtos, we c remove te α term L We c wrte te ew ppromto ormul s L Ts ppromto ormul rses te order o tructo error rom O α to O β wt α < β c Plese red te ook o p. 7 or cocrete emple to sow ow te ppromto ccurcy s mproved usg etrpolto c
Bsc Smpso s Rule Smple trpezod rule uses two pots or ppromtos C we get more ccurte ppromto usg more pots?
Bsc Smpso s Rule A tree pot umercl tegrto rule usg te mddle pot o te tervl s kow s te Smpso s rule wt deret wegts or ec pot d 4 Usg Tylor s epso, we c d te error term o ts ppromto s 5 4 9 For some pot ξ,+. Ts sould e compred to te error term o te smple trpezod rule O It s desrle to sudvde te tervl dptvely so tt reemet s oly plced te re o lrge luctuto o ucto vlue
Bsc Smpso s Rule Comprg composte trpezod rule d Smpso s rule 4
Bsc Smpso s Rule Usg Tylor seres or t, we ve '! "! I we replce te tervl sze y, we ot ' By comg tese two epsos, we get 4 6 6' 4 "' 4! 4 4 " F"' 4! " 4 4 4 4 "' 4! 4 4 5
Bsc Smpso s Rule O te oter d, dee We epd F+ s F t dt 4 F F F' F" F"' Note tt F =, F=, F =, F =, we ve d 4 ' From te prevous pge, we ve [ 4 ] " 4 4 "' 4 4 5 "' 54! ' 4! 5 5 F 5! 4 5 5 4 4 " 5 6
7 Bsc Smpso s Rule Sutrctg te prevous two equtos, we ve We ve te Smpso s rule s Te error term o te Smpso s rule s d 4 5 9 ] 4 [ 9 4 5 d ] 4 [ 6
Adptve Smpso s Algortm Reduce te sze o te tervls to get more ccurte ppromtos 8
Adptve Smpso s Algortm Gve tervl [,], we c use te sc Smpso s rule to compute ppromto to te tegrl s were te ppromto prt s I d S, E, S, 4 6 d te error term s E, 5 4 9 For smplcty, we ssume 4 rems costt o,. Let =, we ve I S E For te rst step ppromto wt S S, 9
Adptve Smpso II Ad E 9 5 4 We te sudvde te tervl [,] d pply te sc Smpso s rule o te sutervls [,c] d [c,] respectvely. We ve ew ppromto o [,] s te sum o two seprte ppromtos were c = + / wt d E 9 S / 5 I S Ts s certly etter ppromto sce te sutervls re smller t te orgl tervl S, c 4 9 E S c, / 5 4 6 E
Adptve Smpso III Sutrctg te two ppromtos yelds Hece te umercl tegrto c e I S S S E E Te error term s te computle d c e used or uldg te dptve process S S 5 I ts test sows tt te error s lrger t ε, te tervl [,] c e splt to two sutervls [,c] d [c,] wt c = + /. Te prevously descred procedure s replced y ε/ to mke sure tt te error sum s smller t ε S E 5 5E S S 4
Adptve Smpso s Algortm Ree te tervls t te plces were te ucto cges quckly 4
Adptve Smpso IV Numercl tegrto o sutervls c I d d d c I L I Let S e tesumo S L o[,c] d S R o[ c,],weve I S I I S S L R L R R I L S L I R S R I we wt to ve 5 S L I S L S 5 S R S R It s more t eoug to ve d S 5 L S L S 5 R S R 4
Adptve Smpso s Algortm Oe decso to mke s to coose were to ree te tervl 44
45 Computtol Procedure Te tervl [,] s dvded to our sutervls o equl legt. Two Smpso ppromtos re computed usg two doule wdt sutervls d our sgle wdt sutervls I S -S 5ε, we ve doe, d set Oterwse te tervl [,] s dvded l d te recursve procedure s ppled o te two sutervls [,c] d [c,], utl eter te error tolerce s stsed or te mmum umer o sudvsos s reced 4 4 4 6 c c c S S 6 5 S S S
Adptve Smpso s Algortm Wc su tervl or ot to dvde or reemet 46
Guss Qudrture Formuls A geerl umercl tegrto ormul s It suces to kow te odes,,, d te wegts A, A,, A. For mportt specl uctos, tey re lsted some reerece ooks Suppose set o odes s gve, ow to d te wegts. Ts c e doe usg Lgrge terpolto polyoml s Wt d A A A p l jo, j I p s good ppromte to, we tcpte ppromte to d l j j p d s good 47
We tegrte over p s were we c compute d p d Guss Qudrture l l A d Note tt te polyoml terpolto s ect or polyoml o degree t most. It ollows tt te tegrto wll e ect or suc polyomls I te odes c e cose creully, t s possle to crese te order o polyoml wt te ect tegrto remrkly. Ts ws dscussed y Krl Guss d A 48
49 A Emple Determe qudrture ormul we te tervl s [,] d te odes re,, d. We rst eed to compute te crdl uctos Te wegts re computed y tegrtg tese uctos, l j j j 8 d d l A, l j j j, l j j j
A Emple Smlrly, we ve A l d d A l d d So te qudrture ormul deed o te tervl [,] d usg te ode,,, s d 8 4 8 It c e vered tt ts ormul gves ect vlues or te tree uctos,, 8 4 5
Guss Qudrture Teorem Let q e otrvl polyoml o degree + suc tt q d k Let,,, e zeroes o q. Te we dee te ormul Wt tese s s odes, te ppromto wll e ect or ll polyomls o degree t most +. All tese odes le te ope tervl, We c rst gure out te qudrture or k d A t dt Te use te trsormto t = [ ]/ or Guss qudrture o te geerl tervl [,], A A t l d 5
Te trsormed tegrl s d Guss Teorem Proo Proo o Guss Qudrture Teorem: Let e y polyoml o degree t most +. Dvdg y q wt quotet p d remder r p q r Bot p d r re polyomls o degree t most t dt By ypotess, we ve q p d Sce re roots o q, we ve p q r r 5
Guss Teorem Proo II Sce te degree o r s t most, te tegrto r d s ect d r d p q d A r Guss Qudrture Teorem gurtees g ccurcy umercl tegrto wt just ew odes. However, dg tese odes s ot esy tsk. Te roots o Legedre polyomls re te odes or Guss qudrture o te tervl [-,]. Wt q =, q =, we ve or q q r d A q 5