МИНИСТЕРСТВО ОБРАЗОВАНИЯ И НАУКИ РОССИЙСКОЙ ФЕДЕРАЦИИ. Численные методы. Учебно-методическое пособие

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1 МИНИСТЕРСТВО ОБРАЗОВАНИЯ И НАУКИ РОССИЙСКОЙ ФЕДЕРАЦИИ Нижегородский государственный университет им. Н.И. Лобачевского Численные методы К.А.Баркалов Учебно-методическое пособие Рекомендовано методической комиссией факультета ВМК для иностранных студентов, обучающихся в ННГУ по направлению подготовки «Фундаментальная информатика и информационные технологии» бакалавриат -е издание Нижний Новгород

2 Баркалов К.А. Численные методы: Учебно-методическое пособие. Нижний Новгород: Нижегородский госуниверситет,. 8 с. В настоящем пособии изложены учебно-методические материалы по курсу «Численные методы» для иностранных студентов, обучающихся в ННГУ по направлению подготовки «Фундаментальная информатика и информационные технологии» бакалавриат. Нижегородский государственный университет им. Н.И. Лобачевского,

3 Mstr o Educto d Scece o te Russ Federto Stte eductol sttuto o ger educto «Locevsk Stte Uverst o Nz Novgorod» Computg mtemtcs К.А.БАркалов Tutorl Recommeded te Metodcl Commsso o te Fcult o Computer Scece or tertol studets, studg t te B.Sc. progrm Fudmetl Iormtcs d Iormto Tecologes Nz Novgorod

4 Itroducto Computg mtemtcs s te stud o lgortms tt use umercl ppromto s opposed to geerl smolc mpultos or te prolems o mtemtcl lss d lger s dstgused rom dscrete mtemtcs. Computg mtemtcs turll ds pplctos ll elds o egeerg d te pscl sceces, ut te st cetur, te le sceces d eve te rts ve dopted elemets o scetc computtos. Ordr deretl equtos pper te movemet o evel odes plets, strs d gles; optmzto occurs portolo mgemet; umercl ler lger s mportt or dt lss; stocstc deretl equtos d Mrkov cs re essetl smultg lvg cells or medce d olog. Beore te dvet o moder computers umercl metods ote depeded o d terpolto lrge prted tles. Sce te md t cetur, computers clculte te requred uctos sted. Te terpolto lgortms everteless m e used s prt o te sotwre or solvg deretl equtos. Te eld o umercl lss predtes te veto o moder computers m cetures. Ler terpolto ws lred use more t ers go. M gret mtemtcs o te pst were preoccuped umercl lss, s s ovous rom te mes o mportt lgortms lke Newto's metod, Lgrge terpolto poloml, Guss elmto, or Euler's metod. To cltte computtos d, lrge ooks were produced wt ormuls d tles o dt suc s terpolto pots d ucto coecets. Usg tese tles, ote clculted out to 6 decml plces or more or some uctos, oe could look up vlues to plug to te ormuls gve d ceve ver good umercl estmtes o some uctos. Te ucto vlues re o loger ver useul we computer s vlle, ut te lrge lstg o ormuls c stll e ver d. Te meccl clcultor ws lso developed s tool or d computto. Tese clcultors evolved to electroc computers te 94s, d t ws te oud tt tese computers were lso useul or dmstrtve purposes. But te veto o te computer lso lueced te eld o umercl lss, sce ow loger d more complcted clcultos could e doe. Course progrm. Fte-dgt rtmetc.. Postol or rd otto.. Normlzed scetc otto.. Mce epslo.4. Asolute d reltve errors. Soluto o oler equtos.. Bsecto metod... Error lss

5 .. Newto s metod... Grpcl terpretto o Newto s metod... Error lss.. Fed-Pot Iterto... Error lss. Iterpolto d poloml ppromto.. Prolem sttemet.. Ler terpolto.. Geerl terpolto prolem 4. Numercl deretto 4.. Prolem sttemet 4.. Two-pot ormuls or rst dervtve 4... Error lss 4.. Tree-pot ormul or te rst dervtve 4.4. Tree-pot ormul or te secod dervtve 4.5. Deretto v terpolto 5. Numercl tegrto 5.. Prolem sttemet 5.. Itegrto v terpolto 5.. Trpezod rule 5.4. Smpso s rule 6. Itl-vlue prolems or ODE 6.. Prolem sttemet 6.. Tlor-seres metod 6... Error lss 6.. Euler's metod 6.4. Secod-order Ruge-Kutt metods 6.5. Fourt-order Ruge-Kutt metod 7. Curve ttg 7.. Prolem sttemet 7.. Lest squres le 7.. Lest squres poloml. Fte-dgt rtmetc.. Postol or rd otto Cosder deret ottos o oe umer. Decml sstem: = = Br sstem:. = Teorem wtout proo. For teger p, rel umer m e represeted te orm k k k k p... p p p p. p...

6 were coecets β re teger d sts te equltes β p. Let's reduce te otto. d wrte te umer X s te sequece o coecets β : kk.. Te reduced orm. s clled postol represetto o te umer te umer sstem otto wt rd p... Normlzed scetc otto I te decml sstem, rel umer c e epressed ormlzed scetc otto. Ts mes tt te decml pot s sted d pproprte powers o re suppled so tt ll te dgts re to te rgt o te decml pot, d te rst dgt dspled s ot zero. r. were /r<, d s teger. I ectl te sme w, we c use scetc otto te r sstem m q. were /r<, d m s teger. Te umer q s clled te mtss d te teger m te epoet; ot q d m wll e represeted s se umers... Mce epslo I we store rel umer -t memor 4 tes d use t or te umer sg, 7 ts or te epoet prt d 4 ts or te represetto o te lotg pot coecet te we c represet te umers wt 6-7 sgct dgts te tervl rom 8 to 8 ppromtel or postve umers d smmetrcl tervl or egtve oes. Suc ormt or represetto o rel umers dees te dt tpe lot te C lguge. For more precse computtos esurg more ccurc te tpe doule s used. It requres 8 tes memor d eteds te qutt o sgct dgts up to 5-6 d te rge o possle postve rel umers rom -8 to 8 egtve umers re cluded smmetrcll. I computer opertes wt se d crres plces te mtss o ts lotg-pot umers, te lot,. Te umer s te ut roudo error d s crcterstc o te computer, ts opertg sstem, d te mode o computto weter sgle- or douleprecso.

7 To compute ppromte vlue o o mce, te ollowg lgortm c e used, eter sgle- or doule-precso. It determes te smllest postve umer o te orm k suc tt. +. te mce. Algortm. Step. eps =.; k = ; Step. eps = eps/; Step. k = k+; Step 4. I +eps= te STOP; else GOTO Step ; Step 5. Output eps;.4. Asolute d reltve errors I te prctce o umercl lss t s mportt to e wre tt computed solutos re ot ect mtemtcl solutos. Te precso o umercl soluto c e dmsed severl ws. Deto. Suppose tt p s ppromto to p. Te solute error s d te reltve error s provded tt p. E p = p p, R p = p p / p, Te error s smpl te derece etwee te true vlue d te ppromte vlue, weres te reltve error s porto o te true vlue. Emple. Fd te error d reltve error te ollowg tree cses.. Let =.459 d =.4. Te E = = =.59, R = / =.59 /.459 =.57.. Let = d = Te E = = =4, R = / = 4 / =.4. c. Let z=. d z=.9. Te E z = z z =..9 =. R z = z z / z =. /. =.5. I p moves w rom greter t or less t te reltve error R p s etter dctor o te ccurc o te ppromto t E p. Reltve error s preerred or lotg-pot represettos sce t dels drectl wt te mtss.

8 . Soluto o oler equtos Ts prt s devoted to te prolem o determg roots o equtos or zeros o uctos. It s prolem o requet occurrece scetc work. For emple, te teor o drcto o lgt, we eed te roots o te equto t =. I te clculto o pletr orts, we eed te roots o Kepler's equto or vrous vlues o d s =. Te geerl questo, posed te smplest cse o rel-vlued ucto o rel vrle, s ts: gve rel-vlued ucto, d te vlues o or wc =. We sll cosder severl o te stdrd procedures or solvg ts prolem... Bsecto metod I s cotuous ucto o te tervl [,] d <, te must ve t lest oe zero te tervl,. Te secto metod eplots ts de te ollowg w. I < te we compute c = + / d test weter c<. I so, te s zero [,c]. So we reme c s d strt g wt te ew tervl [,], wc s l s lrge s te orgl tervl. I c> te c<, d ts cse we reme c s. I eter cse, ew tervl cotg zero o s ee produced, d te process c e repeted. I c= te c= d zero s ee oud. However, t s qute ulkel tt c wll e zero te computer ecuse o roudo errors. Tus, te stoppg crtero sould ot e weter c=. A resole tolerce must e llowed, suc s c < or <, were d s rter smll. To llustrte te process o root dg we cosder two deret cses. I te rst cse gure secto metod selects let sutervl.

9 Fgure. Bsecto metod selects let sutervl I te secod cse gure secto metod selects rgt sutervl. Fgure. Bsecto metod selects let sutervl O gure we llustrte tree tertos o secto metod. Fgure. Itertos o secto metod

10 Now we re red to wrte te lgortm. Iput o te lgortm re edpots d ; tolerce TOL; mmum umer o tertos N. Output o te lgortm re soluto ppromto c or messge o lure. Step. Set = ; F =. Step Wle < N do Steps -6. Step Set c= + -/; //Compute c Fc = c. Step 4 I Fc = or - / < TOL te OUTPUT c; //Procedure completed successull STOP. Step 5 Set = +. Step 6 I F-Fc > te set = c; F = Fc else set = c. Step 7 OUTPUT 'Metod led ter N tertos, N =', N; //Te procedure ws usuccessul. STOP.... Error lss To lze te secto metod, let us deote te successve tervls tt rse te process [, ], [, ], d so o. Here re some oservtos out tese umers: + + =.5 I we ppl repetedl, we d tt = - Tus I we put te, tkg lmt te eqult, we ot [r], wece r =.

11 Suppose tt, t cert stge te process, te tervl [, ] s just ee deed. I te process s ow stopped, te root s cert to le ts tervl. Te est estmte o te root t ts stge s ot or ut te mdpot o te tervl: c = + / Te error s te ouded s ollows Teorem. I [, ], [, ],, [, ] deote te tervls te secto metod, te te lmts d est, re equl, d represet zero o. I d c = + /, te. Emple. Suppose tt te secto metod s strted wt te tervl [5,6]. How m steps sould e tke to compute root wt reltve ccurc o oe prt? Soluto. Te stted requremet o reltve precso mes tt We kow tt r 5, d tus t wll suce to secure te eqult B mes o te precedg teorem, we er tt te ollowg codto wll e sucet Solvg ts or, we coclude tt 7... Newto s metod Newto's metod s geerl procedure tt c e ppled m dverse stutos. We speclzed to te prolem o loctg zero o rel-vlued ucto o rel vrle, t s ote clled te Newto-Rpso terto. We ve ucto wose zeros re to e determed umercll. Let r e zero o d let e ppromto to r. I ests d s cotuous, te Tlor's teorem... were =r. I s smll tt s, s er r, te t s resole to gore te O -term d solve te remg equto or. Tereore, te result s = /. I s ppromto to r, te / sould e etter ppromto to r. Newto's metod egs wt estmte o r d te dees ductvel.

12 Te stoppg crtero: r < or + <, were d re rter smll.... Grpcl terpretto o Newto s metod From te descrpto lred gve, we c s tt Newto's metod volves lerzg te ucto. Tt s, ws replced ler ucto. Te usul w o dog ts s to replce te rst two terms ts Tlor seres. Tus te te lerzto t c produces te ler ucto. Notce tt l s good ppromto to te vct o c, d ct we ve lc= c d lc= c. Tus, te ler ucto s te sme vlue d te sme slope s t te pot c. So Newto's metod we re costructg te tget to te -curve t pot er r, d dg were te tget le tersects te -s see gure 4. Fgure 4. Lerzg te ucto Keepg md ts grpcl terpretto, we c esl mge uctos d strtg pots or wc te Newto terto wll l see gure 5. Fgure 5. Newto terto ls

13 I ts emple, te spe o te curve s suc tt or cert strtg vlues, te sequece [ ] wll dverge. Tus, orml sttemet out Newto's metod must volve ssumpto tt s sucetl close to zero, or tt te grp o s prescred spe. Now we re red to wrte te lgortm. Iput o te lgortm re tl ppromto ; tolerce TOL; mmum umer o tertos N. Output o te lgortm re soluto ppromto or messge o lure. Step Set =. Step Wle < N do Steps -6. Step Set = - /'. //Compute p Step 4 I l < TOL te OUTPUT //Te procedure ws successul. STOP. Step 5 Set = +. Step 6 Set = // Updte. Step 7 OUTPUT 'Te metod led ter N tertos, N =', N; //Te procedure ws usuccessul. STOP.... Error lss B errors, we me te quttes e = r. Let us ssume tt s cotuous d r s smple zero o, so tt rr. From te deto o te Newto terto, we ve B Tlor's teorem, we ve were < <r. From ts equto, we ve d Ts equto tells us tt e + s rougl costt tmes e. Ts desrle stte o rs s clled qudrtc covergece. It ccouts or te ppret doulg o precso wt ec terto o Newto's metod. Emple. Fd ecet metod or computg squre roots sed o te use o Newto's metod. Soluto. Let R>, d. Te s root o te equto R. I we use Newto's metod o te ucto R, te terto ormul c e wrtte s

14 Ts ormul s ver cet, d s credted to Hero, Greek egeer d rctect wo lved sometme etwee B.C. d A.D. I, or emple, we ws to compute d eg wt = 4, te successve ppromts re s ollows Fed-Pot Iterto Deto. A umer p s ed pot or gve ucto g gp=p. Root-dg prolems d ed-pot prolems re equvlet clsses te ollowg sese. Gve root-dg prolem, we c dee uctos g wt ed pot t p, e.g., g. Coversel, te ucto g s ed pot t, te te ucto s zero t p. Cosder s emple ucto = see gure 6. Tere re two ed pots o ts ucto p =, p = Fgure 6. Fed pots Te ollowg teorem gves sucet codtos or te estece d uqueess o ed pot. Teorem. I gc[,] d g[,] or ll [,], te g s ed pot p[,]. I, ddto, g' ests o, d postve costt k< ests wt g' k, or ll,, te te ed pot p[,] s uque. Proo. I g or g, te g s ed pot t edpot. I ot, te g> d g<. Te ucto g s cotuous o [,], wt =g> d =g <. Te Itermedte Vlue Teorem mples tt tere ests p, or wc p. Ts umer p s ed pot or g sce pgpp mples tt gpp. Suppose, ddto, tt g' k< d tt p d q re ot ed pots [,]. l pq, te te Me Vlue Teorem mples tt umer ests, p<<q, wt.

15 Tus, pq gpgq g' pq k pq < pq wc s cotrdcto. Ts cotrdcto must come rom te ol supposto, pq. Hece, p=q d te ed pot [,] s uque. Ts teorem llustrted o gure 7. Fgure 7. Fed-pot teorem Te m de o te ed-pot terto metod s ollowg. To ppromte te ed pot o ucto g, we coose tl ppromto p d geerte te sequece {p } lettg p =gp, or ec >. I te sequece coverges to p d g s cotuous, te d soluto to g s oted. Ts tecque s clled ed-pot terto. Now we re red to wrte te lgortm. Iput o te lgortm re tl ppromto p ; tolerce TOL; mmum umer o tertos N. Output o te lgortm re soluto ppromto p or messge o lure. Step Set =. Step Wle < N do Steps -6. Step Set p = gp. //Compute p Step 4 I p p l < TOL te OUTPUT //Te procedure ws successul. STOP. Step 5 Set = +. Step 6 Set p = p // Updte p. Step 7 OUTPUT 'Te metod led ter N tertos, N =', N; //Te procedure ws usuccessul. STOP.

16 Process o dg ed pot s llustrted o ollowg gure. Fgure 8. Fed-pot tertos... Error lss Teorem. I gc[,] d g[,] or ll [,], d, ddto, g' ests o, d costt <k< ests wt g' k, or ll,, te, or umer p [,], te sequece deed p =gp, coverges to te uque ed pot p[,]. Corollr. I g stses te poteses o ts teorem, te ouds or te error volved usg p to ppromte p re gve p p k m{p,p } d p p p p k /k. Proo teorem. Teorem mples tt uque ed pot p [,] ests. Usg te ct tt g' k d te Me Vlue Teorem, we ve, or ec, p p gp gp g' p p k p p. were,. Applg ts eqult ductvel gves p p k p p k p p k p p. Sce <k<, we ve d Proo corollr. Sce p [,], te rst oud ollows rom eqult p p k p p k m{p,p }. For >, te procedure used te proo o teorem mples tt p + p gp gp k p p k p p. Tus, or m>, p m p = p m p m +p m +p + p p m p m + p m p m + + p + p k m p p +k m p p + +k p p = =k p p +k+k + +k m. B teorem,, so

17 But s geometrc seres wt rto k d <k<. Ts sequece coverges to, wc gves te secod oud Bot equltes te corollr relte te rte t wc {p } coverges to te oud k o te rst dervtve. Te rte o covergece depeds o te ctor k. Te smller te vlue o k, te ster te covergece, wc m e ver slow k s close to. Emple. For g 4 +, we ve gl 6 d g, so g does ot mp [,] to tsel. Moreover, g' 8, so g' > or ll [,]. Altoug Teorem does ot gurtee tt te metod must l or ts coce o g, tere s o reso to epect covergece. Emple. For g=/4+.5, we ve or ll [,] Te oud o te mgtude o 5 5 g / / s smll, wc epls te rpd covergece.. Iterpolto d poloml ppromto.. Prolem sttemet A cesus o te populto o te Uted Sttes s tke ever ers. Te ollowg tle lsts te populto, tousds o people, rom 94 to 99. Yer Populto ts 94, , , 97, 98 6, ,6

18 We mgt sk weter te could e used to provde resole estmte o te populto, s, 965 or 995. Predctos o ts tpe c e oted usg ucto tt ts te gve dt. Ts process s clled terpolto d s te suject o ts cpter. Oe o te most useul d well-kow clsses o uctos mppg te set o rel umers to tsel s te clss o lgerc polomls, te set o uctos o te orm were s oegtve teger d,,..., re rel costts. Oe reso or ter mportce s tt te uorml ppromte cotuous uctos. Gve ucto, deed d cotuous o closed d ouded tervl, tere ests poloml tt s s close to te gve ucto s desred. Teorem. Weerstrss Appromto Teorem wtout proo. Suppose tt s deed d cotuous o [,]. For ec >, tere ests poloml P, wt te propert tt P <, or ll [,]. Weerstrss Appromto Teorem s llustrted o te ollowg gure. Fgure 9. Weerstrss Appromto Teorem.. Ler terpolto Te prolem o determg poloml o degree oe tt psses troug te dstct pots, d, s te sme s ppromtg ucto or wc d mes o rst-degree poloml terpoltg, or greeg wt, te vlues o t te gve pots. We rst dee te uctos d te dee d. Sce L, L, L, L, we ve

19 P + P + So P s te uque ler ucto pssg troug, d,. Ts ct s llustrted o te ollowg gure. Fgure. Ler terpolto.. Geerl terpolto prolem To geerlze te cocept o ler terpolto, cosder te costructo o poloml o degree t most tt psses troug te + pots,,,,,,. Ts prolem s llustrted o gure. Fgure. Poloml terpolto I ts cpter, we solve te ollowg prolem: we re gve tle o + dt pots, : d we seek poloml p o lowest possle degree or wc p =, were.

20 Suc poloml s sd to terpolte te dt, or p s clled terpoltg poloml. Teorem. I,,, re dstct rel umers, te or rtrr vlues,,, tere s uque poloml p o degree t most suc tt p =, were. Proo. Uct Suppose tere were two suc polomls, p d q. Te te poloml p q would ve te propert tt p q or. Sce te degree o p q c e t most, ts poloml c ve t most zeros t s ot te zero poloml. Sce te, re dstct, p q s + zeros; t must tereore e zero. Hece, p q. Estece For, te estece s ovous sce costt ucto p poloml o degree c e cose so tt p. Now suppose tt we ve oted poloml p k o degree k wt p k or k. We tr to costruct p k te orm p k p k +c k Note tt ts s uquestol poloml o degree t most k. Furtermore, p k terpoltes te dt tt p k terpoltes, ecuse p k p k, k. Now we determe te ukow coecet c rom te codto p k k k. Ts leds to te equto p k k +c k k k k k. Ts equto c certl e solved or c ecuse te ctors multplg c re ot zero. Teorem s prooed. Teorem. I,,, re dstct rel umers d s ucto wose vlues re gve t tese umers, te uque poloml P o degree t most ests wt P k = k, were k. Ts poloml s gve were or We wll wrte L,k smpl s L k we tere s o couso s to ts degree. Teorem. wtout proo.

21 Suppose tt,,, re dstct umers te tervl [,] d C + [,]. Te, or ec [,], umer, ests wt were P s te terpoltg poloml. Corollr. Error o terpolto s 4. Numercl deretto 4.. Prolem sttemet I te vlues o ucto re gve t ew pots, s,,...,, c tt ormto e used to estmte dervtve c? Te swer s quled Yes. Let us eg oservg tt rom te vlues loe t s mpossle to er ver muc out uless we re ormed lso tt elogs to some reltvel smll ml o uctos. Tus, s llowed to rge over te ml o ll cotuous rel-vlued uctos, te vlues re lmost useless. Fgure llustrtes severl cotuous uctos tkg te sme vlues t s pots. Fgure. Severl cotuous uctos I we kow tt s poloml o degree t most, te te vlues t + pots completel determe, te teor o terpolto. I ts cse, we recover P precsel, d c te compute Pc. Ad te we c use Pc s estmto o c. 4.. Two-pot ormuls or rst dervtve Frst we cosder ormul or umercl deretto tt emerges drectl rom te deto o : lm Ts ormul gves ovous w to geerte ppromto to : smpl compute

22 * For ler ucto, =+, te ppromte ormul s ect; tt s, t elds te correct vlue o : or ozero vlue o. Formul * s kow s te orwrd-derece ormul > d te ckwrd-derece ormul <. Fgure llustrtes te two-pot ormul grpcll. Fgure. Two pot ormul 4... Error lss Te strtg pot s Tlor's Teorem ts orm. Here s pot te ope tervl etwee d +. For te vldt o equto, d ' sould e cotuous o te closed tervl etwee d +, d '' sould est o te correspodg ope tervl. A rerrgemet o equto elds Now error term s vlle log wt te sc umercl ormul. Notce tt te error term equto s two prts: power o d ctor volvg some gerorder dervtve o. Te -term te error mkes te etre epresso coverge to zero s pproces zero. 4.. Tree-pot ormul or te rst dervtve Te precso o suc umercl deretto ormule s ote judged smpl te power o preset te error term. Te ger te power o te etter, or s lws smll umer. I ts ssessmet, ormul * res poorl, s te error s O. A superor ormul s

23 Ts s derved rom two cses o Tlor's Teorem, mel 6, 6. O sutrctg oe o tese rom te oter, we ot Sce s cotuous some pot ests [, +] suc, tt.5 We ts epresso s susttuted equto prevous, te result s 6. Tree-pot ormul s llustrted o te ollowg gure. Fgure 4. Tree-pot ormul 4.4. Tree-pot ormul or te secod dervtve A mportt ormul or secod dervtves s oted t te sme w rom Telor s teorem O ddg oe o tese to te oter, we ot

24 4 were [, +] d Fll, we ve Deretto v terpolto A geerl pproc to umercl deretto d tegrto c e sed o poloml terpolto. Suppose tt we ve + vlues o ucto t pots,,,. A poloml tt terpoltes t te odes c e wrtte te Lgrge orm s! w L were w. Tkg dervtve ts equto we ve!! d d w w L I s oe o te odes, s, te te precedg equto smples, sce w, d te result s L! d dervtve c e clculted s L X L? L, d so o Eercses. Gve te eplct orm o equto X we d.. Gve te eplct orm o equto X we d.. Estmte error o ppromto usg Telor s teorem.

25 5. Numercl tegrto 5.. Prolem sttemet A seet o corrugted roog s costructed pressg lt seet o lumum to oe wose cross secto s te orm o se wve. Fgure 5. A seet o corrugted roog A corrugted seet 4 t. log s eeded, te egt o ec wve s. rom te ceter le, d ec wve s perod o ppromtel. Te prolem o dg te legt o te tl lt seet s oe o determg te legt o te curve gve s rom to 48. From mtemtcl lss we kow tt ts legt s 48 d cos L d, so te prolem reduces to evlutg ts tegrl. Altoug te se ucto s oe o te most commo mtemtcl uctos, te clculto o ts legt volves ellptc tegrl o te secod kd, wc cot e evluted ordr metods. Metods o ppromto te soluto to prolems o ts tpe re cosdered ts cpter. Numercl tegrto s te process o producg umercl vlue or te tegrto o ucto over set. For emple, ollowg tegrto prolems re ot mele to te tecques lered elemetr clculus. 48 s ep dd cos cos d Tose tecques depeds o tderetto. Tus, to d te vlue o te tegrl, we rst must produce ucto F wt te propert tt F'=. Te, we ve d F F. Tere re m elemetr uctos tt do ot ve smple tdervtves. A good emple s.

26 5.. Itegrto v terpolto Oe powerul strtgem or computg te tegrl umercll s to replce oter ucto g tt ppromtes well d s esl tegrted. Te, we smpl s to ourselves tt rom g t ollows tt d g d Polomls re good cddtes or te ucto g, d deed, g c e poloml tt terpoltes to t cert set o odes. Suppose tt we wt to evlute te tegrl [,]. We c select odes [,] d set up Lgrge poloml L P, were j j j j L,. Te, s metoed prevousl, we smpl wrte d L d P d I ts w, we ot ormul tt c e used o. A d, were d L A. Ts ormul s clled Newto-Cotes ormul, te odes re equll spced. 5.. Trpezod rule Te smplest cse results d te odes re,. I ts cse L, L. Cosequetl,.5 A d L d L A Te correspodg qudrture ormul s d. Ts s clled te trpezod rule ecuse we s ucto wt postve vlues, tegrl s ppromted te re trpezod, s sow gure.

27 Fgure 6. Trpezod rule Error lss o trpezod rule s preseted elow. As r s L! te error o tegrto s d d P d E! I te cse o trpezod ormul,,, d error s! d E Smpso s rule Smpso's rule results rom tegrtg over [,] te secod Lgrge poloml wt odes,, d +, were /. I ts cse L, L, L d A d L d L A, 4 d L A. Cosequetl, 4 d. Error o ts ormul s E, were [,]. Smpso's rule results rom tegrtg over [,] te secod Lgrge poloml.

28 Fgure 7. Smpso's rule 6. Itl-vlue prolems or ODE 6.. Prolem sttemet Te moto o swgg pedulum uder cert smplg ssumptos s descred te secod-order deretl equto were L s te legt o te pedulum, g s te grvttol costt o te ert, d s te gle te pedulum mkes wt te vertcl. I, ddto, we spec te posto o te pedulum we te moto egs, t, d ts veloct t tt pot,, we ve wt s clled tl-vlue prolem. For smll vlues o, te ppromto s c e used to smpl ts prolem to te ler tl-vlue prolem Ts prolem c e solved stdrd deretl-equto tecque. For lrger vlues o, ppromto metods must e used. A tetook o ordr deretl equtos detls umer o metods or eplctl dg solutos to rst-order tl-vlue prolems. I prctce, owever, ew o te prolems orgtg rom te stud o pscl peome c e solved ectl. Te rst prt o ts cpter s cocered wt ppromtg te soluto to prolem o te orm d, or d suject to tl codto.

29 Lter te cpter we del wt te eteso o tese metods to sstem o rst-order deretl equtos. We lso eme te reltosp o sstem o ts tpe to te geerl t-order tl-vlue prolem. 6.. Tlor-seres metod To llustrte te metod we tke cocrete emple cos s,. At te ert o te procedure s te Tlor seres or, wc we wrte s 4 O 6 Te dervtves pperg ere c e oted rom te deretl equto. Te re s cos, cos cos s. We decde to use ol terms up to d cludg te ormul. Te terms tt we ve ot cluded strt wt term 4, d te costtute collectvel te tructo error eret our procedure. Te resultg umercl metod s sd to e o order. Te order o te Tlor-seres metod s terms up to d cludg re used. We could perorm vrous susttutos to ot ormule or,,... cotg o dervtves o o te rgt-d sde Error lss At ec step, te locl tructo error s O 4 sce we ve ot cluded terms volvg 4, 5,... rom te Tlor seres. Tus, s, te evor o te locl errors sould e smlr to C 4. Uortutel, we do ot kow C. But 4 s 8 sce =. So wt good luck, te error ec step sould e rougl o te mgtude 8. Ater severl udred steps, tese smll errors could ccumulte d spol te umercl soluto. At ec step ecept te rst, te estmte k o k lred cots errors, d urter computtos cotue to dd to tese errors. Te locl tructo error s te error mde oe step we we replce te process te oe. I te Tlor-seres metod, we replce te te Tlor seres or + prtl sum. Te ccumulto o ll tese m locl tructo errors gves rse to te glol tructo error. I te locl tructo errors re O +, te te glol tructo error must e O ecuse te umer o steps ecessr to rec rtrr pot, vg strted t, s /.

30 6.. Euler's metod Te Tlor-seres metod wt s clled Euler's metod. It looks lke ts,. Ts ormul s te ovous dvtge o ot requrg deretto o,. Locl tructo error o Euler's metod s O, te te glol tructo error must e O. Euler's metod s rst-order metod. To terpret Euler's metod geometrcll, ote tt we s close ppromto to, te ssumpto tt te prolem s well-posed mples tt,, Oe step Euler's metod s sow let gure, d seres o steps ppers rgt gure. Fgure 8. Euler's metod 6.4. Secod-order Ruge-Kutt metods Let s eg wt te Tlor seres or + O From te deretl equto, we ve, Here suscrpts deote prtl dervtves. Te rst tree terms Tlor seres c e wrtte ow te orm O O * We re le to elmte te prtl dervtves wt te d o te rst ew terms te Tlor seres two vrles

31 ,,, O O Tlor seres c e rewrtte s, O Hece, te ormul or dvcg te soluto s,,, Ts ormul c e used repetedl to dvce te soluto oe step t tme. It s clled secod-order Ruge-Kutt metod. It s lso kow s Heu's metod. I geerl, secod-order Ruge-Kutt ormuls re o te orm, O w w were w, w,, re prmeters t our dsposl. Ts equto c e rewrtte wt te d o te Tlor seres two vrles s O w w ** Comprg * wt **, we see tt we sould mpose tese codtos Oe soluto s w w.5,, wc s te oe correspodg to Heu's metod. Te sstem o equtos s solutos oter t ts oe, suc s te oe oted lettg w, w,.5. Te resultg ormul s clled te moded Euler metod,.5, Fourt-order Ruge-Kutt metod Te ger-order Ruge-Kutt ormuls re ver tedous to derve, d we sll ot do so. Te ormule re rter elegt, owever, d re esl progrmmed oce te ve ee derved. Here re te ormule or te clsscl ourt-order Ruge- Kutt metod 6 4 F F F F were,.5,.5.5,.5, 4 F F F F F F F

32 Ts s clled ourt-order metod ecuse t reproduces te terms te Tlor seres up to d cludg te oe volvg 4. Te locl error s tereore O 5, glol error s O Curve ttg 7.. Prolem sttemet Hooke's lw sttes tt we orce s ppled to sprg, te legt o te sprg s ler ucto o tt orce. We c wrte te ler ucto s Fl=klE, were Fl s te orce requred to stretc te sprg l uts, te costt E s te legt o te sprg wt o orce ppled, d te costt k s te sprg costt. Suppose we wt to determe te sprg costt or sprg tt s tl legt 5.. We ppl orces o, 4, d 6 to d d tt ts legt creses to 7., 9.4, d.. A quck emto sows tt te pots,5.,,7., 4,9.4, d 6,. do ot qute le strgt le. Altoug we could smpl use oe rdom pr o tese dt pots to ppromte te sprg costt, t would seem more resole to d te le tt est ppromtes ll te dt pots to determe te costt. Ts tpe o ppromto wll e cosdered ts cpter. Te stud o ppromto teor volves two geerl tpes o prolems. Oe prolem rses we ucto s gve eplctl, ut we ws to d smpler tpe o ucto, suc s poloml, tt c e used to determe ppromte vlues o te gve ucto. Te oter prolem ppromto teor s cocered wt ttg uctos to gve dt d dg te est ucto cert clss to represet te dt. Bot prolems ve ee cosdered prevous cpters. Te Tlor poloml o degree out te umer s ecellet ppromto to +l-tmes deretle ucto smll egorood o. Te Lgrge terpoltg poloml o degree s used ot s ppromtg poloml d s poloml to t cert dt. Now we cosder te prolem o dg te est ucto cert clss to represet te cert dt. Cosder te prolem o estmtg te vlues o ucto t otulted pots, gve te epermetl dt ollowg tle. A grp o te vlues tle s sow o gure.,,5 4, 4 5, 4 8

33 5 7, 6 8,8 7, 8,5 9, 5,6 From ts grp, t ppers tt te ctul reltosp etwee d s ler. Te lkel reso tt o le precsel ts te dt s ecuse o errors te dt. So t s uresole to requre tt te ppromtg ucto gree ectl wt te dt. I ct, suc ucto would troduce osclltos tt were ot orgll preset. For emple, we c ppromte ts dt wt 9-t degree terpoltg poloml t s sow o te ollowg gure. Fgure 9. Iterpoltg poloml Ts poloml s clerl poor predctor o ormto etwee umer o te dt pots. A etter pproc would e to d te est some sese ppromtg le, eve t does ot gree precsel wt te dt t pot. Mm prolem. Let + deote te t vlue o te ppromtg le d e te t gve -vlue. Te prolem o dg te equto o te est ler ppromto te solute sese requres tt vlues o d e oud to mmze Ts s commol clled mm prolem d cot e dled elemetr tecques. Asolute devto. Aoter pproc to determg te est ler ppromto volves dg vlues o d to mmze

34 Ts qutt s clled te solute devto. To mmze ucto o two vrles, we eed to set ts prtl dervtves to zero d smulteousl solve te resultg equtos. I te cse o te solute devto, we eed to d d wt Te dcult s tt te solute-vlue ucto s ot deretle t zero, d we m ot e le to d solutos to ts pr o equtos. 7.. Lest squres le Te lest squres pproc to ts prolem volves determg te est ppromtg le we te error volved s te sum o te squres o te dereces etwee te -vlues o te ppromtg le d te gve -vlues. Hece, costts d must e oud tt mmze te lest squres error Te lest squres metod s te most coveet procedure or determg est ler ppromtos. Te mm pproc geerll ssgs too muc wegt to t o dt tt s dl error, weres te solute devto metod does ot gve sucet wegt to pot tt s cosderl out o le wt te ppromto. Te lest squres pproc puts susttll more wegt o pot tt s out o le wt te rest o te dt ut wll ot llow tt pot to completel domte te ppromto. Te geerl prolem o ttg te est lest squres le to collecto o dt {, },,,m, volves mmzg te totl error wt respect to te prmeters d. For mmum to occur, we eed Tese equtos smpl to te orml equtos

35 Te soluto o ts orml equtos sstem s oted Krmer s rule or Guss metod. Emple. Let s clculte te lest squres le or te dt rom te prevous emple.,,,,,,5 4, 7,, 4, 9,,6 4, 5, 6,, 5, 7, 5, 5, 6, 8,8 6, 5,8 7,, 49, 7,7 8,,5 64,, 9,, 8, 7,, 5,6, 56, Sum 55, 8, 85, 57,4 Te orml equtos mpl tt So P.58.6 Te grp o ts le d te dt pots re sow gure. Te ppromte vlues gve te lest squres tecque t te dt pots re ollowg tle. P,,,8,,5,7, 4, 4,5 4, 5, 5,79

36 5, 7, 7, 6, 8,8 8,87 7,,,4 8,,5,94 9,,,48, 5,6 5, 7.. Lest squres poloml Te geerl prolem o ppromtg set o dt {, },,,m, wt lgerc poloml o degree <m, usg te lest squres procedure s dled smlr mer. We coose te costts,,..., to mmze te lest squres error As te ler cse, or E to e mmzed t s ecessr tt E or ec j,, m. Tus, or ec j, j Ts gves + orml equtos te + ukows j Tese orml equtos ve uque soluto provded tt te j re dstct. Error o ppromto c e wrtte te ollowg orm E m P. Home eercses Fte dgt rtmetc. Asolute d reltve error.. Fd tervls cotg solutos to te ollowg equtos.. =. 4 =. Sow tt te ollowg equtos ve t lest oe soluto te gve tervls.. cos + =, [.,.] d [.,.]

37 . l =, [,] d [e,4] c. cos =, [, ] d [,4]. Fd te lrgest tervl wc p * must le to ppromte p wt reltve error t most 4 or ec vlue o p.. p=. p=e c. p= 4. Suppose p * must ppromte p wt reltve error t most. Fd te lrgest tervl wc p * must le or ec vlue o p.. p=5. p=9 c. p=5 d. p=9 5. Compute te solute error d reltve error ppromtos o p p *.. p =, p * = /7. p =, p * =.46 c. p = e, p * =.78 d. p =, p * =.44 e. p = e, p * =. p =, p * = 4 Soluto o oler equtos. Bsecto metod.. Let = + +. To wc zero o does te Bsecto metod coverge we ppled o te ollowg tervls?. [-.5,.5]. [-.5,.4] c. [-.5,] d. [-,-.5]. Let = + +. To wc zero o does te Bsecto metod coverge we ppled o te ollowg tervls?. [-,.5]. [-.5,] c. [-.75,.5] d. [-.5,.75]. Use Teorem to d oud or te umer o tertos eeded to ceve ppromto wt reltve error to te soluto o + 4 = lg te tervl [, 4]. 4. Use Teorem to d oud or te umer o tertos eeded to ceve ppromto wt solute error 6 to te soluto o = lg te tervl [, ]. 5. Wrte progrm tt mplemets te Bsecto metod d d solutos ccurte to wt or te ollowg prolems:. = or

38 . e + = or c. cos + = or d d. cos + = or.. d.. Soluto o oler equtos. Newto s metod.. I Newto's metod s used o strtg wt, wt s?. Devse Newto terto ormul or computg R / were R>.. Wrte progrm tt mplemets te Newto s metod d d solutos ccurte to wt or te ollowg prolems:. or. e + or c. cos + or d d. cos + or.. d.. Soluto o oler equtos. Fed pot metod.. Use Teorem to sow tt g= +.5 s/ s uque ed pot o [,].. Use Corollr to estmte te umer o tertos requred to ceve 4 ccurc.. Wrte progrm tt mplemets te ed-pot terto metod d d soluto ccurte to wt 4 d compre teoretcl estmte to te umer ctull eeded. Iterpolto. Use pproprte Lgrge terpoltg poloml o degree oe, two d tree to ppromte ollowg dt: 5 6. Costruct te Lgrge terpoltg polomls or te ollowg uctos, d d oud or te solute error o te tervl [, ]:., =., =.4, =., =., =., =.4, =. e cos, =, =., =.6, =. cos, =, =., =.6, = Numercl deretto. Derve te ollowg two ormuls or ppromtg dervtves d sow tt te re ot O 4 estlsg ter error terms.

39 Derve te ollowg two ormuls or ppromtg te trd dervtve d ter error terms. Wc s more ccurte? Numercl tegrto. Appromte te ollowg tegrls usg trpezod rule.5. d 4.5. l d c..5 4 d. Fd oud or te error Eercse usg te error ormul, d compre ts to te ctul error.. Repet Eercse usg Smpso's rule. 4. Repet Eercse usg Smpso's rule d te results o Eercse. Itl-vlue prolems or ODE. Use Euler's metod to ppromte te solutos or ec o te ollowg tlvlue prolems..,,,. 5. /,,,. 5. Te ctul solutos to te tl-vlue prolems Eercse re gve ere. Compre te ctul error t ec step to te error oud.. /. l. Repet Eercses d usg Heu's metod. 4. Repet Eercses d usg moded Euler s metod. 5. Repet Eercses d usg ourt-order Ruge-Kutt metod. 6. Use te Euler s metod d te secod order Ruge-Kutt metod to ppromte te soluto o te ollowg sstem o rst-order deretl equtos 4 cos 4s,,,,. s, Compre te result to te ctul soluto

40 e e s, e e. 7. Use te Euler s metod d te secod order Ruge-Kutt metod to ppromte te soluto o te ollowg ger-order deretl equto e,,,. Compre te result to te ctul soluto e /6 e e. Curve ttg. Fd te lest squres polomls o degrees d or te dt te ollowg tle. Compute te error E ec cse Fd te lest squres polomls o degrees, d or te dt te ollowg tle. Compute te error E ec cse Questo Emto questos. Lgrge terpolto poloml. Iterpolto teorem. Error lss.. Usg ucto =ep d pots =, =.5, =, =7.4, =4.5, =.7, clculte umercll rst dervtve te pot ; d oud or te solute error o clcultos. Questo. Numercl deretto. Forwrd-derece, ckwrd-derece d treepot ormuls or te rst dervtve. Error lss.. Costruct te Lgrge terpoltg poloml o degree or te ucto =ep usg pots =, =.5, =, =7.4, =4.5, =.7,

41 d d oud or te solute error o te tervl [, ]. Questo. Numercl deretto. Tree-pot ormul or te secod dervtve. Deretto v terpolto. Error lss.. Costruct te Lgrge terpoltg poloml o degree or te ucto =s/ usg pots =, =.5, =, =, =., =.5, d d oud or te solute error o te tervl [, ]. Questo 4. Numercl tegrto. Itegrto v terpolto. Trpezod rule, error lss. Mdpot d Smpso s rule.. Usg ucto =ep d pots =, =.5, =, =7.4, =4.5, =.7, clculte umercll secod dervtve te pot ; d oud or te solute error o clcultos. Questo 6. Lest squres metod o terpolto. Lest squres poloml. Norml equtos.. Use Euler's metod to ppromte te soluto or te ollowg tl-vlue prolem.,,,.5 Estmte te error oud. Questo 7. Itl-vlue prolems or ordr deretl equtos. Euler's metod. Error lss. Secod-order Ruge-Kutt metods.. Fd te lest squres poloml o degree or te dt te ollowg tle. Compute te error o ppromto.

42 Questo 8. Itl-vlue prolems or ordr deretl equtos. Secod-order Ruge- Kutt metods. Error lss... Fd te lest squres poloml o degree or te dt te ollowg tle. Compute te error o ppromto. Questo 9. Fte-dgt rtmetc. Rel dt tpes C++. Mce epslo. Asolute error d reltve error o clcultos.. Fd te lest squres poloml o degree or te dt te ollowg tle. Compute te error o ppromto. Questo. Soluto o oler equtos. Bsecto metod. Error lss.. Usg ucto =ep d pots =, =.5, =, =7.4, =4.5, =.7, clculte umercll secod dervtve te pot ; d oud or te solute error o clcultos. Questo. Soluto o oler equtos. Newto s metod. Error lss.. Use secod order Ruge-Kutt metod to ppromte te soluto or te ollowg tl-vlue prolem.,,,.5 Estmte te error oud.

43 Reereces. Hlderd, F. B Itroducto to Numercl Alss d edto ed.. McGrw-Hll.. Güter Hämmerl d Krl-Hez Hom 99. Numercl Mtemtcs, Sprger-Verlg.. Dvd R. Kcd d E. Wrd Cee. 99 Numercl lss: mtemtcs o scetc computg. Brooks/Cole Pulsg Comp. 4. J. Stoer d R. Bulrsc 99. Itroducto to Numercl Alss, Sprger- Verlg. 5. Leder, Jeer J. 4. Numercl Alss d Scetc Computto. Addso Wesle. 6. Jo H. Mtews d Kurts D. Fk 4. Numercl Metods usg Mtl, 4t Ed., Perso Pretce Hll. 7. Wllm H. Press Numercl Recpes C. Te rt o scetc computg. Cmrdge Uverst Press.

44 Tle o cotets Itroducto... 4 Course progrm Fte-dgt rtmetc Postol or rd otto Normlzed scetc otto Mce epslo Asolute d reltve errors Soluto o oler equtos Bsecto metod Error lss..... Newto s metod Grpcl terpretto o Newto s metod Error lss..... Fed-Pot Iterto Error lss Iterpolto d poloml ppromto Prolem sttemet Ler terpolto Geerl terpolto prolem Numercl deretto Prolem sttemet Two-pot ormuls or rst dervtve Error lss Tree-pot ormul or te rst dervtve Tree-pot ormul or te secod dervtve Deretto v terpolto Eercses Numercl tegrto Prolem sttemet Itegrto v terpolto Trpezod rule Smpso s rule Itl-vlue prolems or ODE Prolem sttemet Tlor-seres metod Error lss Euler's metod Secod-order Ruge-Kutt metods Fourt-order Ruge-Kutt metod Curve ttg Prolem sttemet Lest squres le... 4

45 7.. Lest squres poloml... 6 Home eercses... 6 Emto questos... 4 Reereces... 4 Tle o cotets... 44