SISTEME DE ECUATII LINIARE

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Ἀλφαῖος Ηλιόπουλος
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1 NLIZ NUMERIC- SISTEME DE ECUTII LINIRE ( SISTEME DE ECUTII LINIRE. Itodc Mtod d zov sstmo d ct d fom () s gpz g do ctgo: mtod dct, zt p pocd d m s mtod dct (ttv) x 2 2 x () B (') Poptt p c of tt fct s ct st tzt c mt d z pt mtod d zov zt p pocd d m coscto. Ect sstm t s pot mtpc p costt s s com c jto opto spmt otd-s f fom smpfct sstm cosdt. L g mtod d cc, pt zov mt pom p sstm d cc dt, t vt vd o s d ct cm f: m scvto d cc, pcz ztto, postt todc o tst d pcz, scvt, p pcs dsfs gotm. I cd mtodo dct, c s cosd f mtod xct, s- mps mtod c z gotm d m Gss, c vt Gss-Jod, Doot, Cot s Chosky. Mtod xct pmt ot sot xct sstm d ct fcd stct d o d tp od-off sstm d cc, d o d otj s tch foosd m ft d opt mt. I cd mtodo ttv s- mps mtod Jco, Gss-Sd, mtod xo sccsv s mtod Lczos. Mtod ttv s cctzz p fpt c sot sstm cosdt s ot c mt s d vo c pzt sot pt dvs tt sccsv. I cd csto mtod, s p pom d g c mtod, c sg c m m vtz d covgt soto pt o poxm t dcvt s. Dt mtod mc dct, tzt zov sstm d ct gc omog s vo pzt: mtod m dct Gss, mtod fctoz LU s mtod Chosky.

2 NLIZ NUMERIC- SISTEME DE ECUTII LINIRE ( 2. Pocd mc d zov sstmo d ct Dt mtod mc dct, tzt zov sstm d ct gc omog s vo pzt: mtod m dct Gss, mtod fctoz LU s mtod Chosky, c s mtod dct (ttv) d zov sstmo d ct v f pztt mtod Jco. 2.. Mtod m Gss I pm tp mt tgz (ssttt t) s m sccsv - ps cosct x,,2,- d ct sstm c c c ztt m mto stt s dgo pcp mtc s tsfom csto t-o mtc spo tgh. I ps s ot: s c ott zt: ( ) j () j 2 2 j 2 2 x x 2 2 x, 2,3, ; j j,2, 2 (2) (3) 2 () () () () 2 x () 2 2 () x (4) I ps - m sstm d ct dv: 2 x () () () () ( ) ( ) ( ) (5) x ( ) ( ) ( ) x s fom sts: ( ) B (6) ( ) [ ] [ ] [ ] 2

3 NLIZ NUMERIC- SISTEME DE ECUTII LINIRE ( I ps s ot c ( ) ( ) j () ( ) j () () [ ] [ ] B ( ) ( ) ( ) ( ) [ ] (7) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) j j m m j,, j +,, I f dp cosm ps -, sstm st compt tgzt: 2 3 x () () () () ( 2) ( 2) ( 2) 33 3 x3 3 ( ) ( ) x s tzd o sc smoc: [ \ S][ ] [ B ] ' (8) (9) () Osvt () I pocs d cc tomt, mtc sccsv [ ] s costsc spt d mmo t oct pt mtc [ ]. Pocd s pzt gotmt fom mto:,2,, +,, m m j +,, j j m j 3

4 NLIZ NUMERIC- SISTEME DE ECUTII LINIRE ( Pt dtm sot sstm () oc pocs d m vs (ssttt po) c pot f dscs stf: x x x ( ) ( ) ( 2) ( ) ( ), ( 2), ( 2), x x ( ), + x+ ( ), 2, 3,, () spctv fom gotmc: x, 2,, x j j + x j c j st mt mtc tgh S mt mtc B'. I cocz, m pc m Gss dp - tp d cc s ot o mtc S spo tgh, soct sstm () c codt c mtc s f sg, mt tsfomt d p dgo pcp s zt fc ps m: (),,2,,;,2,,-. Vo dtmt mtc st dt d pods mto d p dgo pcp mtc S: () ( 2) ( ) dt( ) dt( S) 33 (2) 2.2. Dscomp LU O modfc mtod m, mt dscomp s fctoz LU st fcvt tzt. I cst mtod, mtc cofcto st tfomt t- pods d do mtc L s U d L st mtc fo tgh s U st mtc spo tgh vd mt g c p dgo pcp. Oc mtc pot f scs c pods mtc pmto P mtc fo tgh L s spo tgh U, t-o ftt d mod. D tg gm d postt vom g vt c U p dgo.i cot v f xmpfct pc mtod cz mtc d dms 4x4. 4

5 NLIZ NUMERIC- SISTEME DE ECUTII LINIRE ( F mto dscomp mtc pods do mtc L s U: (3) Imtd L c pm coo U: ; 2 2; 3 3; 4 4 (4) Imtd pm L c coo U: 2 2; 3 3; 4 4; (5) d d ; 3 ; 4 (6) I cst mtod s tz gs coo L s U, stf c mz gs mto coo L, p mt o L c c coo do U: ; ; (7),, Pocdd og otm cot: ; ; (8) Fom g pt ot mto mtco L s U st: j j j j j k k k k Pt j, s pt., kj ; kj j,,2,, (9), > j, j 2,3,, j (2) j 5

6 NLIZ NUMERIC- SISTEME DE ECUTII LINIRE ( Osvt: Stoc mtc Doc mtc L s U pot f stoct zo zvt mtc cst mtod st cosct s s dm d schm compct: (2) 2 2 Rzov sstm d ct [ Α ][ ] [ B] dp fctoz [ L ][ U][ ] [ B] Notm [ U ][ ] [ B' ] tc vm: [ L][ B' ] [ B] [ B' ] p ssttt t dp c c B' cosct, p ssttt po otm U B' [ ][ ] [ ] [ ] U vtj spc mtod fctoz LU st postt zov sstm c m mt coo d tm c o m coom d fot d cc. D tt c sp dos d cz csc m Gss, c tm t cosct vs, cst mtod st cs s costm cst vcto momt dc fom LU. Nm opto tmtc cs f sot pt fc o tm, st xct m opto cs fct mt dt o mtc x c vcto d dms. 2.3 Fctoz Chosky I cz mtco smtc ( T ) s poztv dft ( T >, pt oc vcto ) dt c m fct mtod d zov sstmo st fctoz Chosky. Mtod cost dscomp mtc L s L T stf ct LL T d L st mtc fo tgh. cst dscomp s pot fc pt oc mtc smtc s poztv dft: T L L () 2 mt L 2 (23) 6

7 NLIZ NUMERIC- SISTEME DE ECUTII LINIRE ( Emt mtc L s ot fct d c mtc p dtfc tm c tm zov pods mtc (). S ot mto t d ct: ; j j 2 j j ; < j ( ), > j >,, < j jj Rt (24) t c tm d p dgo pcp zt s fom dc ptt d dft dt mt dgo spctv s sm ptto tto mto d p cs coo cct to. Tm scd s ccz c dft t mt spctv j mtc s sm o pods d ct do fcto, pztd mt cooo s j, tot mptt tm cospzto d p dgo pcp. Rzov sstm d ct: C ott [ ][ ] [ ] mz: T [ L ][ L][ ] [ B] j (24) ; (25) L B' pocd d zov do tp s pc dp cm Ssttt t: L B' B B' (26) [ ][ ] [ ] [ ] ' ( ) (27) ' ' j j, 2,3,, j Ssttt po L B' (28) [ ][ ] [ ] [ ] x ', 2,, ' x j j + x j (29) 7

8 NLIZ NUMERIC- SISTEME DE ECUTII LINIRE ( Osvt: Rmc st fpt c fctoz Chosky vcto B st fctt s cosct mtc L o dt ott pot f tzt s pt t stt, vcto dft. 2.4 Mtod ttv Jco d zov sstmo d ct ( ) I pcp mtod ttv cost cosd vo t pt vcto sot s po p pc gotm d cc ttv s dtm () s d poxmt sccsv, ( 2) ( k ),, c pcp t s covg ct sot xct sstm. F sstm d ct: : x x3 + + x 2x x x 2 (3) x x3 + + x x x3 + + x Expctd d fc t sstm (3) p x,2,, otm mto st d t: 2 3 x x3 x x x3 x (3) 2 x x x s c pot f scs fom mtc stf: x g m2 m3 m x g 2 m 2 m23 m2 + (32) x g m m 2 m3 m x x m m m m g 2 3, x d j ;. g mj 8

9 NLIZ NUMERIC- SISTEME DE ECUTII LINIRE ( Cc s pot codc mtc cosdd t (32) scs s fom codst: g + M (33) gsd smt fc tt poxmt tto coscto. Cosdd vcto d po: ( ) T ( ) ( ) ( ) ( ) ( ) x x x x ( ) 2 (34) dp pm tt gsm: () ( ) g + M (35) po: ( 2) ( ) g + M Rt (36) s m pot sc stf: () ( ) g + M. (36) ( m+) ( m) ( 2) () ( ) g + M g + M ( 3) ( 2) ( ) ( ) g + M g + M g g + M ( m+ ) 2 m m+ ( ) ( ) 2 ( ) ( + M ) ( I + M) g + M ( ) ( I + M g + M ) ( I + M + M ) g + M I + M + M + + M g + M Cm pocs ttv, dc covgt, t s dpd d vcto d po (), v zt c: + M (38) m m dc mtc M t s f covgt, s c M <. Foosd pt om, om zt c ptm sc: j m j < ; < ; j < (39) j j j j j j d tm gtt ddcd-s c, pt c mtod Jco s f covgt, t c mtc sstm s f dgo domt. Pocs d t st cott z to (37) p cd pcz dot: ( k+ ) ( k ) < ε (4) ( k+ ) m d tt fd ftt d ctt poxmt t () s d mod c st codtot mtc cofcto sstm. (37) 9

10 NLIZ NUMERIC- SISTEME DE ECUTII LINIRE ( 3. Codto sstmo d ct Sstm d d ct pot f codtot (c sot st) s codtot (c sot st). Sttt sot s jdc pot c mc ptt vcto. Dc modfc smfctv tm, sot sstm s modfc cs ms, tc sstm st codtot, vd sot st; dc sot s modfc st, sstm st codtot s sot st. U sstm st codtot pt o opt dtmt, fctt pt-o mtod dtmt, tc cd cst sstt st m. cst dft todc dstct t do tp d d codto: sstm pot f codtot pt o mtod ptc s codtot pt o opt ptc, oc f mtod foost. S cosdm sstm: (4) F ' + s ' + d pzt vcto c podc pt st vcto pt sot. vm: (42) ' ' c m pot f scs: ( ' ) ' (43) s (44) s: D (46) zt ; ; std cot c cod( ) ; ; (45) (46) (47) mt m d codt mtc otm: cod ( ) cod ( ) (48)

11 NLIZ NUMERIC- SISTEME DE ECUTII LINIRE ( Rt (48) pot fz fomt sp sttt sstmo d ct. stf dc mtc sstm m d codt cod() tc modfc sot xpmt p st d cs od d mm c s modfc tm dft d. tf sps dc mc modfc s podc tm, sot s modfc s cs ms. Sstm c mtc cofcto vd m d codt cod() mc st sstm codtot, st sot. Exmp mc F sstm d ct:.57x 3.42y 2.35 (49).6x 3.4y 2.4 c sot x.64, y.6 fost ott fctd-s cc c t cf smfctv. Sstm st s codtot doc modfc smt cofcto podc vt m soto. fmt st stt gfc Fg Y x-3.42y2.35.6x-3.4y Fg. O mtt codto s ot dc s scd pm ct d c d do s ztt s mtpc c : 3.x + 2.4y 5. (5).6x 3.4y 2.4 Sot cst o sstm x.63, y.6 st pt dft d c pcdt, d modfc cofcto ftz mod smt (Fg. 2).

12 NLIZ NUMERIC- SISTEME DE ECUTII LINIRE ( Y x+2.4y5..6x-3.4y Fg. 2 2

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