Numerical Analysis Third Class Chemical Engineering Department University of Technology Assoc. Prof. Dr. Zaidoon Mohsin shakor

Transcription

1 Numercl Alss Thrd Clss Chemcl Egeerg Deprtmet Uverst o Techolog Assoc. Pro. Dr. Zdoo Mohs shkor

2 Itroducto to Numercl Alss U. Alss versus Numercl Alss The word lss mthemtcs usull mes who to solve problem through equtos. The solvg procedures m clude lgebr, clculus, deretl equtos, or the lke. Numercl lss s smlr tht problems solved, but the ol procedures tht re used re rthmetc: dd, subtrct, multpl, dvde d compre. Dereces betwee ltcl solutos d umercl solutos: A ltcl soluto s usull gve terms o mthemtcl uctos. The behvor d propertes o the ucto re ote ppret. However, umercl soluto s lws ppromto. It c be plotted to show some o the behvor o the soluto. A ltcl soluto s ot lws megul b tsel. Emple: s oe o the roots o. Whle the umercl soluto s ppromto, t c usull be evluted s ccurte s we eed. Actull, evlutg ltc soluto umercll s subject to the sme errors. U. Computers d Numercl Alss Numercl MethodsProgrms Computers Numercl Alss As ou wll ler eough bout m umercl methods, ou wll be ble to wrte progrms to mplemet them. Progrms c be wrtte computer lguge. I ths course ll progrms wll be wrtte Mtlb evromet. Actull, wrtg progrms s ot lws ecessr. Numercl lss s so mportt tht etesve commercl sotwre pckges re vlble. Numercl Alss/Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

3 U. Tpes o Equtos The equtos s dvded to three m ctegores such s below gure:- U. Kds o Errors Numercl Procedures The totl error comprses o: UModel ErrorU: due to the msmtch betwee the phscl stuto d the mthemtcl model. UDt ErrorU: due to the mesuremets o doubtul ccurc. UHum ErrorU: due to hum bluders. U Propgted ErrorU: the error the succeedg steps o process due to occurrece o erler error. 5 UTructo ErrorU: the oto o tructo error usull reers to errors troduced whe more complcted mthemtcl epresso s replced wth more elemetr ormul. Ths ormul tsel m ol be ppromted to the true vlues, thus would ot produce ect swers. Numercl Alss/Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

4 Emple.: Tructo o te seres to te seres to te umber o terms leds to the tructo error. For emple, the Tlor seres o epoetl ucto e...!!! I ol our terms o the seres re used, the e!! e !! The tructo error would be the uused terms o the Tlor seres, whch the re E t 5 5! 5!! 5!.565 Check ew Tlor seres ppromtos o the umber e, or,, d. Gve tht e.788. Order o Appromto Percet reltve Absolute error or e error % % % 6 URoud-O ErrorU: AT T TroudT-To errort, lso clledt T Troudg errort, s the derece betwee the clculted ppromto o umber d ts ect mthemtcl vluet TdueT Tto roudg Emple.: Numbers such s π, e, or cot be epressed b ed umber o decml plces. Thereore the cot be represeted ectl b the computer. Cosder the umber π. It s rrtol,.e. t hs tel m dgts ter the perod: π The roud-o error computer represetto o the umber π depeds o how m dgts re let out. Numercl Alss/Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

5 Let the true vlue or π s.59. Number o dgts Appromto Percet reltve Absolute error Decml dgt or π error..59.9% %..7.% U.5 Errors Numercl Procedures There re two commo ws to epress the sze o the error computed result: bsolute error d reltve error. Absolute error true vlue ppromte vlue, whch s usull used whe the mgtude o the true vlue s smll. true vlue - ppromte vlue Reltve error, whch s desrble oe. true vlue Whle true vlue ppromte vlue Percet reltve error, ε t % true vlue Numercl Alss/Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

6 Soluto o Frst-Order Ordr Deretl Equtos A equto tht cossts o dervtves s clled deretl equto. Deretl equtos hve pplctos ll res o scece d egeerg. Mthemtcl ormulto o most o the phscl d egeerg problems led to deretl equtos. So, t s mportt or egeers d scetsts to kow how to set up deretl equtos d solve them. Deretl equtos re o two tpes Ordr deretl equto ODE. Prtl deretl equtos PDE. A ordr deretl equto s tht whch ll the dervtves re wth respect to sgle depedet vrble. Emples o ordr deretl equto clude: d,, d d d d,, d d d d d d d 5 s d d d d s, d, d, Frst order ordr deretl equtos re o the orm: d t, wth dt O the let hd sde s the dervtve o the depedet vrble wth respect to the depedet vrble t. O the rght hd sde, there s ucto tht m deped o both d t. M deretl equtos cot be solved ectl. Numercl methods hve bee developed to ppromte solutos. Numercl lss s eld mthemtcs tht s cocered wth developg ppromte umercl methods d ssessg ther ccurc, or stce or solvg deretl equtos. We wll dscuss the most bsc method such Tlor, Euler d Ruge-Kutt methods. U. Tlor Seres Method Fucto c be epded over smll tervl usg the Tlor seres rom strt or reerece pot h h h h h!!! Where h h, costt. Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

7 P P P Emple. Solve the ollowg ordr deretl equto ODE usg Tlor s method o order wth h. d d P Soluto d d R R RR, or, wth.5 d P P d RR RRP h RR RRP RR hp U.. Euler s Method Euler s method s the smplest d lest useul o these three methods. I we re solvg rst-order deretl equto o the orm d dt t, wth the tl codto A, Euler s method begs b ppromtg the rst dervtve s d dt t t t t Settg ths equl to t, d solvg or t t elds the ollowg lgorthm or dvcg the umercl soluto o ordr deretl equto: d d, h, Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

8 Usg Euler s method we hve the ollowg cosderto: h h h h h, h,, Eercse.: Appl Euler s method to ppromte the soluto o the tl vlue problem d d wth 5 Soluto We kow tht the ltcl soluto o equto s, 5ep. We umercll solve equto usg Euler s method wth h. the tme tervl [,.5], d the check how well ths method perorms. We hve. The 5 Ad 5 5 h.. h... h... h... h...5 h h h h h h We summrze ths the ollowg tble. I h., the Ect Derece Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

9 The thrd colum cots the ect vlues, 5ep. The lst colum cots the bsolute error ter ech step, computed s -REctR. We see tht whe h., the umercl ppromto s ot ver good ter ve steps. I we repet the sme ppromto wth smller vlue or h, s h., the ollowg tble results or the rst ve steps: Ect Derece Dog ve steps ol gets us to.5. We c do more steps utl we rech.5. We d tht the l pot wll be: Ect Derece Choosg smller vlue or h resulted better ppromto t.5 but lso requred more steps. Oe source o error the ppromto comes rom the ppromto tsel.. Fourth order Ruge-Kutt Method To d umercl soluto to the tl vlue problem,, usg Ruge-Kutt method we hve the ollowg cosderto: k k k k h 6 k, k k h, kh h, kh h k h k, Ths method gves more ccurte result compred to Euler s method Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor d d

10 Emple.: Solve the ollowg ordr deretl equto ODE usg ourth order Ruge-Kutt method to clculte. d ;, h. d Soluto: krr krr.5.5. krr krr h o k k k k krr... krr krr krr t..8 Emple.: A bll t K s llowed to cool dow r t mbet temperture o K. Assumg het s lost ol due to rdto, the deretl equto or the temperture o the bll s gve b dt.67 T 8 8, T K dt where T s K d t secods. Fd the temperture t t 8 secods usg Ruge-Kutt th order method. Assume step sze o h secods. Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

11 Soluto dt.67 dt 8 T 8 8 t,t.67 T 8 h T T 6 k k k k For, t, T K k t,, T t h,t k h,.5579 k 8, t h,t k h,.87 k 8, t h,t k h,.89,65. k h T T k k k k K.6975 T s the ppromte temperture t ttr t t h For, t,t K k t,, T t h,t k h, k 8 6, Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

12 .7 t h,t k h, ,68. k t h,t k h, ,59.9 k h T T k k k k K T s the ppromte temperture t tme tr t h 8 t Tble d Fgure show the eect o step sze o the vlue o the clculted temperture t t 8 secods. Tble Vlue o temperture t tme, t 8s or deret step szes Step sze, h T8 E t ε t % Emple.5 Usg Mtlb Commds solve the ollowg equto usg both Eulr d Ruge-Kutt method d to ppromte the soluto o the tl vlue problem d d, Soluto: wth step sze h.. Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

13 Eulr cler ll, clc,ormt short ; ; h.5 or :5 h; d; h*d; ed _ect --*ep; error_ect- tble[',',_ect',error'] tble Ruge Kutt cler ll, clc,ormt short ;;h.; le''; %, or :5 h; k,; k h/,k*h/; k h/,k*h/; k h,k*h; /6*h*k *k *k k; ed _ect --*ep; error_ect- tble[',',_ect',error'] tble Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

14 Solvg Smulteous Frst-Order Ordr Deretl Equtos. Itegrto two smulteous rst-order ordr deretl equtos Cosder the ollowg sstem o rst-order ODE s descrbg the depedece o two depedet vrbles d z o oe depedet vrble : d,, z d dz g,, z d These two deretl equtos re coupled d must be tegrted smulteousl becuse both equtos volve both depedet vrbles. Itl codtos re requred gvg the vlues o d z t the tl vlue o. The lgorthm or th-order Ruge-Kutt tegrto o two coupled ODEs s: z k k k k k k k k h k 6 h z k 6 k k k k k k,,z g,,z.5h,.5hk,z.5hk g.5h,.5hk,z.5hk.5h,.5hk,z.5hk g.5h,.5hk,z.5hk h, hk,z hk g h, hk,z hk As emple eothermc recto usted-stte cotuous strred tk rector d eothermc recto plug low rector wth het echge through the rector wll. From the oe d two ODE emples, ou c eted the method to tegrto o three coupled ODE s. Three coupled ODE s would be ecoutered, or emple, or recto o gses sted-stte o-sotherml plug low rector wth sgct pressure drop dc/d, dt/d, d dp/d. Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

15 . Itegrto o sstem o rst-order ordr deretl equtos d,,,, d d,,,, d dm d m,,,, m The soluto o the bove equtos s: m m h k 6, k, k, k, Where,,..., m d k,, m,, k k h, hk hk,,,,, m h, hk hk,,,,, m k, h, hk,, hk,, m hk hk,m hk,m, m The de o the soluto to sstem o deretl equtos s smlr to soluto o sgle deretl equto. Emple.: Usg ourth order Ruge-Kutt method wth step sze h. solve d, d d d, - To clculte. d. Soluto At,, - k, - - k, - k,.5hk,.5hk,.5h Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

16 k,.5h.5hk,.5hk, k,.5hk,.5hk,.5h k,.5h.5hk,.5hk, k, hk, hk, h k, h hk, hk, t.. h/6k, k, k, k,../6 [ ].99. h/6 k, k, k, k,. - [ ] -.99 Emple. Use ode5 Mtlb Commd to solve the ollowg rst order sstem or d. t d, d d d, - Usg ourth order Ruge-Kutt method wth step sze h. Soluto The Mtlb routes ode5 c be used to solve the sstem. A Mtlb ucto must be creted to evlute the slopes s colum vector. The ucto me ths emple s eode, whch must be sved rst the hrd drve wth the sme me eode.m. ucto dd eode, dd,*; dd,*; The commd ode5 s the evluted rom the commd wdows. Mtlb wll set the step sze to cheve preset ccurc tht c be chged b user. The depedet vrble c lso be speced t cert loctos betwee the tl d l vlues d Mtlb wll provde the depedet vlue t these loctos. Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

17 sp:.:; [,]ode5'eode',sp,[, -] Eercse.: Let s cosder smple emple o model o plug low rector tht s descrbed b sstem o ordr deretl equtos. A plug low rector s operted s show Fgure 9. below. A k B k C Fgure 9. Isotherml plug low rector The plug low tll hs ol rectt A, the compoets A rect to orm compoet B. The mole blce or ech compoet s gve b the ollowg deretl equtos dca u kca dz dcb u kca k CB dz dcc u k CB dz Wth the ollowg tl vlues C A z kmol/m C B z C C z d k k Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

18 I u.5 m/s d rector legth z m. Solve the deretl equtos d plot the cocetrto o ech speces log the rector legth Soluto: We ll strt b wrtg the ucto deg the rght hd sde RHS o the ODEs. The ollowg ucto le emple s used to set up the ode solver. ucto dc Emple z, C u.5; k; k; dc, -k *C / u; dc, k *C-k *C / u; dc, k *C/ u; Now we ll wrte m scrpt le to cll ode5. CA, CB d CC must be deed wth the sme mtr, d so b cllg CA s C, CB s C d CC s c, the re lsted s commo to mtr C. The ollowg ru le s creted to obt the soluto: cler ll, clc [z, C] ode5' Emple', [:.:], [ ] plot z,c:,,'k-',z,c:,,'k*:',z,c:,,'kd-.' lbel 'Legth m'; lbel 'Cocetrtos kmol/m^ '; leged 'A', 'B', 'C' The produced plot s s Fgure 9. Cocetrtos kmol/m A B C Legth m Fgure 9.: A, B d C cocetrtos log plug low rector Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

19 . Solvg Hgher Order Ordr Deretl Equtos We hve lered Euler s d Ruge-Kutt methods to solve rst order ordr deretl equtos o the orm d d,, Wht do we do to solve deretl equtos tht re hgher th rst order? For th emple order deretl equto o the orm d d d o d d d wth tl codtos c be solved b ssumg z d dz z d d d dz z d d d dz d d d dz d d z d d d d z z z The bove Equtos rom to represet rst order deretl equtos s ollows dz z d dz d dz d z z z,,, z z,,, z z z Ech o the rst order ordr deretl equtos s ccomped b oe tl codto. These rst order ordr deretl equtos re smulteous ture but c be solved b the methods used or solvg rst order ordr deretl equtos tht we hve lred lered. Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

20 Hgher Order Ordr Deretl Equtos d d e s d d wth d -.,. 6 d d d d d d wth d, d d,, d d d d wth d d d.5, d, d.5, d d d, Sstem o rst Order Ordr Deretl Equtos d z,. d dz d e s z, z. 6 d z, d dz u, z d du u z, u d d z, d dz u, z d du v, u. 5 d dv z, v. 5 d Emple. Re-wrte the ollowg deretl equto s set o rst order deretl equtos. d d 5 e d d, 5, 7 Soluto The ordr deretl equto would be rewrtte s ollows. Assume d z, The d dz d d d Substtutg ths the gve secod order ordr deretl equto gves dz z 5 e d dz e z 5 d The set o two smulteous rst order ordr deretl equtos complete wth the tl codtos the s Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

21 d d 5 z, e z 5, z 7 dz d. Now oe c ppl o the umercl methods used or solvg rst order ordr deretl equtos. Emple.5 Gve the thrd-order ordr deretl equto d ssocted tl codtos d d d d d d 5,, d.6, d d d.. Wrte ths deretl equto s sstem o rst-order ordr deretl equtos b. Usg ourth order Ruge-Kut method to estmte. d. tkg Δ. d z, d dz u, z. 6 d du u 5z, u. d b Soluto Frst step o tegrto,, z.6, u., Δ. k z.6 k u. k- u-5 z kz.5 Δ k ku.5 Δ k k.5 Δ - u.5 Δ k-5 z.5 Δ k-.5 Δ k kz.5 Δ k ku.5 Δ k k.5 Δ - u.5 Δ k-5 z.5 Δ k-.5 Δ k Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

22 kzδ k kuδ k kδ - uδ k-5 zδ k-δ k Δ.. Δ/6 k k kk./ zzδ/6*k*k*kk.6./ uuδ/6*k*k*kku./ d dz d The t.,.599, z.587, u -.7 d d d Secod step o tegrto., z.587, u-.7, Δ. k z.587 k u -.7 k- u-5 z kz.5 Δ k ku.5 Δ k k.5 Δ- u.5 Δ k-5 z.5 Δ k-.5 Δ k kz.5 Δ k ku.5 Δ k k.5 Δ - u.5 Δ k-5 z.5 Δ k-.5 Δ k kzδ k kuδ k kδ- uδ k-5 zδ k-δ k Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

23 Δ... Δ/6 k k kk.599./ zzδ/6*k*k*kk.587./ uuδ/6*k*k*kk -.7./ d dz d The t.,.55, z.575, u d d d Soluto Usg Mtlb: We ll strt b wrtg the ucto deg the rght hd sde RHS o the ODEs. The ollowg ucto le E s used to set up the ode solver. ucto dq E,q q;zq;uq; dq,z; dq,u; dq,^-*u-5*z-; The ollowg ru le s creted to obt the soluto: cler ll,clc,ormt compct [,q]ode5'e',[:.:.],[,.6,.] q:, The produced results wll be.. q Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

24 P Iterpolto d Appromto Wht s terpolto? M tmes, dt s gve ol t dscrete pots such s RR, RR, RR, RR,. RR, RR, RR, RR. So, how the does oe d the vlue o t other vlue o? Well, cotuous ucto m be used to represet the dt vlues wth pssg through the pot Fgure.. The we c d the vlue o t other vlue o. Ths s clled terpolto. O course, lls outsde the rge o or whch the dt s gve, t s o loger terpolto, but sted, s clled etrpolto.,,,, Fgure. Iterpolto o dscrete dt For dt pots, there s oe d ol oe poloml o order tht psses through ll the pots. For emple, there s ol oe strght le tht s, rst-order poloml tht coects two pots. Smlrl, ol oe prbol coects set o three pots. Poloml Iterpolto cossts o determg the uque P th order poloml tht ts dt pots. Ths poloml the provdes ormul to compute termedte vlues. Oe o the methods used to d ths poloml s clled the Lgrge method o terpolto. Other methods clude Newto s dvded derece poloml method d the drect method. Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

25 RR 9. 9.RR 9.RR. Lgrge Iterpoltg Poloml Cosder ucto tht psses through the two dstct pots RR, RR d RR, RR s show Fgure.. The rst order poloml tht ppromtes the ucto betwee these two pots c be epressed s b Where d b re costts. c lso be wrtte Lgrg orm s crr RR crr RR Fgure. Frst d secod order poloml ppromto. I. Ler B weghtg the verge o the two vlues used to produce the coordtes o the le the ormul: L L where: L L Emple. Compute -decml plce vlue o l 9. rom l 9..97, l b ler Lgrge terpolto d determe the error, usg the ect vlue o l Soluto: 9., RR 9.5, RR l 9..97, RR l ; hece we get 9.5 L. 9.5,.5 9. L. 9.,.5 l 9. prr LRR LRR L L 9... The bsolute error s Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

26 II. Qudrtc B weghtg the verge o the three pots tht produce the prbol we c derve the ormul: L L L where: L L L Emple. Compute l 9. rom the dt the prevous emple. d the ddtol thrd vlue l Soluto: 9.5. L L L L 9..5 L 9..8 L 9.. l 9. prr The bsolute error s III. Geerl Lgrge Iterpoltg Poloml I geerl, the Lgrge poloml c be represeted s: L where L j j Emple. Fd the Lgrge terpolto poloml tht tkes the vlues prescrbed below Rk RkR 5 Soluto PRR L, k RkR PRR k 5 j j Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

27 P Whe workg wth grds hvg lrge umbers o tervls oe tpcll ssgs set o low degree,, or bss uctos to ech djcet set o,, or odes. Emple. The ollowg tble gves the vlue o dest o sturted wter or vrous tempertures o sturted strem. o Temp P PC T 5 5 Dest kg/mp d Use thrd order Lgrge terpoltg polomls to correlte dest s ucto o temperture. o Fd the destes whe the tempertures re P PC. Soluto T 5T T L -. T T 5 5 T T T L T -. T.8T T T 5T L - T T -.T T T 5T -6 - L. T 6 T L L L L - T - T -.566T T Emple.5 Use Lgrge globl terpolto b oe poloml d pecewse poloml terpolto wth qudrtc or the ollowg odes. Rk 5 RkR Soluto Globl terpolto b oe poloml: P L, k RkR PRR Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor k

28 Pecewse poloml terpolto wth qudrtc PRR PRR ; ; 5. Newto Dvded Derece Iterpoltg The Lgrg terpolto polomls re useul dscussos o umercl tegrto. A ltertve terpolto s Newto s Dvded Derece Iterpolto. It volves ewer rthmetcl opertos. Aother dvtge o Newto s rests wth the ollowg scero. Suppose we eed to mprove the ccurc d crese the umber o grd pots. From the orms o Lgrge terpolto polomls, ll the terms hve to be evluted oce g, d ths s huge mout o work the umber o pots s lrge. Newto s does ot suer rom ths drwbck, d just oe ddtol term eeds to be computed. I. Ler Iterpolto Cosder the dgrm below whch curve s modeled poorl b : Usg smlr trgles the slopes re the sme d hece: Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

29 to d d we c Ad thus the coordte o the curve t RR bove to become: be ppromted b rerrgg the Emple.6 Estmte the commo logrthm o usg ler Newto s terpolto. Iterpolte betwee log 8.99 d log.798. b Iterpolte betwee log d log.97. For ech o the terpoltos, compute the percet reltve error bsed o the true vlue. USoluto ε t %.886% b ε t %.8% II. Qudrtc Iterpolto To reduce the error, qudrtc terpolto tht troduces some curvture to the terpolto s used. The orm: Let RR Let RR b b b produce: b use the prevous dett to produce: b Ad g b substtuto o brr b brr Numercl Alss /Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor derve tht: b b b

30 RR RR RR brr Emple.7 Ft secod-order Newto s Iterpoltg poloml to estmte log usg the dt rom Emple.7 t 8, 9, d. Compute the true percet reltve error. Soluto Frst, order the pots b b 9 RR.955 RR.97 8 RR Substtutg these vlues elds the qudrtc ormul whch c be evluted t or ε t. %.% III. Geerl orm o Newto Dvded Derece Iterpoltg Poloml I geerl, we d the te dereces deed s: j [, j] [, [, j] [ j, k ] [, j, k ] [,,..., ] j k,..., ] [,,..., The the geerl Newto Iterpoltg Poloml o order wth dt pots s deed s: ] b b b... b... Numercl Alss /Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

31 P C P P P Where b b b b [ [ [, ],,, ],..., ] b b [, ] b [,, ] b [, ] [,,, ] [,, ] [, ] For emple o thrd order poloml, gve,,,,,, d,, [ ] [ [,, ],, [ ],, ] Emple.8 The ollowg tble gves the vlue o dest o sturted wter or vrous tempertures o sturted strem. o TempP T : 5 5 Dest kg/mp P d : Usg Newto dvded derece terpoltg d the destes whe the tempertures re o o P PC d 75P PC respectvel. Soluto T D [RR,RR] [RR,RR,RR] [RR,RR,RR,RR] [RR,RR,RR,RR,RR] P P P Numercl Alss /Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

32 P P T - -. T -*T T -*T -5*T - *T T -*T -5*T - P T -.5T.667 T T P kg/m P kg/m Or b drect substtuto P kg/m P P P P P kg/mp P P P P kg/mp -6-6 Numercl Alss /Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

33 SRrR s d re Curve Fttg Wht s regresso? Regresso lss gves ormto o the reltoshp betwee respose vrble d oe or more depedet vrbles to the etet tht ormto s coted the dt. The gol o regresso lss s to epress the respose vrble s ucto o the predctor vrbles. Oce regresso lss reltoshp s obted, t c be used to predct vlues o the respose vrble, det vrbles tht most ect respose, or ver hpotheszed csul models o the respose.. Ler regresso Ler regresso s the most populr regresso model. I ths model we wsh to predct respose to dt pots RR,RR, RR,RR,..., RR, RR dt b regresso model gve b. Where RR RR the costts o the regresso model. A mesure o goodess o t, tht s, how predcts the respose vrble s the mgtude o the resdul, ε t ech o the dt pots. ε Idell, ll the resduls ε re zero, oe m hve oud equto whch ll the pots le o the model. Thus, mmzto o the resdul s objectve o obtg regresso coecets. The most populr method to mmze the resdul s the lest squres method, where the estmtes o the costts o the models re chose such tht the sum o the squred resduls s mmzed, tht s mmzeε. Let us use the lest squres crtero where we mmze S ε r clled the sum o the squre o the resduls. Numercl Alss /Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

34 Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor Fgure. Ler regresso o vs. dt showg resduls t tpcl pot, RR. To d RR d RR, we mmze SRrR wth respect to RR d RR: r S r S Gvg Notg tht..... Solvg the bove equtos gves: Or rom equto.,,,,, ε

35 o Emple. The ollowg vs. dt s gve vs Fgure. Dt pots o the vs dt Although s ect t to the dt, scetst thks tht c epl the dt. Fd costts o the model,, d, Soluto Frst d the costts o the ssumed model Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

36 RR d c P TRR DR Ths gves P UEmple. The ollowg tble gves the vlue o dest o sturted wter or vrous tempertures o sturted strem. o Temp PC T 5 5 Dest kg/mp D Use curve ttg to t the results to rst-order poloml D A BT. b Fd the destes whe the tempertures re P PC d 75P PC respectvel. USoluto: RR be computed b costructg the ollowg tble: TRRP TR DR D9.-. T Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor o o

37 P To compre the predcted vlues to the epermetl vlues: TR DR DRRestmted D9.-. T D D Poloml Models Gve N dt pots RR, RR, RR, RR.., RNR, RNR use lest squres method to regress th the dt to P order poloml. I the developmet, we use s the degree o the poloml d N s the umber o dt prs,. We wll lws hve N > the ollowg. Assume the uctol reltoshp or ttg Y wth errors deed b e Y, whch,,,, N. We mmze the sum o error squres, N N. S e At the mmum, ll the rst prtl dervtves wth respect to S N N S N S,,, S N, Rerrge them to get N N N N N, s vsh. We hve Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

38 N N N N N, N N N N N, N N N N N, or, mtr orm, N N N N N N N N N N N N N N N N N N N N.. Equtos. represet ler sstem. However, ths sstem s usull ll-codtoed d roud-o errors c dstort the soluto o s. Up to degree- or, the problem s ot too gret. It s ver requet to use degree hgher th. Emple. Rotmeter clbrto dt low rte versus Rotmeter redg re s ollows: Rotmeter Redg R Flow rte VL/m Usg curve ttg to t the clbrto dt to secod order poloml. b Clculte the lowrte V t rotmeter redg R7. Soluto: d P P order poloml S r ds d r o ds d r ds d r ε o o o o Re rrgg bove equtos Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

39 RP RP RP o o o Mkg requred tble R V RP RV P B substtuto equto 5 o o o Solvg bove equto smulteousl gves; o.8786,.598,.857 P the V B R.857 V R Numercl Alss /Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

40 . Noler Dt Wheever dt rom epermetl tests re ot ler, we eed to t to them some ucto other th rst-degree poloml. Populr orms tht re tred re the power orm b or the epoetl orm b e. Sce such oler equtos re much more dcult to solve th ler equtos, the re usull lerzed b tkg logrthms beore determg the prmeters: l l bl, or l l b. ler o-ler Fgure. Ler vs o-ler dt I cses whe such lerzto o the ucto s ot desrble, or whe o method o lerzto c be dscovered, grphcl methods re requetl used; oe merel plots the epermetl vlues d sketches curve tht seems to t well. Emple. The progress o homogeeous chemcl recto s ollowed d t s desred to evlute the rte costt d the order o the recto. The rte lw epresso or the recto s kow to ollow the power ucto orm r kc Use the dt provded the tble to obt d k. C A gmol/l r A gmol/l s Soluto Tkg the turl log o both sdes o Equto, we obt r l k l C l Numercl Alss /Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

41 Let z l r w l C l k mplg tht k e We get z w Ths s ler relto betwee z d w, where w z w w w z z Tble: Ketcs rte lw usg power ucto 7 7 w 7 z 7 7 w C r w z w z w z w.998 From bove equtos w Numercl Alss /Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

42 The 7 k e Fll, the model o progress o tht chemcl recto s r.78 C.9 Emple.5 It s suspected rom theoretcl cosdertos tht the rte o wter low rom rehouse s proportol to some power o the ozzle pressure. Assume pressure dt s more ccurte. You re trsormg the dt. Flow rte, F gllos/m Pressure, p ps b Wht s the epoet b o the ozzle pressure the regresso model F p Numercl Alss /Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

43 Soluto The lerzto o the bove dt s doe s ollows. Where b F p l F l b l p z z l b F l p Implg l e There s ler reltoshp betwee z d. Ler regresso costts re gve b b Sce z z the z z z l55 l5 z l l96 l7 l9 l l5 l5 l5 l l68 l l l l7 l l5 l l55 9. l96 l9 l5 l5 l68 l l b l5 l l Numercl Alss /Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

44 Emple.6 The ollowg dt hve bee obted or the decomposto o bezee dzoum chlorde to chlorobezee: T K k sp P From ths dt, determe the pre-epoetl ctor A d ctvto eerg E, ssumg tht the rte costt ollows Arrheus orm. E k Aep RT Soluto: l k l A l k /T l A o E R We get o E RT T K - k sp P /T l k P e e e e e e o.7 o l A.7 A ep.7 8. E / R E R Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

45 A Mtlb progrm or solvg emple.6 s lsted Tble.. Tble. Mtlb code d results or soluto emple.6 Mtlb T[,9,,8,]; Code K[.,.,.8,.55,.77]; Results E./T; logk; Polpolt,,; E-Pol*8. AoepPol.8e5 Ao 8.e7 The comprso betwee epermetl d predcted k vlues s show below gure: Recto rte costt k k Ep. k Pred Temperture K Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

46 Numercl Itegrto Numercl Itegrto Appromto. Itegrto s the process o mesurg the re uder ucto plotted o grph. Sometmes, the evluto o epressos volvg these tegrls c become dutg, ot determte. For ths reso, wde vret o umercl methods hve bee developed to d the tegrl. Here we dscuss s deret methods or ppromtg the vlue o dete tegrl. Ech method revolves roud ssoctg dete tegrl wth re uder curve. The rst three use res o rectgles, the ourth uses res o trpezods, d the l ppromto techque uses res o shpes tht clude porto o prbol.. Let-Edpot Appromto O ech o the our subtervls show below, we crete rectgle whose wdth s the legth o the subdvso d whose heght s determed b the ucto vlue t the let edpot o ech subdvso. wdth:, heght: wdth:, heght: / e wdth:, heght: / wdth:, heght: / The sum o the res o the our rectgles represets our ppromto or the re uder the curve d thereore represets ppromto or the vlue o the dete tegrl: Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

47 Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor d e Ths sme sequece o steps c be geerlzed or let-edpot ppromto o the dete tegrl d b usg subdvsos: b d. Rght-Edpot Appromto Ag we crete rectgles whose wdths re ech the legth o subdvso, but here ech heght s determed b the ucto vlue t the rght edpot o ech subtervl. The sum o the res o these our rectgles represets rght-edpot ppromto or the re uder the curve d thereore s ppromto or the vlue o the dete tegrl: wdth:, heght: e wdth:, heght: / wdth:, heght: / wdth:, heght: /

48 Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor d e Ths sme sequece o steps c be geerlzed or rght-edpot ppromto o the dete tegrl b d usg subdvsos: b d. Mdpot Appromto For thrd tme we crete rectgles ech o whose wdth s the legth o the subdvso, but ow ech heght s determed b the ucto vlue t the mdpot o ech subdvso. The sum o the res o these our rectgles represets mdpot ppromto or the re uder the curve d thereore s other ppromto or the vlue o the dete tegrl: wdth:, heght: / 7 / 8 e wdth:, heght: / / 8 wdth:, heght: / / 8 wdth:, heght: / 5 / 8

49 Numercl Alss /Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor d e Ths sme sequece o steps c be geerlzed or mdpot ppromto o the dete tegrl b d usg subdvsos: b d. Trpezodl Rule Trpezodl rule s bsed o the Newto-Cotes ormul tht oe ppromtes the tegrd b P th P order poloml, the the tegrl o the ucto s ppromted b the tegrl o tht P th P order poloml. Itegrtg polomls s smple d s bsed o the clculus ormul. The heght o ech trpezod s the legth o the subdvso. The two bses o ech trpezod correspod to the vlues o the ucto t the edpots o the subtervl o whch the trpezod hs bee drw. heght:, bses: / d e heght:, bses: d / heght:, bses: / d / heght:, bses: / d /

50 Numercl Alss /Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor It m be useul to remove the rst o these trpezods d rotte t to more covetol oretto s we clculte ts re. The sum o the res o these our trpezods represets ppromto or the re uder the curve d thereore s oe more ppromto or the vlue o the dete tegrl: d e Ths sme sequece o steps c be geerlzed or trpezod ppromto o the dete tegrl d b usg subdvsos: b d Sgle Segmet Trpezodl Rule b d Multple Segmets Trpezodl Rule b d heght: bse legth: bse legth: / Are / /

51 Emple. Evlute the tegrl I b trpezodl rule dvdg the tervl [, ] to ve equl prts. Soluto 5 5 / d From Trpezodl Rule; I [ 5 6]. [ ].886 Emple. Use Multple-segmet Trpezodl Rule to d the re uder the curve rom to. e Soluto Usg two segmets, we get 5 e e.6 e 5 Are [ 5 ] [.9.6] Numercl Alss /Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

52 TP CRpR PTP P P TP P P e So wht s the true vlue o ths tegrl? d Mkg the reltve true error % t 79.56% 6.59 Tble: Vlues obted usg Multple-segmet Trpezodl Rule or Appromte Vlue E t d e % % % % % % % t Emple. The verge vlues o ucto c be determed b:- Cp mh T T CpdT T T CRpR P - PT9.75 P P -.95 P KJ/Kg K Use ths reltoshp to vert the verge vlue o specc het o dr r the rge rom K to 5 K: Altcll Numercll usg ve pots Trpezodl Rule Soluto Cp mh T T T.95 - T dt - Numercl Alss /Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

53 Cp mh Cp mh T T T T T T 5 5 T Cp Cp mh dt/ *Cp *Cp Cp Cp Cp5 T T 7.5/*.89 * AltclSoluto - Numercl Soluto Reltve Error % % % % AltclSoluto.67.5 Smpso s Rule / Smpso s Rule The l ppromto techque we develop ths secto s clled Smpso s Rule. It s deret rom the rst our methods becuse we re ot cretg polgos o ech subtervl but rther we crete gure wth o-strght compoet to t. For ths method, t s requred tht the umber o subtervls be eve umber. A prbol s creted tht cots the pots,,,, d,. e Aother prbol s creted tht cots the pots,, Numercl Alss /Lec. - - Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

54 Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor Smpso s Rule uses prs o subdvsos d cretes over ech pr prbol tht cots the pots R-R, R-R, R-R, R-R, d RR, RR or gog rom to P P/RR. A shpe s creted usg the resultg prbol, two vertcl segmets oe rom R-R, to R R, R-R d oe rom RR, to R R, RR d the segmet o the -s wth edpots R-R, d RR,. The re o the resultg shpe such s o the red-shded gure bove or the gree-shded gure bove-s clculted usg the ormul. The sum o the res o these shpes represets ppromto or the re uder the curve d thereore s ppromto or the vlue o the dete tegrl: d e Ths sme sequece o steps c be geerlzed or the Smpso s Rule ppromto o the dete tegrl d b usg subdvsos: b d

55 P Sgle Segmet / Smpso s Rule b d Multple Segmet / Smpso s Rule b d Emple. Evlute the tegrl.8 d I b / Smpso s rule dvdg the tervl [,.8] to equl sub-tervls. Soluto From Smpso s /P.8 rd Rule I d [ ] [ [ ]. [. [ ] ].689 ].6 Smpso s Rule /8 Smpso s Rule Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

56 P I we coect the pots o the curve usg P uder the curve c be ppromted b the ollowg ormul: b d [ rd order Lgrge poloml, the re ] 5 Sgle Segmet /8 Smpso s Rule b d [ ] 8 Multple Segmet /8 Smpso s Rule b d 8 Emple.5 Evlute the tegrl o the ollowg tbulr dt wth The trpezodl rule. b Smpso s rules F USoluto Trpezodl rule 5:. I [ ].5 b Smpso s rules 5:. I [ 8 ]. [.5 5 ] Emple.6 The volume o s gve b ollowg epresso: V F CA A o.9 d A k A 7 wth k.7 ep 65/ T m d T 9A 5 usg.5.75 A - F A 5mol/m, CA.5 mol LP Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

57 Clculte the volume o the rector usg Smpsos rule wth ve pots steps. Soluto X T k k A V 5 /.5*.5 / *. *7.6 *.95 * L A Mtlb progrm or solvg emple.5 s lsted Tble.. Tble. Mtlb code d results or soluto emple.5 Mtlb Code Results X:.5:.9 T59*X./.5*X.75 k.7e7*ep-65./t./k.*-x V5/.5*.5/****5 X T k V.878e Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

58 Numercl Deretto Numercl Deretto s method used to ppromte the vlue o dervtve over cotuous rego [,b]. Let s cotuous ucto wth step sze h. There re orwrd, bckwrd d cetered derece methods to ppromte the dervtves o t pot RR. 5. Forwrd Derece Appromto o the Frst Dervtve We kow lm For te ' '. Fgure 5.: Grphcl represetto o orwrd derece ppromto o rst dervtve So ou wt to d the vlue o t ' Δ ' hed s. Ths gves Where Δ, we m choose other pot Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

59 Emple 5. t The veloct o rocket s gve b ν t l 9.8t, t Where 'ν' s gve m/s d 't' s gve secods. Use orwrd derece ppromto o the rst dervtve o ν t to clculte the ccelerto t t 6s. Use step sze o t s. Soluto t t 6 Δ t t t t ν ν t t t 6 8 Hece ν ν ν 8 l ν 6 l ν 8 ν m / s 9.7m / s.75m / s The ect vlue o 6 c be clculted b derettg dt d t l 9.8t t t t t d 9.8 dt t m / s The bsolute reltve true error s True Vlue - Appromte Vlue t.699% True Vlue 9.67 Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

60 5. Bckwrd Derece Appromto o the Frst Dervtve We kow For te I lm ' Δ ', ' Δ ' s chose s egtve umber, Ths s bckwrd derece ppromto s ou re tkg pot bckwrd rom. To d the vlue o t, we m choose other pot ' Δ ' behd s. Ths gves where Fgure 5. Grphcl represetto o bckwrd derece ppromto o rst dervtve Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

61 Emple 5. The veloct o rocket s gve b ν t l 9.8t, t t Use bckwrd derece ppromto o the rst dervtve o ν t to clculte the ccelerto t t 6 s. Use step sze o t s. Soluto t t ν 6 Δ t t t ν t t t t ν ν ν 6 l ν l ν ν The bsolute reltve true error s 9.7m / s.m / s m / s t.557% 9.67 Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

62 5. Cetrl Derece Appromto o the Frst Dervtve As show bove, both orwrd d bckwrd dvded derece ppromto o the rst dervtve re ccurte o the order o Δ. C we get better ppromtos? Yes, other method to ppromte the rst dervtve s clled the Cetrl derece ppromto o the rst dervtve. From Tlor seres Δ Δ Δ!!!! Subtrctg equto rom equto!! Hece showg tht we hve obted more ccurte ormul s the error s o the order o. Fgure 5. Grphcl Represetto o cetrl derece ppromto o rst dervtve. Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

63 Emple 5. The veloct o rocket s gve b ν t l 9.8, t t. t Use cetrl dvded derece ppromto o the rst dervtve o ν t to clculte the ccelerto t Soluto t t t t 6 ν t t ν t t Δ 6 8 t t t 6 6 ν 8 ν ν t 6s. Use step sze o t s 8 ν ν 8 l ν l ν 8 ν 5.. The bsolute reltve true error s t.7769% m / s.m / s 9.695m / s. The results rom the three derece ppromtos re gve Tble. Tble Summr o 6 usg deret dvded derece ppromtos. Tpe o Derece 6 t % Appromto m / s Forwrd Bckwrd Cetrl Clerl, the cetrl derece scheme s gvg more ccurte results becuse the order o ccurc s proportol to the squre o the step sze. Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

64 Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor 5. Hgher Order Dervtves Emple: Secod order dervtve: Note tht or the cetered ormulto, t s dervto o dervtve: '' Forwrd '' Bckwrd '' Cetered '' I Forwrd Derece Methods Frst Dervtve ' Secod Dervtve '' Thrd Dervtve Fourth Dervtve 6 II Bckwrd Derece Methods Frst Dervtve ' Secod Dervtve ''

65 Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor Thrd Dervtve Fourth Dervtve 6 III Cetrl Derece Methods Frst Dervtve ' Secod Dervtve '' Thrd Dervtve Fourth Dervtve 6

66 Solvg Sstem o Ler Equtos 6. Ler Equto m s equto, whch vrble s epressed terms o d the costt m, s clled Ler Equto. I Ler Equto epoets o the vrble s lws oe. 6. Ler Equto vrbles:... b,,..., Where, re vrbles d,,,..., d b re costts. 6. Sstem o Ler Equtos: A Ler Sstem o m ler equtos d ukows s:... b... b... b b m Where m m,,,..., m m re vrbles or ukows d s d b s re costts. 6. Augmeted Mtr Sstem o ler equtos: b b b C be wrtte the orm o mtrces product b b b Or we m wrte t the orm AXb, Where A Augmeted mtr s [ A : b], X, b b b b b b b Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

67 Emple 6.: Wrte the mtr d ugmeted orm o the sstem o ler equtos 6z 6 z z Soluto: Mtr orm o the sstem s 6 6 z Augmeted orm s [ : b] 6 6 A. 6.5 Methods or Solvg Sstem o Ler Equtos. Guss Elmto Method. Guss -Jorde Elmto Method 6.5. Guss Elmto. Guss elmto s geerl method o dg possble solutos to ler sstem o equtos. Guss Elmto Method Step. B usg elemetr row opertos b b b Step. Fd soluto b bck substtutos. A A A B B B Emple 6.: Solve the sstem o ler equtos b Gusso- Elmto method 6 7 Soluto: Step. Augmeted mtr s R R r r r r Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

68 d b R R r r r r R 5 Equvlet sstem o equtos orm s: Step. Bck Substtuto Solutos re,, / / Emple 6.: For the below gure clculte the vlues o the ukow low rtes FRR, FRR usg Gusso- Elmto method FRR. F? 99% Bezee % Toluee F? 5% Bezee 9% Toluee % Xlee F kg/hr % Bezee % Toluee % Xlee Tow er Tow er F? % Toluee 9% Xlee Compoet mterl blce gves these three equtos o three vrbles Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

69 F F F.99F.5F F.F.9F.F BAugmeted mtr s RRRrR R.99/ rr RRRrR R./ rr RRRrRR-.9/-.9rR RRRrRR/-.9 RRRrRR/ Equvlet sstem o equtos orm s: FR R FRR FR R FR R.5FR R FR R 8.65 Step. Bck Substtuto FR R 8.65 FR R R R FR R7.95 FR R- FRR-FR R Guss - Jorde Elmto Method Guss - Jorde Method B usg elemetr row opertos b b b B B B Emple 6.: Solve the sstem o ler equtos b Guss-Jorde elmto method R R RR R R 8 - R R- R R R R R-7RR R R R Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

70 Soluto: Augmeted mtr s RRR rrrrr RRR rrr-rr RRR-rR RRRrRR-rR RRR-rRR/5 8 RRRrRR-rR 5 9 RRRrRR5rR RRRrRR-rR Equvlet sstem o equtos orm s: R R R R R R s the soluto o the sstem. Emple 6.5: Totl d compoet mterl blce o sstem o dstllto colums gves the lowg equtos:- FR R FRR FRR FRR 69.FR R.5FRR.5FRR.FRR.5.5FR R.8FRR.FRR.5FRR 7.8FR R.5FRR.5FRR.FRR 87. Use Guss - Jorde method to compute the our u-kow's bove equtos:- Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

71 Soluto: Augmeted mtr s RRR rrr-./rr RRRrRR-.5/rR RRRrRR-.8/rR RRRrRR-.55/-.5rR RRRrRR--./-.5rR RRRrRR-.88/-.8rR RRR rrr/-.5 RRRrRR/-.8 RRRrRR/ RRR rrr-rr RRRrRR-.8/rR RRRrRR-.857/rR 8 RRR rrr-rr.6 75 RRRrRR--.6/rR RRRrRR-rR Equvlet sstem o equtos orm: 75 FR R 6, FR R, FR R 75 d FR R 8 s the soluto o the sstem. 8 Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

72 RR PRRIRR PRR HRRO PHRRI HRRPOR Emple 6.6 Blce the ollowg chemcl equto: RR RR RR R5R USoluto: P blce: RR RRRR R5R I blce: RRRR R5R H blce: RRRRR5R O blce: RRR5R Re-wrte these s homogeeous equtos, ech hvg zero o ts rght hd sde: RR RR- RR- R5R RR- R R- RR- RR- R5R RR- R5R At ths pot, there re our equtos ve ukows. To complete the sstem, we dee ulr equto b rbtrrl choosg vlue or oe o the coecets: RR We c esl solve the bove equtos to blce ths recto usg MATLAB such tble 6. Tble 6. Mtlb code d results or soluto emple 6.6 Mtlb Code Results A [ ]; B [;;;;]; X A\B X Ths does ot eld tegrl coecets, but multplg b wll do the trck: The blced equto wll be: PRRIRR PRR 8 HRRO PHRRI HRRPOR Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

73 RR RR RR Itertve Methods or Solvg Sstem o Ler Equto 7. Jcob Method Let the gve equto be b cz d b cz d b cz d I the gve sstem o equto s dgoll domt the d b c z d b c z d c b z 7.. Codto or Jcob method o coverges: The sucet codto s b c b c c b The bsolute vlue o the dgol elemet ech row o the coecet mtr must be greter th the sum o the bsolute vlues o the o-dgol elemets the sme row. Emple 7.: Use the Jcob terto method to obt the soluto o the ollowg equtos: 6RR - RR RR -5RR - -R R7 RR 5 Soluto Step : Re-wrte the equtos such tht ech equto hs the ukow wth lrgest coecet o the let hd sde: 6RR RR-RR 7RR 5R R-R 5RR RR RR 6 Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

74 P Step : Assume the tl guesses the clculte, d : Step : Use the vlues obted the rst terto, to clculte the vlues or the P terto: ` d so o or the et tertos so tht the et vlues re clculted usg the curret vlues: The results or 9 tertos re: Ukows Iter. R R R d Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

75 RR P P Emple 7.: Solve the equtos b Jcob method RR RR RR 7 RR RR RR 8 RR RR RR 5 Soluto Rewrte the gve equto the orm: Usg, we obt Puttg these vlues o the rght o equtos to obt These d urther tertes re lsted the tble below: The vlues 5P solutos re: th d 6P, RR th Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor tertos beg prctcll the sme, we c stop. Hece the d RR

76 RR -5RR RR P 7. Guss-Sedel Method I the gve sstem o equto s dgoll domt the d b c z d c z b z d b c Emple 7.: Use the Guss-Sedel method to obt the soluto o the ollowg equtos: 6RR - RR RR RR - -R R7 RR 5 Soluto Step : Re-wrte the equtos such tht ech equto hs the ukow wth lrgest `coecet o the let hd sde: rom eq. 6 5 rom eq. 7 rom eq. 5 Step : Assume the tl guesses, the clculte Use the updted vlue 8 d to clculte 7 Smlrl, use d to clculte.6 Step : Repet the sme procedure or the P d.69. terto : Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

77 d so o or the et tertos so tht the et vlues re clculted usg the curret vlues: d cotue the bove tertve procedure utl [RkRP. The procedure elds the ect soluto ter 5 tertos ol: P- RkRP P]/ RkRP P < ЄR Ror, d Ukow Iter. R R R Emple 7.: Solve b Guss Sedel method, the equtos: RR RR RR 7 RR RR RR 8 RR RR RR 5 Soluto As beore, we strt wth tl estmte. We wrte the gve equto the orm These d urther tertes re lsted the tble below: Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

78 P P The vlue the P the solutos s RR d rd d P tertos beg prtculrl the sme, we c stop. Hece d RR., RR Emple 7.5: For the below gure clculte the vlues o the ukow low rtes F, F d F b usg Guss-Sedel Method. F? 99% Bezee % Toluee F? 5% Bezee 9% Toluee % Xlee Compoet mterl blce gves these three equtos o three vrbles.99f.5f F.F.9F.F F.F.9F Re-rrgg the bove equtos F.5F /.99 F.F.F /.9 F.F /.9 F kg/hr % Bezee % Toluee % Xlee Tow er Strtg wth FFF/ Iterto F F F F? % Toluee 9% Xlee A Mtlb progrm or solvg the bove equtos usg Guss-Sedel method s Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor Tow er

79 lsted Tble 7. Tble 7. Mtlb code d results or soluto emple 7.5 Mtlb Code Results F.; F.; F. or : F-.5*F/.99; F-.*F-.*F/.9; F-.*F/.9; dsp[, F, F, F] ed Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

80 Solutos o No-Ler Equtos Oe Vrble Root Fdg U8. Grphcl Methods A smple method or obtg root o the equto s to plot the ucto d observe where t crosses the s. There s much vlble sotwre tht wll cltte mkg plot o ucto. We wll use Mtlb eclusvel or the course otes; however ou c use other sotwre such s Ecel or Mtcd or our work. Emple 8. 6 Solve ep.5 P 5 usg the grphcl method. Soluto 6 The ucto ep.5 P 5 c be plotted Fgure 8. usg the Mtlb sttemets lsted tble 8.. Tble 8. Mtlb code or solvg emple 8. usg grphcl method Mtlb Code :.:; 6*-ep-.5*./-5; plot,,[ ],[ ] lbel''; lbel'' grd o; zoom o Fgure 8. The grphcl method or roots dg. Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor

81 d The Mtlb Zoom o sttemet llows the ucto to be zoomed t the cursor wth let mouse clck rght mouse clck wll zoom out. Ech tme ou clck the es lmts wll be chged b ctor o or out. You c zoom s m tmes s ecessr or the desred ccurc. Fgure 8. shows the ppromte root to be Fgure 8. The grphcl method or roots dg wth Mtlb Zoom o. The plot o ucto betwee RR R Rs mportt or uderstdg ts behvor wth ths tervl. More th oe root c occur wth the tervl whe RR d RR re o opposte sdes o the s. The roots c lso occur wth the tervl whe RR d RR re o the sme sdes o the s. Sce the uctos tht re tget to the s sts the requremet t ths pot, the tget pot s clled multple root. U8. The Bsecto Method The bsecto method or tervl hlvg c be used to determe the soluto to o tervl [RR, RR b] s rel d cotuous o the tervl d RR d RR hve opposte sgs. We ssume or smplct tht the root ths tervl s uque. The locto o the root s the clculted s lg t the mdpot o the subtervl wth whch the uctos hve opposte sgs. The process s repeted to speced ccurc. The procedure c be summrzed the ollowg steps Numercl Alss /Lec Wrtte b Assoc. Pro. Thrd Clss Dr. Zdoo M. Shkoor