12 Iterative Methods. Linear Systems: Gauss-Seidel Nonlinear Systems Case Study: Chemical Reactions
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1 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 usg systemtc method to obt reed estmte o the root. DM869/Computtol Numercl Alyss/_c.doc Avlble t
![HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 er Systems: Guss-Sedel the most commoly used tertve method or solvg ler lgebrc equtos or set o equtos [ A ]{ { b wth the dgol elemets re ll ozero b](/docs-images/84/89560451/images/2-0.jpg "b b - the preset terto - - the prevous terto tl gu")
2 HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 er Systems: Guss-Sedel the most commoly used tertve method or solvg ler lgebrc equtos or set o equtos [ A ]{ { b wth the dgol elemets re ll ozero b b b - the preset terto - - the prevous terto tl guesses or the 's - e.g. zeros covergece chec ε % ε s ech ew vlue s mmedtely used the et equto to determe other vlue: Guss-Sedel rther th usg the ltest vlble 's those re reted or the et terto: Jcob terto <Guss-Sedel vs. Jcob b tertos> DM869/Computtol Numercl Alyss/_c.doc Avlble t
3 HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 Emple Use the Guss-Sedel method to obt the soluto or T Note tht the soluto s {.5 7. Sol DM869/Computtol Numercl Alyss/_c.doc Avlble t
4 Covergece d dgol domce ote tht the Guss-Sedel method s smlr to the techque o smple ed-pot terto HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 covergece crtero: > dgolly domt systems MATAB M-le: GussSedel ew ew ew b b b ew ew old ew old old mtr orm: { { d [ C]{ b / { d b / b / [ C] / / / / / / DM869/Computtol Numercl Alyss/_c.doc Avlble t 4
5 HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 Relto desged to ehce covergece moded by weghted verge o the results o the prevous d the preset tertos weghtg ctor λ: λ - λ relto method Guss-Sedel method - λ uderrelto typclly employed to me ocoverget system coverge or to hste covergece by dmpeg out osclltos - λ overrelto ew etr weght o the preset vlue pushg the estmte closer to the truth desged to ccelerte the covergece o lredy coverget system lso clled successve overrelto SOR - choce o proper λ ~ hghly problem-specc ote determed emprclly DM869/Computtol Numercl Alyss/_c.doc Avlble t 5
6 HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 DM869/Computtol Numercl Alyss/_c.doc Avlble t 6 Noler Systems Noler systems o equtos: 57 Geerl epresso: K M K K Successve substtuto the sme strtegy or ed-pot terto d the Guss-Sedel method covergece ote depedg o the mer whch the equtos re ormulted dvergg the tl guesses re sucetly close to the true soluto lmted utlty or solvg oler systems
7 HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 Emple Use successve substtuto to determe the roots o. 57 Note tht correct pr o roots s d. Itte the computto wth guesses o.5 d.5. Sol DM869/Computtol Numercl Alyss/_c.doc Avlble t 7
8 HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 Newto-Rphso employg the dervtve.e. the slope o ucto to estmte ts tercept wth the s o the depedet vrble.e. the root rom the rst-order Tylor seres epso tl guess t the root the pot t whch the slope tercepts the s Accoutg or two vrble Tylor seres epso By deto the we hve Sce ll vlues subscrpted wth 's re ow correspodg to the ltest guess or ppromto the oly uows re d. ler equtos Applyg the Crmer's rule Note tht the deomtor o ech o these equtos s ormlly reerred to s the determt o the Jcob o the system. DM869/Computtol Numercl Alyss/_c.doc Avlble t 8
9 HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 Emple Use the multple-equto Newto-Rphso method to determe roots o wth guesses o.5 d.5.. Itte the computto 57 Sol DM869/Computtol Numercl Alyss/_c.doc Avlble t 9
10 HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 DM869/Computtol Numercl Alyss/_c.doc Avlble t Geerlzg the two-equto Newto-Rphso pproch to solve smulteous equtos let's ccout or multvrble Tylor seres epso. For -th equto: For the cocseess epressg mtr otto ]{ [ { ]{ [ J J where [J] Jcob mtr evlutg the prtl dervtves t J ] [ M M M tl vlues ucto vlues t ve Eq. by the verse o b the bove; T { l vlues { T T { Multplyg the bo the Jco ; { ] [ { { J Recll the N-R ormul or the sgle Eq. d compre wth ' the Jcob s logous to the dervtve o multvrte ucto oler optmzto chque dg the vlue o tht mmzg slower th N-R but hvg better covergece property N te [ ] F
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