CS321. Numerical Analysis
|
|
- Charles Simpson
- 6 years ago
- Views:
Transcription
1 CS Numercl Alyss Lecture 4 Numercl Itegrto Proessor Ju Zg Deprtmet o Computer Scece Uversty o Ketucky Legto, KY Octoer 6, 5
2 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
3 Numercl Itegrto
4 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
5 Prtto
6 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 ; ;
7 Upper d Lower Bouds
8 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
9 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
10 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 [ ]
11 Trpezod Rule
12 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 [ ]
13 Trpezod Rule wt Uorm Spcg
14 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
15 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
16 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
17 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[,,...,,
18 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
19 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
20 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 [ ] "
21 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 " It ollows tt.7. Te umer o sutervls s [/] = 58
22 Recursve Trpezod Ide Wt c we do te tl prtto o tervl s ot e eoug?
23 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
24 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
25 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
26 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,,,,,
27 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
28 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, d R, 6 R, R, [ R, R,] 6 8
29 More Romerg Algortm We could pply te etrpolto de repetedly to get were 6 d R, 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
30 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
31 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
32 Bsc Smpso s Rule Smple trpezod rule uses two pots or ppromtos C we get more ccurte ppromto usg more pots?
33 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 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
34 Bsc Smpso s Rule Comprg composte trpezod rule d Smpso s rule 4
35 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 5
36 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 "' "' 54! ' 4! 5 5 F 5! " 5 6
37 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 [ d ] 4 [ 6
38 Adptve Smpso s Algortm Reduce te sze o te tervls to get more ccurte ppromtos 8
39 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, For smplcty, we ssume 4 rems costt o,. Let =, we ve I S E For te rst step ppromto wt S S, 9
40 Adptve Smpso II Ad E 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, / E
41 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
42 Adptve Smpso s Algortm Ree te tervls t te plces were te ucto cges quckly 4
43 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
44 Adptve Smpso s Algortm Oe decso to mke s to coose were to ree te tervl 44
45 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 c c c S S 6 5 S S S
46 Adptve Smpso s Algortm Wc su tervl or ot to dvde or reemet 46
47 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
48 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 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
50 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 It c e vered tt ts ormul gves ect vlues or te tree uctos,, 8 4 5
51 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
52 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
53 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
CS321. Introduction to Numerical Methods
CS Itroducto to Numercl Metods Lecture Revew Proessor Ju Zg Deprtmet o Computer Scece Uversty o Ketucky Legto, KY 6 6 Mrc 7, Number Coverso A geerl umber sould be coverted teger prt d rctol prt seprtely
More informationMathematically, integration is just finding the area under a curve from one point to another. It is b
Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] CHAPTER VI Numerl Itegrto Tops - Rem sums - Trpezodl rule - Smpso s rule - Rrdso s etrpolto - Guss qudrture rule Mtemtlly, tegrto s just dg te re uder urve rom
More informationChapter 12-b Integral Calculus - Extra
C - Itegrl Clulus Cpter - Itegrl Clulus - Etr Is Newto Toms Smpso BONUS Itroduto to Numerl Itegrto C - Itegrl Clulus Numerl Itegrto Ide s to do tegrl smll prts, lke te wy we preseted tegrto: summto. Numerl
More informationIn Calculus I you learned an approximation method using a Riemann sum. Recall that the Riemann sum is
Mth Sprg 08 L Approxmtg Dete Itegrls I Itroducto We hve studed severl methods tht llow us to d the exct vlues o dete tegrls However, there re some cses whch t s ot possle to evlute dete tegrl exctly I
More informationChapter Simpson s 1/3 Rule of Integration. ( x)
Cpter 7. Smpso s / Rule o Itegrto Ater redg ts pter, you sould e le to. derve te ormul or Smpso s / rule o tegrto,. use Smpso s / rule t to solve tegrls,. develop te ormul or multple-segmet Smpso s / rule
More informationNumerical differentiation (Finite difference)
MA umercl Alyss Week 4 To tk d wrte portle progrms we eed strct model of ter computtol evromet. Ftful models do est ut tey revel tt evromet to e too dverse forcg portle progrmmers to lot eve te smplest
More informationNumerical Differentiation and Integration
Numerl Deretto d Itegrto Overvew Numerl Deretto Newto-Cotes Itegrto Formuls Trpezodl rule Smpso s Rules Guss Qudrture Cheyshev s ormul Numerl Deretto Forwrd te dvded deree Bkwrd te dvded deree Ceter te
More informationMTH 146 Class 7 Notes
7.7- Approxmte Itegrto Motvto: MTH 46 Clss 7 Notes I secto 7.5 we lered tht some defte tegrls, lke x e dx, cot e wrtte terms of elemetry fuctos. So, good questo to sk would e: How c oe clculte somethg
More informationArea and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]
Are d the Defte Itegrl 1 Are uder Curve We wt to fd the re uder f (x) o [, ] y f (x) x The Prtto We eg y prttog the tervl [, ] to smller su-tervls x 0 x 1 x x - x -1 x 1 The Bsc Ide We the crete rectgles
More informationthis is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]
Atervtves The Itegrl Atervtves Ojectve: Use efte tegrl otto for tervtves. Use sc tegrto rules to f tervtves. Aother mportt questo clculus s gve ervtve f the fucto tht t cme from. Ths s the process kow
More informationIntegration by Parts for D K
Itertol OPEN ACCESS Jourl Of Moder Egeerg Reserc IJMER Itegrto y Prts for D K Itegrl T K Gr, S Ry 2 Deprtmet of Mtemtcs, Rgutpur College, Rgutpur-72333, Purul, West Begl, Id 2 Deprtmet of Mtemtcs, Ss Bv,
More informationDifferential Entropy 吳家麟教授
Deretl Etropy 吳家麟教授 Deto Let be rdom vrble wt cumultve dstrbuto ucto I F s cotuous te r.v. s sd to be cotuous. Let = F we te dervtve s deed. I te s clled te pd or. Te set were > 0 s clled te support set
More informationunder the curve in the first quadrant.
NOTES 5: INTEGRALS Nme: Dte: Perod: LESSON 5. AREAS AND DISTANCES Are uder the curve Are uder f( ), ove the -s, o the dom., Prctce Prolems:. f ( ). Fd the re uder the fucto, ove the - s, etwee,.. f ( )
More information12 Iterative Methods. Linear Systems: Gauss-Seidel Nonlinear Systems Case Study: Chemical Reactions
HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 Itertve Methods er Systems: Guss-Sedel Noler Systems Cse Study: Chemcl Rectos Itertve or ppromte methods or systems o equtos cosst o guessg vlue d the
More information14.2 Line Integrals. determines a partition P of the curve by points Pi ( xi, y
4. Le Itegrls I ths secto we defe tegrl tht s smlr to sgle tegrl except tht sted of tegrtg over tervl [ ] we tegrte over curve. Such tegrls re clled le tegrls lthough curve tegrls would e etter termology.
More informationChapter 2 Intro to Math Techniques for Quantum Mechanics
Wter 3 Chem 356: Itroductory Qutum Mechcs Chpter Itro to Mth Techques for Qutum Mechcs... Itro to dfferetl equtos... Boudry Codtos... 5 Prtl dfferetl equtos d seprto of vrbles... 5 Itroducto to Sttstcs...
More information4. Runge-Kutta Formula For Differential Equations. A. Euler Formula B. Runge-Kutta Formula C. An Example for Fourth-Order Runge-Kutta Formula
NCTU Deprme o Elecrcl d Compuer Egeerg Seor Course By Pro. Yo-Pg Ce. Ruge-Ku Formul For Derel Equos A. Euler Formul B. Ruge-Ku Formul C. A Emple or Four-Order Ruge-Ku Formul
More informationUNIVERSITI KEBANGSAAN MALAYSIA PEPERIKSAAN AKHIR SEMESTER I SESI AKADEMIK 2007/2008 IJAZAH SARJANAMUDA DENGAN KEPUJIAN NOVEMBER 2007 MASA : 3 JAM
UNIVERSITI KEBANGSAAN MALAYSIA PEPERIKSAAN AKHIR SEMESTER I SESI AKADEMIK 7/8 IJAZAH SARJANAMUDA DENGAN KEPUJIAN NOVEMBER 7 MASA : 3 JAM KOD KURSUS : KKKQ33/KKKF33 TAJUK : PENGIRAAN BERANGKA ARAHAN :.
More informationStrategies for the AP Calculus Exam
Strteges for the AP Clculus Em Strteges for the AP Clculus Em Strtegy : Kow Your Stuff Ths my seem ovous ut t ees to e metoe. No mout of cochg wll help you o the em f you o t kow the mterl. Here s lst
More informationICS141: Discrete Mathematics for Computer Science I
Uversty o Hw ICS: Dscrete Mthemtcs or Computer Scece I Dept. Iormto & Computer Sc., Uversty o Hw J Stelovsy bsed o sldes by Dr. Be d Dr. Stll Orgls by Dr. M. P. Fr d Dr. J.L. Gross Provded by McGrw-Hll
More informationCalculating the Values of Multiple Integrals by Replacing the Integrands Interpolation by Interpolation Polynomial
Jour o omputtos & Modeg vo o -5 ISSN: 79-75 prt 79-5 oe Scepress Ltd cutg te Vues o Mutpe Itegrs by Repcg te Itegrds Iterpoto by Iterpoto Poyom S Nzrov d bduzzov bstrct Te or des t te costructo o mutdmeso
More informationRoberto s Notes on Integral Calculus Chapter 4: Definite integrals and the FTC Section 2. Riemann sums
Roerto s Notes o Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 2 Rem sums Wht you eed to kow lredy: The defto of re for rectgle. Rememer tht our curret prolem s how to compute the re of ple rego
More information4. Runge-Kutta Formula For Differential Equations
NCTU Deprme o Elecrcl d Compuer Egeerg 5 Sprg Course by Pro. Yo-Pg Ce. Ruge-Ku Formul For Derel Equos To solve e derel equos umerclly e mos useul ormul s clled Ruge-Ku ormul
More informationChapter Simpson s 1/3 Rule of Integration. ( x)
Cper 7. Smpso s / Rule o Iegro Aer redg s per, you sould e le o. derve e ormul or Smpso s / rule o egro,. use Smpso s / rule o solve egrls,. develop e ormul or mulple-segme Smpso s / rule o egro,. use
More informationDATA FITTING. Intensive Computation 2013/2014. Annalisa Massini
DATA FITTING Itesve Computto 3/4 Als Mss Dt fttg Dt fttg cocers the problem of fttg dscrete dt to obt termedte estmtes. There re two geerl pproches two curve fttg: Iterpolto Dt s ver precse. The strteg
More informationITERATIVE METHODS FOR SOLVING SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS
Numercl Alyss for Egeers Germ Jord Uversty ITERATIVE METHODS FOR SOLVING SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS Numercl soluto of lrge systems of ler lgerc equtos usg drect methods such s Mtr Iverse, Guss
More informationOutline. Finite Difference Grids. Numerical Analysis. Finite Difference Grids II. Finite Difference Grids III
Itrodcto to Nmercl Alyss Mrc, 9 Nmercl Metods or PDEs Lrry Cretto Meccl Egeerg 5B Semr Egeerg Alyss Mrc, 9 Otle Revew mdterm soltos Revew bsc mterl o mercl clcls Expressos or dervtves, error d error order
More informationThe z-transform. LTI System description. Prof. Siripong Potisuk
The -Trsform Prof. Srpog Potsuk LTI System descrpto Prevous bss fucto: ut smple or DT mpulse The put sequece s represeted s ler combto of shfted DT mpulses. The respose s gve by covoluto sum of the put
More information. The set of these sums. be a partition of [ ab, ]. Consider the sum f( x) f( x 1)
Chapter 7 Fuctos o Bouded Varato. Subject: Real Aalyss Level: M.Sc. Source: Syed Gul Shah (Charma, Departmet o Mathematcs, US Sargodha Collected & Composed by: Atq ur Rehma (atq@mathcty.org, http://www.mathcty.org
More informationFORMULAE FOR FINITE DIFFERENCE APPROXIMATIONS, QUADRATURE AND LINEAR MULTISTEP METHODS
Jourl o Mtemtcl Sceces: Advces d Alctos Volume Number - Pges -9 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS QUADRATURE AND LINEAR MULTISTEP METHDS RAMESH KUMAR MUTHUMALAI Dertmet o Mtemtcs D G Vsv College
More informationThe definite Riemann integral
Roberto s Notes o Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 4 The defte Rem tegrl Wht you eed to kow lredy: How to ppromte the re uder curve by usg Rem sums. Wht you c ler here: How to use
More informationi+1 by A and imposes Ax
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 09.9 NUMERICAL FLUID MECHANICS FALL 009 Mody, October 9, 009 QUIZ : SOLUTIONS Notes: ) Multple solutos
More informationMATLAB PROGRAM FOR THE NUMERICAL SOLUTION OF DUHAMEL CONVOLUTION INTEGRAL
Bullet of te Trslv Uversty of Brşov Vol 5 (54-1 Seres 1: Specl Issue No 1 MATLAB PROGRAM FOR THE NUMERICAL SOLUTION OF DUHAMEL CONVOLUTION INTEGRAL M BOTIŞ 1 Astrct: I te ler lyss of structures troug modl
More informationCS473-Algorithms I. Lecture 3. Solving Recurrences. Cevdet Aykanat - Bilkent University Computer Engineering Department
CS473-Algorthms I Lecture 3 Solvg Recurreces Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet Solvg Recurreces The lyss of merge sort Lecture requred us to solve recurrece. Recurreces re lke solvg
More informationLecture 3-4 Solutions of System of Linear Equations
Lecture - Solutos of System of Ler Equtos Numerc Ler Alger Revew of vectorsd mtrces System of Ler Equtos Guss Elmto (drect solver) LU Decomposto Guss-Sedel method (tertve solver) VECTORS,,, colum vector
More informationUNIT I. Definition and existence of Riemann-Stieltjes
1 UNIT I Defto d exstece of Rem-Steltjes Itroducto: The reder wll recll from elemetry clculus tht to fd the re of the rego uder the grph of postve fucto f defed o [, ], we sudvde the tervl [, ] to fte
More informationSome Unbiased Classes of Estimators of Finite Population Mean
Itertol Jourl O Mtemtcs Ad ttstcs Iveto (IJMI) E-IN: 3 4767 P-IN: 3-4759 Www.Ijms.Org Volume Issue 09 etember. 04 PP-3-37 ome Ubsed lsses o Estmtors o Fte Poulto Me Prvee Kumr Msr d s Bus. Dertmet o ttstcs,
More informationSpectrum Estimation by Several Interpolation Methods
IJCSNS Itertol Jourl o Computer Scece d Network Securty VOL.6 No.A Februry 006 05 Spectrum Estmto by Severl Iterpolto Metods Mbu Isr Oym Ntol College o Tecology Oym-S Tocg 33-0806 JAPAN Summry I ts pper
More informationENGI 4430 Numerical Integration Page 5-01
ENGI 443 Numercal Itegrato Page 5-5. Numercal Itegrato I some o our prevous work, (most otaly the evaluato o arc legth), t has ee dcult or mpossle to d the dete tegral. Varous symolc algera ad calculus
More informationAdvanced Algorithmic Problem Solving Le 3 Arithmetic. Fredrik Heintz Dept of Computer and Information Science Linköping University
Advced Algorthmc Prolem Solvg Le Arthmetc Fredrk Hetz Dept of Computer d Iformto Scece Lköpg Uversty Overvew Arthmetc Iteger multplcto Krtsu s lgorthm Multplcto of polyomls Fst Fourer Trsform Systems of
More informationNumerical Methods. Lecture 5. Numerical integration. dr hab. inż. Katarzyna Zakrzewska, prof. AGH. Numerical Methods lecture 5 1
Numeril Methods Leture 5. Numeril itegrtio dr h. iż. Ktrzy Zkrzewsk, pro. AGH Numeril Methods leture 5 Outlie Trpezoidl rule Multi-segmet trpezoidl rule Rihrdso etrpoltio Romerg's method Simpso's rule
More informationAsymptotic Dominance Problems. is not constant but for n 0, f ( n) 11. 0, so that for n N f
Asymptotc Domce Prolems Dsply ucto : N R tht s Ο( ) ut s ot costt 0 = 0 The ucto ( ) = > 0 s ot costt ut or 0, ( ) Dee the relto " " o uctos rom N to R y g d oly = Ο( g) Prove tht s relexve d trstve (Recll:
More informationChapter 1. Infinite Sequences and Series. 1.1 Sequences. A sequence is a set of numbers written in a definite order
hpter Ite Sequeces d Seres. Sequeces A sequece s set o umers wrtte dete order,,,... The umer s clled the rst term, s clled the secod term, d geerl s th clled the term. Deto.. The sequece {,,...} s usull
More informationChapter Gauss-Seidel Method
Chpter 04.08 Guss-Sedel Method After redg ths hpter, you should be ble to:. solve set of equtos usg the Guss-Sedel method,. reogze the dvtges d ptflls of the Guss-Sedel method, d. determe uder wht odtos
More informationМИНИСТЕРСТВО ОБРАЗОВАНИЯ И НАУКИ РОССИЙСКОЙ ФЕДЕРАЦИИ. Численные методы. Учебно-методическое пособие
МИНИСТЕРСТВО ОБРАЗОВАНИЯ И НАУКИ РОССИЙСКОЙ ФЕДЕРАЦИИ Нижегородский государственный университет им. Н.И. Лобачевского Численные методы К.А.Баркалов Учебно-методическое пособие Рекомендовано методической
More informationA Technique for Constructing Odd-order Magic Squares Using Basic Latin Squares
Itertol Jourl of Scetfc d Reserch Publctos, Volume, Issue, My 0 ISSN 0- A Techque for Costructg Odd-order Mgc Squres Usg Bsc Lt Squres Tomb I. Deprtmet of Mthemtcs, Mpur Uversty, Imphl, Mpur (INDIA) tombrom@gml.com
More information3/20/2013. Splines There are cases where polynomial interpolation is bad overshoot oscillations. Examplef x. Interpolation at -4,-3,-2,-1,0,1,2,3,4
// Sples There re ses where polyoml terpolto s d overshoot oslltos Emple l s Iterpolto t -,-,-,-,,,,,.... - - - Ide ehd sples use lower order polyomls to oet susets o dt pots mke oetos etwee djet sples
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2
Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit
More informationSequences and summations
Lecture 0 Sequeces d summtos Istructor: Kgl Km CSE) E-ml: kkm0@kokuk.c.kr Tel. : 0-0-9 Room : New Mleum Bldg. 0 Lb : New Egeerg Bldg. 0 All sldes re bsed o CS Dscrete Mthemtcs for Computer Scece course
More informationINTRODUCTION ( ) 1. Errors
INTRODUCTION Numercl lyss volves the study, developmet d lyss of lgorthms for obtg umercl solutos to vrous mthemtcl problems. Frequetly umercl lyss s clled the mthemtcs of scetfc computg. Numercl lyss
More informationModule 2: Introduction to Numerical Analysis
CY00 Itroducto to Computtol Chemtr Autum 00-0 Module : Itroducto to umercl Al Am of the preet module. Itroducto to c umercl l. Developg mple progrm to mplemet the umercl method opc of teret. Iterpolto:
More informationComplex Variables. Chapter 19 Series and Residues. March 26, 2013 Lecturer: Shih-Yuan Chen
omplex Vrble hpter 9 Sere d Redue Mrch 6, Lecturer: Shh-Yu he Except where otherwe oted, cotet lceed uder BY-N-SA. TW Lcee. otet Sequece & ere Tylor ere Luret ere Zero & pole Redue & redue theorem Evluto
More informationSOME REMARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOMIAL ASYMPTOTE
D I D A C T I C S O F A T H E A T I C S No (4) 3 SOE REARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOIAL ASYPTOTE Tdeusz Jszk Abstct I the techg o clculus, we cosde hozotl d slt symptote I ths ppe the
More information19 22 Evaluate the integral by interpreting it in terms of areas. (1 x ) dx. y Write the given sum or difference as a single integral in
SECTION. THE DEFINITE INTEGRAL. THE DEFINITE INTEGRAL A Clck here for swers. S Clck here for solutos. Use the Mdpot Rule wth the gve vlue of to pproxmte the tegrl. Roud the swer to four decml plces. 9
More informationNumerical Integration - (4.3)
Numericl Itegrtio - (.). Te Degree of Accurcy of Qudrture Formul: Te degree of ccurcy of qudrture formul Qf is te lrgest positive iteger suc tt x k dx Qx k, k,,,...,. Exmple fxdx 9 f f,,. Fid te degree
More information20 23 Evaluate the integral by interpreting it in terms of areas. (1 x ) dx. y 2 2
SECTION 5. THE DEFINITE INTEGRAL 5. THE DEFINITE INTEGRAL A Clck here for swers. S Clck here for solutos. 7 Use the Mdpot Rule wth the gve vlue of to pproxmte the tegrl. Roud the swer to four decml plces.
More informationNumerical Integration
Numericl tegrtio Newto-Cotes Numericl tegrtio Scheme Replce complicted uctio or tulted dt with some pproimtig uctio tht is esy to itegrte d d 3-7 Roerto Muscedere The itegrtio o some uctios c e very esy
More informationIntroduction to Numerical Integration Part II
Introducton to umercl Integrton Prt II CS 75/Mth 75 Brn T. Smth, UM, CS Dept. Sprng, 998 4/9/998 qud_ Intro to Gussn Qudrture s eore, the generl tretment chnges the ntegrton prolem to ndng the ntegrl w
More informationSUM PROPERTIES FOR THE K-LUCAS NUMBERS WITH ARITHMETIC INDEXES
Avlble ole t http://sc.org J. Mth. Comput. Sc. 4 (04) No. 05-7 ISSN: 97-507 SUM PROPERTIES OR THE K-UCAS NUMBERS WITH ARITHMETIC INDEXES BIJENDRA SINGH POOJA BHADOURIA AND OMPRAKASH SIKHWA * School of
More informationThe proof is a straightforward probabilistic extension of Palfrey and Rosenthal (1983). Note that in our case we assume N
PROOFS OF APPDIX A Proo o PROPOSITIO A pure strtey Byesn-s equlbr n te PUprtcpton me wtout lled oters: Te proo s strtorwrd blstc etenson o Plrey nd Rosentl 983 ote tt n our cse we ssume A B : It s esy
More informationApproximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
More informationChapter Runge-Kutta 2nd Order Method for Ordinary Differential Equations
Cter. Runge-Kutt nd Order Metod or Ordnr Derentl Eutons Ater redng ts cter ou sould be ble to:. understnd te Runge-Kutt nd order metod or ordnr derentl eutons nd ow to use t to solve roblems. Wt s te Runge-Kutt
More informationNumerical Analysis Topic 4: Least Squares Curve Fitting
Numerl Alss Top 4: Lest Squres Curve Fttg Red Chpter 7 of the tetook Alss_Numerk Motvto Gve set of epermetl dt: 3 5. 5.9 6.3 The reltoshp etwee d m ot e ler. Fd futo f tht est ft the dt 3 Alss_Numerk Motvto
More informationBasics of Information Theory: Markku Juntti. Basic concepts and tools 1 Introduction 2 Entropy, relative entropy and mutual information
: Markku Jutt Overvew Te basc cocepts o ormato teory lke etropy mutual ormato ad EP are eeralzed or cotuous-valued radom varables by troduc deretal etropy ource Te materal s maly based o Capter 9 o te
More informationSimpson s 1/3 rd Rule of Integration
Simpso s 1/3 rd Rule o Itegrtio Mjor: All Egieerig Mjors Authors: Autr Kw, Chrlie Brker Trsormig Numericl Methods Eductio or STEM Udergrdutes 1/10/010 1 Simpso s 1/3 rd Rule o Itegrtio Wht is Itegrtio?
More informationChapter Trapezoidal Rule of Integration
Cper 7 Trpezodl Rule o Iegro Aer redg s per, you sould e le o: derve e rpezodl rule o egro, use e rpezodl rule o egro o solve prolems, derve e mulple-segme rpezodl rule o egro, 4 use e mulple-segme rpezodl
More informationCHAPTER 6 CURVE FITTINGS
CHAPTER 6 CURVE FITTINGS Chpter 6 : TOPIC COVERS CURVE FITTINGS Lest-Squre Regresso - Ler Regresso - Poloml Regresso Iterpolto - Newto s Dvded-Derece Iterpoltg Polomls - Lgrge Iterpoltg Polomls - Sple
More information6.6 Moments and Centers of Mass
th 8 www.tetodre.co 6.6 oets d Ceters of ss Our ojectve here s to fd the pot P o whch th plte of gve shpe lces horzotll. Ths pot s clled the ceter of ss ( or ceter of grvt ) of the plte.. We frst cosder
More informationLevel-2 BLAS. Matrix-Vector operations with O(n 2 ) operations (sequentially) BLAS-Notation: S --- single precision G E general matrix M V --- vector
evel-2 BS trx-vector opertos wth 2 opertos sequetlly BS-Notto: S --- sgle precso G E geerl mtrx V --- vector defes SGEV, mtrx-vector product: r y r α x β r y ther evel-2 BS: Solvg trgulr system x wth trgulr
More informationRendering Equation. Linear equation Spatial homogeneous Both ray tracing and radiosity can be considered special case of this general eq.
Rederg quto Ler equto Sptl homogeeous oth ry trcg d rdosty c be cosdered specl cse of ths geerl eq. Relty ctul photogrph Rdosty Mus Rdosty Rederg quls the dfferece or error mge http://www.grphcs.corell.edu/ole/box/compre.html
More informationMu Sequences/Series Solutions National Convention 2014
Mu Sequeces/Seres Solutos Natoal Coveto 04 C 6 E A 6C A 6 B B 7 A D 7 D C 7 A B 8 A B 8 A C 8 E 4 B 9 B 4 E 9 B 4 C 9 E C 0 A A 0 D B 0 C C Usg basc propertes of arthmetc sequeces, we fd a ad bm m We eed
More informationChapter 2 Intro to Math Techniques for Quantum Mechanics
Fll 4 Chem 356: Itroductory Qutum Mechcs Chpter Itro to Mth Techques for Qutum Mechcs... Itro to dfferetl equtos... Boudry Codtos... 5 Prtl dfferetl equtos d seprto of vrbles... 5 Itroducto to Sttstcs...
More informationOn Several Inequalities Deduced Using a Power Series Approach
It J Cotemp Mth Sceces, Vol 8, 203, o 8, 855-864 HIKARI Ltd, wwwm-hrcom http://dxdoorg/02988/jcms2033896 O Severl Iequltes Deduced Usg Power Seres Approch Lored Curdru Deprtmet of Mthemtcs Poltehc Uversty
More informationME 501A Seminar in Engineering Analysis Page 1
Mtr Trsformtos usg Egevectors September 8, Mtr Trsformtos Usg Egevectors Lrry Cretto Mechcl Egeerg A Semr Egeerg Alyss September 8, Outle Revew lst lecture Trsformtos wth mtr of egevectors: = - A ermt
More informationAvailable online through
Avlble ole through wwwmfo FIXED POINTS FOR NON-SELF MAPPINGS ON CONEX ECTOR METRIC SPACES Susht Kumr Moht* Deprtmet of Mthemtcs West Begl Stte Uverst Brst 4 PrgsNorth) Kolt 76 West Begl Id E-ml: smwbes@yhoo
More informationINTERPOLATION(2) ELM1222 Numerical Analysis. ELM1222 Numerical Analysis Dr Muharrem Mercimek
ELM Numerl Alss Dr Murrem Merme INTEROLATION ELM Numerl Alss Some of te otets re dopted from Luree V. Fusett Appled Numerl Alss usg MATLAB. rete Hll I. 999 ELM Numerl Alss Dr Murrem Merme Tod s leture
More informationTopic 6b Finite Difference Approximations
/8/8 Course Instructor Dr. Rymond C. Rump Oice: A 7 Pone: (95) 747 6958 E Mil: rcrump@utep.edu Topic 6b Finite Dierence Approximtions EE 486/5 Computtionl Metods in EE Outline Wt re inite dierence pproximtions?
More informationPubH 7405: REGRESSION ANALYSIS REGRESSION IN MATRIX TERMS
PubH 745: REGRESSION ANALSIS REGRESSION IN MATRIX TERMS A mtr s dspl of umbers or umercl quttes ld out rectgulr rr of rows d colums. The rr, or two-w tble of umbers, could be rectgulr or squre could be
More informationA Brief Introduction to Olympiad Inequalities
Ev Che Aprl 0, 04 The gol of ths documet s to provde eser troducto to olympd equltes th the stdrd exposto Olympd Iequltes, by Thoms Mldorf I ws motvted to wrte t by feelg gulty for gettg free 7 s o problems
More informationUNIVERSITY OF BRISTOL. Examination for the Degrees of B.Sc. and M.Sci. (Level C/4) ANALYSIS 1B, SOLUTIONS MATH (Paper Code MATH-10006)
UNIVERSITY OF BRISTOL Exmitio for the Degrees of B.Sc. d M.Sci. (Level C/4) ANALYSIS B, SOLUTIONS MATH 6 (Pper Code MATH-6) My/Jue 25, hours 3 miutes This pper cotis two sectios, A d B. Plese use seprte
More informationFast Fourier Transform 1) Legendre s Interpolation 2) Vandermonde Matrix 3) Roots of Unity 4) Polynomial Evaluation
Algorithm Desig d Alsis Victor Admchi CS 5-45 Sprig 4 Lecture 3 J 7, 4 Cregie Mello Uiversit Outlie Fst Fourier Trsform ) Legedre s Iterpoltio ) Vdermode Mtri 3) Roots of Uit 4) Polomil Evlutio Guss (777
More informationMATRIX AND VECTOR NORMS
Numercl lyss for Egeers Germ Jord Uversty MTRIX ND VECTOR NORMS vector orm s mesure of the mgtude of vector. Smlrly, mtr orm s mesure of the mgtude of mtr. For sgle comoet etty such s ordry umers, the
More informationAsymptotic Formulas Composite Numbers II
Iteratoal Matematcal Forum, Vol. 8, 203, o. 34, 65-662 HIKARI Ltd, www.m-kar.com ttp://d.do.org/0.2988/mf.203.3854 Asymptotc Formulas Composte Numbers II Rafael Jakmczuk Dvsó Matemátca, Uversdad Nacoal
More informationf(t)dt 2δ f(x) f(t)dt 0 and b f(t)dt = 0 gives F (b) = 0. Since F is increasing, this means that
Uiversity of Illiois t Ur-Chmpig Fll 6 Mth 444 Group E3 Itegrtio : correctio of the exercises.. ( Assume tht f : [, ] R is cotiuous fuctio such tht f(x for ll x (,, d f(tdt =. Show tht f(x = for ll x [,
More informationSystems of second order ordinary differential equations
Ffth order dgolly mplct Ruge-Kutt Nystrom geerl method solvg secod Order IVPs Fudzh Isml Astrct A dgolly mplct Ruge-Kutt-Nystróm Geerl (SDIRKNG) method of ffth order wth explct frst stge for the tegrto
More informationChapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures
Chpter 5 The Riem Itegrl 5.1 The Riem itegrl Note: 1.5 lectures We ow get to the fudmetl cocept of itegrtio. There is ofte cofusio mog studets of clculus betwee itegrl d tiderivtive. The itegrl is (iformlly)
More informationMathematics HL and further mathematics HL formula booklet
Dplom Progrmme Mthemtcs HL d further mthemtcs HL formul boolet For use durg the course d the emtos Frst emtos 04 Publshed Jue 0 Itertol Bcclurete Orgzto 0 5048 Mthemtcs HL d further mthemtcs formul boolet
More informationOptimality of Strategies for Collapsing Expanded Random Variables In a Simple Random Sample Ed Stanek
Optmlt of Strteges for Collpsg Expe Rom Vrles Smple Rom Smple E Stek troucto We revew the propertes of prectors of ler comtos of rom vrles se o rom vrles su-spce of the orgl rom vrles prtculr, we ttempt
More informationCURVE FITTING LEAST SQUARES METHOD
Nuercl Alss for Egeers Ger Jord Uverst CURVE FITTING Although, the for of fucto represetg phscl sste s kow, the fucto tself ot be kow. Therefore, t s frequetl desred to ft curve to set of dt pots the ssued
More informationLECTURE 8: Topics in Chaos Ricker Equation. Period doubling bifurcation. Period doubling cascade. A Quadratic Equation Ricker Equation 1.0. x x 4 0.
LECTURE 8: Topcs Chaos Rcker Equato (t ) = (t ) ep( (t )) Perod doulg urcato Perod doulg cascade 9....... A Quadratc Equato Rcker Equato (t ) = (t ) ( (t ) ). (t ) = (t ) ep( (t )) 6. 9 9. The perod doulg
More informationMath 2414 Activity 16 (Due by end of class August 13) 1. Let f be a positive, continuous, decreasing function for x 1, and suppose that
Mth Actvty 6 (Due y ed of clss August ). Let f e ostve, cotuous, decresg fucto for x, d suose tht f. If the seres coverges to s, d we cll the th rtl sum of the seres the the remder doule equlty r 0 s,
More informationTrapezoidal Rule of Integration
Trpezoidl Rule o Itegrtio Mjor: All Egieerig Mjors Authors: Autr Kw, Chrlie Brker Trsormig Numericl Methods Eductio or STEM Udergrdutes /0/200 Trpezoidl Rule o Itegrtio Wht is Itegrtio Itegrtio: The process
More informationTrapezoidal Rule of Integration
Trpezoidl Rule o Itegrtio Civil Egieerig Mjors Authors: Autr Kw, Chrlie Brker http://umericlmethods.eg.us.edu Trsormig Numericl Methods Eductio or STEM Udergrdutes /0/00 http://umericlmethods.eg.us.edu
More informationComputer Programming
Computer Progrmmg I progrmmg, t s ot eough to be vetve d geous. Oe lso eeds to be dscpled d cotrolled order ot be become etgled oe's ow completes. Hrl D. Mlls, Forwrd to Progrmmg Proverbs b Her F. Ledgrd
More information#A79 INTEGERS 16 (2016) ROUND FORMULAS FOR EXPONENTIAL POLYNOMIALS AND THE INCOMPLETE GAMMA FUNCTION
#A79 INTEGERS 6 (06) ROUND FORMULAS FOR EXPONENTIAL POLYNOMIALS AND THE INCOMPLETE GAMMA FUNCTION St Wgo Deprtmet of Mthemtcs d Computer Scece, Mclester College, St. Pul, Mesot wgo@mclester.edu Receved:
More information( a n ) converges or diverges.
Chpter Ifiite Series Pge of Sectio E Rtio Test Chpter : Ifiite Series By the ed of this sectio you will be ble to uderstd the proof of the rtio test test series for covergece by pplyig the rtio test pprecite
More information4 Round-Off and Truncation Errors
HK Km Slgtly moded 3//9, /8/6 Frstly wrtte at Marc 5 4 Roud-O ad Trucato Errors Errors Roud-o Errors Trucato Errors Total Numercal Errors Bluders, Model Errors, ad Data Ucertaty Recallg, dv dt Δv v t Δt
More informationMATH 104 FINAL SOLUTIONS. 1. (2 points each) Mark each of the following as True or False. No justification is required. y n = x 1 + x x n n
MATH 04 FINAL SOLUTIONS. ( poits ech) Mrk ech of the followig s True or Flse. No justifictio is required. ) A ubouded sequece c hve o Cuchy subsequece. Flse b) A ifiite uio of Dedekid cuts is Dedekid cut.
More informationAnalytical Approach for the Solution of Thermodynamic Identities with Relativistic General Equation of State in a Mixture of Gases
Itertol Jourl of Advced Reserch Physcl Scece (IJARPS) Volume, Issue 5, September 204, PP 6-0 ISSN 2349-7874 (Prt) & ISSN 2349-7882 (Ole) www.rcourls.org Alytcl Approch for the Soluto of Thermodymc Idettes
More informationConvergence of the FEM. by Hui Zhang Jorida Kushova Ruwen Jung
Covergece o te FEM by Hi Zg Jorid Ksov Rwe Jg I order to proo FEM soltios to be coverget, mesremet or teir qlity is reqired. A simple pproc i ect soltio is ccessible is to qtiy te error betwee FEMd te
More information