SYSES OF INER EQUIONS Itroducto Emto thods Dcomposto thods tr Ivrs d Dtrmt Errors, Rsdus d Codto Numr Itrto thods Icompt d Rdudt Systms
Chptr Systms of r Equtos /. Itroducto h systm of r qutos s formd y th ddto of th products of vr wth coffct, whch s so costt. h systm of r quto c sovd v mtr pproch. h gr form of st of r quto hvg r qutos d ukows s O (.) whr r vrs or ukows,,,, K j d j r coffct or costt (r or comp). Eq. (.) c wrtt mor compct form: } { } { ] [ j j (.) whr s mtr [ j ] of sz, s vr vctor { j } d s rght-hd sd vctor { j }. h procss of sovg Eq. (.) yd thr poss soutos:. Uqu souto.g.:. No souto.g.:. Ift soutos.g.:
Chptr Systms of r Equtos /. Emto thods h most popur mthod s th Guss mto mthod, whch comprss of two stps:. Forwrd mto to form uppr trgur systm v rowsd trsformto procss,. Bck susttuto to produc th souto of j. Cosdr th foowg systm: O If, for,,,, sustrct th -th quto wth th product of wth th frst quto to produc th frst trsformd systm: () () () () () () O whr () j j j for, j,,, () for,,, h procss c rptd for () tms ut th ()-th trsformd systm s formd s foowd, whch compts th forwrd mtos: (.) () () () () ( ) ( ) ( ) ( ) ( ) O
Chptr Systms of r Equtos / whr ( k) ( k ) ( k ) k j j ( k) kk ( ) ( ) ( k k k ) k ( k) ( k kj ( k k kk ) ) for, j k,, for k,, (.) (.) Bck susttutos c th cutd so tht j r sovd: ( ) (.) ( ) k ( k ) ( k ) ( ) k k kj kk j k j for k,, (.) h ov mthod c f f kk, th row hs to trchgd, whch s rfrrd to s pvotg: Pvotg kk whr th w dgo mt s cd pvot, whch c sctd mog th mmum sout vu of h pvot Guss mto gvs mor ccurt soutos,.g. cosdr ths systms (vus to roudd up to sgfct fgurs): k. Org Guss mto:.6...8 Pvot Guss mto:..8 ( ) () ( ).6.6.6.....8.8.8.6..8.6 ( ) () ( )...6..8..8. Ect souto:.8.8.6.
Chptr Systms of r Equtos / Emp. Sov th foowg systm usg th Guss mto mthod: Souto h systm c rwrtt mtr form s: or [ ] Frst stp of forwrd mto: 6 () () () () () () Scod stp of forwrd mto: () () () Hc, th trsformd uppr trgur systm s: Bck susttutos r s foows
Chptr Systms of r Equtos / 6 Emp. Prform th pvot Guss mto to th systm gv Emp.. Souto h pvot Guss mto c prformd s foowd: () ( ) ( ) () ( ) ( ) () ( ) ( ) ( ) ( ) ( ) ( ) Hc, th uppr trgur systm s: h, ck susttuto c prformd: () ( ) ( ).,,
Chptr Systms of r Equtos /. Dcomposto thods I som css, th ft-hd sd mtr s frquty usd wh th rght-hd sd vctor s chgd dpdg o th cs. h ovr systm c trsformd to uppr trgur form so tht t c usd rptdy for dffrt, thus mtr hs to dcomposd. For gr o-symmtrc systm, th popur mthod s th Doot or U dcomposto: U (.6) whr d U r th owr d uppr trgur mtrcs, rspctvy: u u u u u u u u u u u ( mmory) u h souto stps of th systm r s foowd: By tkg trmdt vctor y: Hc, U U y (.) y (.8) h mts for d U c otd from th Guss mto: U () () ( ) () ()
Chptr Systms of r Equtos / 8 othr vrto of th U dcomposto s th Crout dcomposto, whch mts u for,,, U std of : For th frst row d coum: For j,,,: for,,, (.) j u j for j,,, (.) j j j k k u kj for j, j,, (.c) u jk jk j jj j u k for k j, j,, (.d) d, k k u k (.) If th systm s symmtrc, th Chosky dcomposto c usd, whr mtr c dcomposd such tht: For th k-th row: (.) k k j j kj for,,,k (.) k kk j kk kj (.) hs mthod optmss th us of computr mmory storg th dcomposd form of.
Chptr Systms of r Equtos / Emp. Dcompos th foowg mtr usg th Doott U dcomposto: Souto Wth rfrc to th mtr mts drvd Emp.:., U Emp. Dcompos th foowg mtr usg th Chosky dcomposto: Souto By usg Eq..:.,,,,, k,.88..88..
Chptr Systms of r Equtos /. tr Ivrs d Dtrmt h Guss mto c usd to grt th vrs of squr mtr y rpcg th ft-hd sd vctor wth dtty mtr I. By usg th foowg dtty: I (.) ( ) ( ) ( ) If coums of r wrtt s,, K, d th coums of th I s () ( ) (,, K, ), rspctvy, thus Eq. (.) c rwrtt s: () ( ) ( ) ( ) ( ) ( ) (,, K, ),, K ( ), h, st of r systms c ssmd: ( ) ( ) ( ) ( ) ( ) ( ) (.) Cosquty, th dtrmt of mtr c smpy ccutd usg: dt (.) p () ( ) ( ) p ( ) ( ) ( ) K ( ) whr p s th umr of row trchg oprto durg pvotg.
Chptr Systms of r Equtos / Emp.8 Dtrm th vrs of th foowg mtr usg th Guss mto: Souto h comto of d I c rprstd ugmtd form: mto forwrd Guss Upo ck susttuto: () 8 () () Hc, th vrs of s 8 Emp. Ccut th dtrmt of th mtr gv Emp.8. Pys I Emp., thr s o row trchg prformd, thus p. Hc, ( ) ( ) ( )()() dt
Chptr Systms of r Equtos /. Errors, Rsdus d Codto Numr If s ppromt souto of r systm, th th systm rror s dfd s (.6) O th othr hd, th systm rsdu r s dfd s r (.) or, r For w-codtod systm, th rsdu c rprst th rror. orovr, for comprso, mtr or vctor c prssd form of scr kow s orm. For vctor (, K ), th p-orm s dfd s,, p p p ( ) p p p p (.) If p, t s kow s -orm: (.) If p, t s kow s Eucd orm: (.8) If p, t s kow s mmum orm: {,, K, } m (.) m For mtr [ j ] of sz m, th Frous orm, whch s quvt to th Eucd orm for vctors, s dfd s m j j (.)
Chptr Systms of r Equtos / d, th quvt -orm d mmum orm for mtr r dfd s m mmum sum of coums j (.) j m mmum sum of rows j (.) j h proprts of orms of vctor or mtr r s foowd:. d f, d oy f,.. c c whr c s scr qutty.. B B rgur quty, whr B s vctor or mtr of th sm dmso of.. B B Schwrz quty, whr B s vctor or mtr whch forms vd product wth. h cocpt of orms c usd to ccut th codto umr rprsts th hth of r systm, thr - or w-codtod. If s th rror for th systm, from th rtos r d r, th foowg quty c stshd: r d r r r so, from d : d hus, th comto of oth quty rtos yds th rg of th rtv rror,.. r ( ) r Hc, th codto umr s dfd s ( ) κ (.)
Chptr Systms of r Equtos / whr th rg of th rtv rror s. κ ( ) r κ ( ) r (.6) h chrctrstcs of th codto umr r tht: κ th smr th ttr, d othrws.. ( ) κ, th rtv rsdu r c rprst th rtv. If ( ) rrors. If th rror s soy cotrutd y mtr, th quty coms: E κ ( ) (.) O th othr hd, f th rror s soy cotrutd y vctor, th quty coms: κ ( ) (.8) hrfor, from Eqs. (.6-8), t c s tht th codto umr c dtrm th rg of rror d thus th hth of systm.
Chptr Systms of r Equtos /. Itrto thods For rg systms (sz > ), th mto d dcomposto mthods r ot ffct du to crsg umr of rthmtc oprtos. h umr of rthmtc oprtos c rducd v trto mthods, such s th Jco trto d th Guss-Sd trto mthods. I th Jco trto, Eq. (.) c wrtt for from th -th quto: ( ) ( ) ( ), (.) () () () () Eq. (.) ds t vus ( ),, K, () () () () (, K, ),, whch yd, d th computto cotus s foowd: ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) ( ) ( k ) ( k ) ( k ) ( ) ( k ) ( k ) ( k ) ( ), (.) For k, vctor (k) covrgs to ts ct souto f th dgo dom codto s foowd,.. > j for,, K, j j (.) d th mtr whch foows ths codto s cd dgo dom mtr.
Chptr Systms of r Equtos / 6 o trmt th trto procss, covrgc or trmto crtro c spcfd,.. ( k ) ( k ) < ε (.) h Guss-Sd trto mthod uss th most currt kow souto ftr ch rthmtc oprto ordr to spd up covrgc: ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) ( ) ( k ) ( k ) ( k ) ( ) ( k ) ( k ) ( k ) ( ), (.) s of th Jco mthod, th Guss-Sd mthod must so osrv th dgo dom codto for covrgc to poss (s Fg..). ( 6, ) ( 6, ) () h off-dgo dom systm () h dgo dom systm FIG.. Dvrgc d covrgc th Guss-Sd mthod
Chptr Systms of r Equtos / Emp. Us th Jco trto mthod to sov th foowg systm up to dcm pots: Souto 6 Frst of, form dgo dom systm: 6 h, rwrt th systm ccordg to Eq. (.): ( k ) ( k ) ( k ) 6 ( k ) ( k ) ( k ) ( ) ( k ) ( k ) ( k ) ( ) ( ) By tkg t vus () (,, ), thus th mthod covrgs wth trtos: Itrto o. : () (.8,.6,.), Itrto o. : () (.,.6,.), Itrto o. : () (.,.66,.6), Itrto o. : () (.,.66,.6), Itrto o. : () (.,.66,.6).
Chptr Systms of r Equtos / 8 Emp. Rpt prom gv Emp. usg th Guss-Sd trto mthod. Souto Frst of, form dgo dom systm: 6 By tkg t vus () (,, ), th frst souto th frst trto: () 6 [ ( ) ]. 8 () () Us to ccut d so o,.. [ (.8) ]. 6 () (.8.6 ). () Hc, th mthod covrgs wth trtos: () (.8,.6,.), () (.,.66,.6), () (.,.66,.6), () (.,.66,.6).
Chptr Systms of r Equtos /.8 Icompt d Rdudt Systms If, thr w two stutos: m. m < compt systm.. m > rdudt systm. For compt systm, o souto s poss sc ddto ( m) qutos from othr dpdt sourcs r rqurd ut m. For rdudt systm, uqu souto s ot poss, d th systm hs to optmsd v st squr mthod (so kow s r rgrsso): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ).,,, S Usg th dtty ( ) : B B S msg S: S forms ppromt systm of qutos,.. (.) whr th ft-hd sd mtr s symmtry d th stdrd dvto σ c ccutd from th Eucd orm of,..: m m S σ (.)
Chptr Systms of r Equtos / Emp. Ccut th st ppromt souto for th foowg systm: so, ccut th rsutg stdrd dvto. Souto h ov systm c rwrttd form of s: By usg Eq. (.): 6 8 whr ts soutos r...6,.,
Chptr Systms of r Equtos / h stdrd dvto c otd from th Eucd orm of th rror :.8..68.6...6. ( ) ( ) ( ).8.,.8..68.6. hrfor,.6.8 σ
Chptr Systms of r Equtos / Ercss. Cosdr th foowg systm:..8.. 8. Us th Guss mto mthod to ot th souto of.. Ccut th dtrmt for th ft-hd sd mtr. c. Grt th owr d uppr trgur mtrcs usg th Doott fctorsto.. Cosdr th foowg systm of comp qutos: z z By wrtg, sov th quto usg th Guss-Sd trto mthod usg crosoft Ec ut t covrgs up to dcm pots. y z k k k. Cosdr th foowg st of rdudt qutos:. Drv ppromt systm of r qutos d sov t v th Guss mto.. Ccut th corrspodg stdrd dvto.