Chapter 2 Solving Linear Equation

Transcription

1 EE7 Computer odelg Techques Egeerg Chpter Solvg er Equto A ler equto represets the ler depedece of qutty φ o set of vrbles through d set of costt coeffcets α through α ; ts form s α α... α φ If we replce φ by sgly subscrpted qutty f d the coeffcets α j by doubly subscrpted qutty α j, we my wrte system of such equtos the form α α α α α α α α α j α j α j α j j j α j j α : : : : α : : : : α f f f f (.) The soluto of such systems for the quttes j, whe the coeffcets j d the vlues f re gve, pervdes egeerg pplctos d computtol methods. For emple, systems of ler lgebrc equtos re used drectly mthemtcl models for electrcl, structurl, d ppe etworks, d some computtol methods for fttg curves to dt. I other cses, such s fte dfferece d fte elemet solutos of prtl dfferetl equtos, they represet ppromtos of mthemtcl equtos, whch cot be solved by lytcl methods.. Fudmetls of ler lgebr.. Notto d Deftos A mtr s rectgulr rry of quttes. A emple of mtr s A [ j ] (.) Chpter Pge

2 EE7 Computer odelg Techques Egeerg I Eq. (.), the bold-fced A s the symbol for the mtr, the lterl represetto o the fr rght shows the rrgemet of the elemets, d the form [ j ] s shorter bstrcto of the lterl form. The mtr hs o prtculr meg utl we ssocte the elemets wth gve cocept. A mtr wth m rows d colums hs dmeso (m ) d s referred to s (m ) mtr. A colum vector c d row vector r re show Eq.(.) c ] c ; r [ rj ] [ r r r ] c [ c r c (.).. Opertos Addto d subtrcto for coformble mtrces A d B re summrzed by A ± B C [ c ] [ ± b ( m ) ( m ) ( m ) j j j ] (.) tr multplcto A B s defed oly f the umber of colums A s equl to the umber of rows B. A B P [ p ( m r) ( r ) r k ( m ) p j ( k bkj ) b j b j j ] r b rj (.5) (.5b) tr multplcto s both dstrbutve d ssoctve. If A,B, d C re mtrces wth pproprte dmesos to stsfy the ddto/ subtrcto codtos of Eq. (.) d the multplcto codtos of Eq.(.5), the dstrbutve d ssoctve propertes re epressed Eq.(.6) d Eq.(.7), respectvely, by A ( B C) A B A C; ( B C) A B A C A (.6) A B C A B C) ( A B) C ( (.7) ultplcto of mtr A by sclr q to form product mtr S equl to qa s lso defed operto. The product ths cse s gve by Chpter Pge

3 EE7 Computer odelg Techques Egeerg q A S [ s ] q[ ] [ q ( m ) ( m ) j j j ] (.).. Squre trces A squre mtr s oe tht hs the sme umber of rows s the umber of colums; for emple, mtr wth dmeso ( ). Specl types of squre mtrces re descrbed s follows. A dgol mtr D equl to [d j ] stsfes the codto d j for j (.9) We my replce [d j ] by [d ], mplctly t the correct cotet so tht t s ot cofused wth the vector otto of Eq. (.), d epress the dgol mtr D the form D [ d ] d d O d O d d (.) The detty mtr I s dgol mtr whch every dgol elemet hs ut vlue; t s the mtr of the sclr ut vlue. The zero mtr s oe whose elemets re ll zero; t my be ether squre mtr or geerl rectgulr mtr. ultplctos volvg the geerl dgol mtr D, the detty mtr I, d the zero mtr re summrzed s follows; ech of D, I d s ssumed to be ( ) mtr. c u [ u ] [ d c ] D ( ) ( ) ( ) r D v j ( ) ( ) ( ) D A P j ( ) ( ) ( ) A D Q j ( ) ( ) ( ) I c c ( ) ( ) ( ) [ v ] [ r d ] [ p ] [ d ] j j j [ q ] [ d ] j j (.) (.b) (.c) (.d) (.) Chpter Pge

4 EE7 Computer odelg Techques Egeerg r I r (.b) ( ) ( ) ( ) I A A A ( ) ( ) ( ) ( ) ( ) I (.c) c (.) ( ) ( ) ( ) r ( ) ( ) ( ) A A ( ) ( ) ( ) ( ) ( ) (.b) (.c) The verse of squre mtr A s deoted by A - d s tself squre mtr wth the sme dmeso s A such tht A A A A I (.) The logous sclr relto s ( αα α α ). Two other types of squre mtrces tht wll be useful t lter stge re the lower trgulr d upper trgulr mtrces. The lower trgulr form permts ozero elemets oly o d below the dgol; t my be represeted by [ l ] j l l l O O l l l (.5) The upper trgulr form U permts ozero elemets oly o d bove the dgol; t my be represeted by U [ u ] j u u u u u O O u (.6) Chpter Pge

5 EE7 Computer odelg Techques Egeerg.. The Determt of Squre tr The determt of ( ) squre mtr A s wrtte s A d s defed by ether of or A j A ( j C j ) for y oe vlue of (.7) ( j C j ) for y oe vlue of j (.7b) whch C j s kow s the cofctor of the elemet j. The cofctor C j of ( ) squre mtr s obted by frst removg row d colum j to form ((-) (-)) mtr d the by performg the operto C j j ( ) (determt of A wth row d colum j removed) (.)..5 The tr Equto for er Algebrc Systems The cocept of mtr multplcto epressed Eq.(.5) d (.5b) llows us to represet the system of ler lgebrc equtos gve by Eq.(.) the form j j j j f f f f (.9) or, more compctly, the form A f (.9b) Here, A s the coeffcet mtr formed by coeffcets of the ler system, the orgl rght-hd sdes of the system re ow the compoets of the colum vector f. d the orgl ukow quttes re ow the compoets of the colum vector. A uque soluto for requres A to be squre mtr, d t requres the system of equtos to be lerly depedet. A coeffcet mtr wth zero determt s sgulr, uque soluto for requres osgulr mtr. Chpter Pge 5

6 EE7 Computer odelg Techques Egeerg The soluto of Eq.(.9b) my be derved wth the help of the ssoctve lw from Eq. (.7), the cocept of the verse from Eq.(.), d opertos wth the detty mtr from Eq. (.) s follows: I ( A A) A ( A ) A f (.) Amog the ddtol tools used methods for solvg mtr equtos re row d colum echges. A row echge the coeffcet mtr A of Eq. (.9b) s smply posto echge of two equtos. The sequece of the compoets of s preserved, but those of the rght-hd sde vector f must be echged to correspod to the ew postos. A colum echge the coeffcet mtr does ot lter the sequece of equtos but t requres correspodg echge the compoets of so tht the compoets re multpled by the pproprte coeffcets. Illustrtos of these cocepts re s follows. Echge of rows I d k: k f fk (.) Echge of colums j d k: j j k k j k f (.b)..6 Computtol Techques for Bsc Opertos The procedure of umercl computto: Fd mthemtcl model or wy of presetg the problem. Choose umercl method or clculto formul. Set up the computto procedure. Drw progrm flowchrt. Wrte progrm. u the computto. Check the computto results. Chpter Pge 6

7 EE7 Computer odelg Techques Egeerg Emple : Clculto of Iverse tr ) Fd mthemtcl model or wy of presetg the problem. Problem: Fd the verse mtr from by mtr. The otto of mthemtcl model s A A s the verse mtr A. ) Choose sutble umercl method or formul. Usg formul of verse mtr. A b b b b ) Set up the computto procedure Step Dt put Step Compute the mtr Step Error cotrol Step Compute the verse mtr Step 5 Output the result Chpter Pge 7

8 EE7 Computer odelg Techques Egeerg ) Drw progrm flowchrt STAT,,, DATA INPUT d d: b b b b / d / d / d / d tr s ot No-sgulr b, b, b, b ESUT, OUTPUT END 5) Wrte progrm Numercl computto progrm cludes four prts:. Defe the vrbles d type b. Dt put c. Computto d. Dt output Chpter Pge

9 EE7 Computer odelg Techques Egeerg Progrm emple C /* < POGA> */ # clude <stdo.h> # clude <stdlb.h> Vrbles type Dt put secto m( ) { t, j; double [][]; /* INPUT ATIX */ double b[][]; /* OUTPUT ATIX */ double d; /*** STEP DATA INPUT***/ prtf ( INVESE ATIX / * ); for (; <; ){ for (j; j<; j){ prtf ( %d%d,, j); scf ( %f, &[I][j]); } } /***STEP CACUATE ATIX***/ d[][]*[][]-[][]*[][]; Computg secto /***STEP EO CONTO***/ f (d. ){ prtf( IT IS NON-SING UA ATIX/ ); } /***STEP CACUATE INVESE ATIX***/ b[][][][]/d; b[][]-[][]/d; b[][]-[][]/d; b[][][][]/d; Dt output secto /***STEP 5 ESUT OUTPUT***/ prtf ( <SOUTION>/ ); prtf( B/ ); for (; <; ){ for (j; j<; j){ prtf( % f, b[][j]); } prtf ( / ); } } Chpter Pge 9

10 EE7 Computer odelg Techques Egeerg 6) Compute the problem < B ) Check the computto results Compre wth other method such s ATAB fuctos. * Tp A good progrm for umercl computto should hve followg chrcterstcs: ) short computto tme ) ccurte result ) short progrm steps ) esy to red 5) esy to move to other computers Emple : Compre wth ATAB mtr verse fucto by usg followg mtr: fd A A? usg ATAB progrm d ATAB verse fucto. Chpter Pge

11 EE7 Computer odelg Techques Egeerg..7 Itroducto to Numercl Computto er Equtos. Prelmres esstve Network. V V V V 5 5 Ω Ω Ω Ω Ω V V 5 Ohm s lw: V (.) Krchhoff s lws: I (.) U (.) The mplcto of Krchhoff s d Ohm s lw re tht the currets,, d must stsfy the followg reltos: 5 ) ( ) ( ) ( V V m m m m b b b (.5) Chpter Pge

12 EE7 Computer odelg Techques Egeerg The coeffcets d b re gve rel or comple umbers. Eq. (,5) s system f j m equtos wth vrbles, or ukows, vlues.,,..., Eq. (.5) c be wrtte compctly mtr form s A where ( ) j b A s m mtr of coeffcets, d ( ) vectors of dmeso m d, respectvely. b. Our job s to determe ther b d ( ) re colum j (.6) EXAPE I the 6 m rectgulr rego show Fgure.(), the electrc potetl s zero o the boudres. The chrge dstrbuto, however, s uform d gve by p v ε. Solve Posso s equto to determe the potetl dstrbuto the rectgulr rego. Soluto To determe the potetl dstrbuto the rectgulr rego, we use Posso s equto. y ρ ε v wth zero potetl o the boudres. Fgure. Geometry of the 6 m rectgulr rego d the h m mesh. Chpter Pge

13 EE7 Computer odelg Techques Egeerg By estblshg the rectgulr grd show Fgure.(b), we relze tht we hve s odes d, hece, s ukow potetls for whch to solve. eplcg by ts fte dfferece represetto, we obt ) (,,,,, j j j j j h It should be oted tht lthough the lthough the mesh sze ws ot eplctly used solvg plce s equto the prevous emple, h s cluded s prt of the mtr formto solvg Posso s equto. I SI system of uts, h should be meters. By pplyg the precedg dfferece equto t the vrous odes Fgure.(b), we obt the followg mtr equto: 6 5 Isted of solvg the resultg s equtos, we my ote some symmetry cosdertos Fgure.(b). It s cler tht 6 5 d tht Tkg these symmetry cosdertos to ccout, the umber of equtos reduces to two, d we obt the followg soluto:.56, 7 5. Chpter Pge

14 EE7 Computer odelg Techques Egeerg. Drect ethods for er Systems.. Guss Elmto Emple of Elmto Solve the system of Eqs (.7) By subtrctg the multple of the frst equto from the secod equto d the frst equto from the thrd equto, the frst derved system s obted: 6 (.) The frst equto, fter beg multpled by, becomes 6 (.9) Subtrct ths from the mddle equto Eq.(.7) to see tht result s ) ( ) (7 6) ( The sme rtole yelds the thrd equto the frst derved system. Cotue the computto by subtrctg the multple of the secod equto of the frst derved system from the thrd equto to obt the secod derved system: The dervto of ths upper trgulr system of equto s clled the forwrd elmto process. From the thrd equto of the the system bove, we mmedtely clculte The the secod equto, wth substtuted for, mples tht Chpter Pge

15 EE7 Computer odelg Techques Egeerg Flly, from the frst equto, wth ther umercl vlues replcg d, we see tht ( ) d thereby obt the soluto, whch,,. The computto of the ukows from the upper trgulr system s kow s bck substtuto. et us ssume tht Eq.(.5) the mtr of coeffcets s osgulr; tht s, ssume m, d tht soluto ests d s uque. The Eq.(.5) c be rewrtte s ,,, (.) where, for lter otto coveece, we hve defed b ( ) Assume tht, d subtrct the multplyg,. of the frst equto from the th equto for,,. The coeffcet of the th equto the becomes, d we thereby obt the frst derved system whch hs the form () () () (), (), (), (.) where () j j j (, j ) (.) () () () Assume et tht, d subtrct the multple / of the secod equto of th (.) from the equto (,, ). We thereby obt the secod derved system: Chpter Pge 5

16 EE7 Computer odelg Techques Egeerg () () () () () () (), (), (), (), (.) By repetg ths process utl the (-)st derved system hs bee costructed, we obt () () () () () ( ), (), (), ( ), (.) where the relto for obtg the coeffcets of the k th derved system from the coeffcets of precedg system hs the geerl form ( k,, ; j k,, ) (.5) ( k ) ( k ) k k ( k ) j j ( k ) kj kk I Eq. (.5), k rges from to -. The process s strted by ssgg () (,, ; j,, ) j j By specto of Eq.(.), we see tht the coeffcet mtr of the (-)st derved system s upper trgulr form. The remg step solvg ths system s esy. The vlue of c be obted from the fl equto of Eq.(.)sce osgulrty of A mples tht ecessrly ( ). Specfclly, ( ), ( ) d the (bckwrd) recursve formul for obtg the vlues of the ukows k terms of the prevously clculted vlues j (j > k) s k ( k ) k ( k ), j k kk ( k ) kj j (k,, ) (.6) Guss elmto: Formul (.6) s clled Bck Substtuto. Process (.6-.5) s referred to s Forwrd Elmto. Chpter Pge 6

17 EE7 Computer odelg Techques Egeerg The coeffcets of the successve derved systems ssocted wth Emple. re gve Tble.. TABE. Derved System Forwrd Elmto Orgl System A () Frst Derved System A () 6 Secod Derved System A ().. Guss Elmto wth Pvotg Becuse of the effect of propgted roudg errors, tve Guss elmto s sometmes ustsfctory. Emple. Assume tht the bsece of roud off error, the lst two equtos of the ( ) d derved system re where the zero coeffcets s obted from prevous clcultos. The soluto s. As we kow, resultts of rthmetc opertos usully hve roud off errors: Therefore, the computed derved system of equtos s ctully ε Sce ε, the computto of the ( )st derved system the tve elmto process s cotued wth the ozero elemet ε, d the lst two equtos of the fl derved system re Chpter Pge 7

18 EE7 Computer odelg Techques Egeerg ε ( ) ε ε Bck substtuto ppled to these equtos results the vlues ε ε ε For ε zero, d, whch s the correct vlue. But - wll hve ε ε ε cosderble error becuse, beg very close to, wll duce subtrctve ccellto umertor. The method of bck substtuto (.6) mples tht the vlues of ll the other ukows -,, obted by the use of the erroeous vlue of - re lso suspect. The dffculty the system dscussed bove s ot due smple to ε beg smll but rther to ts beg smll reltve to other coeffcets the sme colum. ( k ) The elemet kk used for elmto s termed pvot. The subtrctve ccellto dffculty just observed c be remeded f we choose s the k th pvot elemet the ( k ) ( coordte hvg the lrgest mgtude mog ll k ), k, k j. The * j* ( k ) elemet s the put the dgol posto by terchgg rows * d k d * j* colums j* d k. Tht s, equtos * d k, d ukows j* d k re terchged. Ths operto s clled pvotg for mml sze or, more smply, mml pvotg, see Fg.. We eed to fd the mmum of (-k) umbers before computg the k th derved system. At the epese of some crese roud off error propgto, populr ltertve procedure s to perform prtl pvotg, where the mml elemet s chose from oly the k th colum. Tht s, we choose the k th ( k ) pvot elemet, to be y coordte tht mmzes ( k ) k, k. The the k th d * th rows re terchged to put the pvot elemet the dgol posto. Fg.. llustrtes the prtl d mml pvotg strteges. kk * k Chpter Pge

19 EE7 Computer odelg Techques Egeerg () O ( k ) k, k Serch ths porto Of k th colum for pvot ( k ) kk ( k ) k () ( k ) k, ( k ) k ( k ) Prtl Pvotg () O Serch ths porto Of mtr for pvot ( k ) k, k ( k ) kk ( k ) k () ( k ) k, ( k ) k ( k ) Fg.. Prtl d ml Pvotg.. Itertve ethods for er Systems.. Jcob Iterto Cosder g (.?) wth m. Assume tht A s osgulr d the rows hve bee echged, s ecessry, so tht the dgol elemets re ozero. Eq. (.?) c the be rewrtte so tht the th equto s eplct for : ( b ) ( b ) ( b ), (.5) Chpter Pge 9

20 EE7 Computer odelg Techques Egeerg Assume tht tl ppromto of the soluto hs bee gve, or smply choose () () () T rbtrry vector. et X (,, ) deote ths tl ppromto. ) Substtute t to the rght-hd sde of Eq.(.5) d evlute. The elemets of the resultg vector gve the et ppromtos of the ukows. ( T () () () et vector X,, deote these ppromtos, d the substtute the ew () vector X, to the rght sde of Eq.(.5) to get further ppromto, X () ( ( ),, () ) T, d so o. The geerl step s gve by ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) ( b ) ( k ) ( k ) ( k ) ( b ) ( k ) ( k ) ( k ) ( b ), (.6) Scheme (.6) s clled the Jcob Iterto ethod. It c be prove tht uder cert codtos, for k, the sequece of vector (k) coverges to the ect soluto of equtos (.). Oe such codto s ech dgol elemet of the mtr of coeffcet stsfy the codto: j j j,,,. (.7) If ths codto s stsfed, the A s sd to be dgolly domt. The crter for stoppg the terto process re usully ether:. The umber of tertos hs eceeded some predetermed mmum K, or. The dfferece betwee successve vlues of ll s re less th some predetermed tolerce, ε. Oe terto stop dvso, multplcto d ddtos (or subtrctos.).. Guss Sedl Iterto Cosder g the recursve Jcob terto scheme (.6). Observe tht clcultg ( k ) ( k ) the "ew vlue of, the prevous vlue of s used o the rght-hd sde lthough the ew vlue, ( k ) s lredy kow. Chpter Pge

21 EE7 Computer odelg Techques Egeerg ( k ) ( k ) ( k ) Smlrly, for obtg the ew vlue, the odd vlues d re used, ( k ) ( k ) lthough the ew, d presumbly more ccurte vlues d of these vrbles re lredy vlble. A modfcto typclly (but ot lwys) gvg fster ( k ) covergece c be devsed f the clculto of ( ), the updted ew ( k ) ( k ) ( k ) ( k ) vlues,, re used phse of the erler vlues,, (.6). Ths modfcto results the Guss Sedl terto method, whch s defed by the recursve scheme. ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) ( b ) ( k ) ( k ) ( k ) ( b ) ( k ) ( k ) ( k ) ( b ), (.) Itertve methods re prtculrly populr for soluto of bd systems rsg umercl method for prtl dfferetl equtos. For such equtos, my cses the wdth k s reltvely wde-ofte proportol to / - but the bd tself hs reltvely few ozero etres. et p deote boud to the effort requred by elmto for soluto of bded systems s proportol to k, d for tertve methods, proportol to p, where s the umber of tertos ecessry for cceptble ccurcy. Thus tertve methods re preferble f tht s p < k, k < ~ p p Ufortutely, t s typclly dffcult to sses utl computtos hve lredy begu. Chpter Pge

22 EE7 Computer odelg Techques Egeerg Tble. Soluto of er Equtos. Computto ethods tr Codto Chrcterstcs *Guss Elmto Squre tr For ler systems of smll *Guss-Jord Squre tr or moderte sze, ether Elmto Guss Elmto or u *Cholesky s ethod Symmetrc tr Decomposto s effectve *u Decomposto Symmetrc tr d effcet. ethod (Sze < zero) Drect ethods Itertve ethods *Jcob Iterto *Guss-Sedl ethod *Successve over- elto (SO ethod) Dgolly Domt. j, > j j,,. For lrge d hgh-order ler equtos, for emple, solvg dfferetl equto, Itertve ethods re ttrctve. Fst covergece. Chpter Pge

23 EE7 Computer odelg Techques Egeerg EXAPE **********(wll be deleted)*************** I the 6 m rectgulr rego show Fgure.(), the electrc potetl s zero o the boudres. The chrge dstrbuto, however, s uform d gve by p v ε. Solve Posso s equto to determe the potetl dstrbuto the rectgulr rego. Soluto To determe the potetl dstrbuto the rectgulr rego, we use Posso s equto. y ρ ε v wth zero potetl o the boudres. Fgure. Geometry of the 6 m rectgulr rego d the h m mesh. By estblshg the rectgulr grd show Fgure.(b), we relze tht we hve s odes d, hece, s ukow potetls for whch to solve. eplcg by ts fte dfferece represetto, we obt (,,,,, ) j j j j j h Chpter Pge

24 EE7 Computer odelg Techques Egeerg It should be oted tht lthough the lthough the mesh sze ws ot eplctly used solvg plce s equto the prevous emple, h s cluded s prt of the mtr formto solvg Posso s equto. I SI system of uts, h should be meters. By pplyg the precedg dfferece equto t the vrous odes Fgure.(b), we obt the followg mtr equto: 6 5 Isted of solvg the resultg s equtos, we my ote some symmetry cosdertos Fgure.(b). It s cler tht 6 5 d tht Tkg these symmetry cosdertos to ccout, the umber of equtos reduces to two, d we obt the followg soluto:.56, 5.7 To mprove the ccurcy of the potetl dstrbuto, fer mesh such s the oe show Fgure. s requred. Becuse of the lrge umber of odes ths cse, symmetry should be used, d soluto for oly oe-qurter of the rectgulr geometry s desred. The pplcto of the dfferece equto t odes,,, d 5 should proceed routely, wheres specl cre should be eercsed t the boudry odes, 6, 9,, d, d lso t the corer ode. For emple, pplyg the dfferece equto t ode 6 yelds ) ( h Or ) ( h Chpter Pge

25 EE7 Computer odelg Techques Egeerg I equto ( ), symmetry ws used to y h complete the fve-pot str dfferece equto. Specfclly the potetl t ode to the rght of 6 ws tke equl to. Smlrly t the corer ode, we obt 5 ( 9 ) b c h Becuse of symmetry, b d 9 c, hece, ( 9 ) h Fg.. The fer mesh soluto d symmetry cosderto of emple. The mtr equto for the twelve odes show Fgure. s the Chpter Pge 5

26 EE7 Computer odelg Techques Egeerg The coeffcet the coeffcet mtr (to the left) of equto ppers wheever symmetry cosderto s used t boudry d corer odes. It should be oted tht the coeffcet mtr equto s the sme for both plce s d Possos equtos. The costt vector o the rght-hd sde of equto, however, depeds o the chrge dstrbuto wth d the potetl t the boudres of the rego of terest. Furthermore, f sted of uform chrge dstrbuto we hve gve chrge dstrbuto ), ( y v ρ, the costts vector o the rght-hd sde of equto should reflect the vlue of ), ( y v ρ clculted t ech ode. Soluto of equto gves , Chpter Pge 6