Е.В. Чижонков - Конспект лекций по методам решения симметричных линейных систем (1162400), страница 6
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ek+1 = TSORSORT T®âªã¤ TSSOR = TSORSOR .⢥थ¨¥. DZãáâì A | ᨬ¬¥âà¨ç ï ¯®«®¨â¥«ì®®¯à¥¤¥«¥ ï ¬ âà¨æ : A =AT > 0 . ¥â®¤ SSOR á室¨âáï , 0 < w < 2 .®ª § ⥫ìá⢮:DZãáâì ¬¥â®¤ SSOR á室¨âáï. ®£¤ (TSSOR ) < 1 ¨«¨, ¤à㣨¬¨á«®¢ ¬¨, jdetTSSORj < 1 . ¬¥â¨¬, çâ® detTSSOR = det2 TSOR = (1 w)2n , § ç¨âj1 wj2n < 1 , ®âªã¤ ¯®«ãç ¥¬, çâ® w 2 (0; 2) .®áâ â®ç®áâì. DZãáâì 0 < w < 2 . áᬮâਬ ®¡®¡é¥ë© ¬¥â®¤ ¯à®á⮩ ¨â¥à 樨:¥®¡å®¤¨¬®áâì.Bxk+1 xk+ Axk = b; T = IB 1 A:31 ¯®¬¨¬, çâ® ¥á«¨11A > 0 , B > A;22Bâ® ®¡®¡é¥ë© ¬¥â®¤ ¯à®á⮩ ¨â¥à 樨 á室¨âáï.DZ®ª ¥¬, çâ® ¬ âà¨æ TSSOR ¯à¥¤áâ ¢¨¬ ¢ ¢¨¤¥ TSSOR = I Q 1 A , £¤¥Q = w(2 w)1D+L Dw1D+R :w«ï í⮣® ¯à®¤¥« ¥¬ á«¥¤ãî騥 ¢ëª« ¤ª¨:TSOR = wD 1 L + I 1wD 1 R = 1 1 DD1(1 w) I wD 1 R == wD L + Iww 1 1D1 wDD=+LD R =+L+L A ;wwww «®£¨ç®(1 w) ITTSOR®âªã¤ T TTSSOR = TSORSOR ==D=+RwD+Rw 1 1D+R A ;wD+R Aw 1D+LwD+L A ;w®á¯®«ì§®¢ ¢è¨áì ¯à¨¢¥¤¥ë¬¨ ¢ëª« ¤ª ¬¨, ¯®«ãç ¥¬:TSSOR = IQ 1A ,, A = Q (I TSSOR) , A = 2 w w Dw + L D 1"D+RwD+R Aw"D+Lw 1#D+L Aw=#1DDD=+L D 1 I ++R A+LA,2 w www 1 1 12 wDDDDD+L=I++R+LA+L,wwwww, 2 w w D = Dw + L + Dw + R A = w2 D D:w,â.¥.
¯®«ã稫¨ ¢ १ã«ìâ â¥ íª¢¨¢ «¥âëå ¯¥à¥å®¤®¢ ⮤¥á⢥®¥ à ¢¥á⢮.®ª ¥¬ ⥯¥àì, çâ®Q1A > 0 , 8x 6= 0 (Qx; x)21(Ax; x) > 0:232DZ८¡à §ã¥¬ ¢ëà ¥¨¥ (¨á¯®«ì§ã¥¬, çâ® L = RT ):1(Ax; x) =2DD1w1+L D+ R x; x(fL + D + Rg x; x) ==2 www2 wDD11=+ R x;+R x(Dx; x) (Rx; x) =D2 www21w1 1 11= (Dx; x) +D Rx; Rx + (Rx; x) ++(Dx; x) =22 w4 4 w2 wDD2 w1=(Dx; x) +D+ R x;+R x :4w2 w22(Qx; x)祢¨¤®, çâ® ¯à¨ w 2 (0; 2) ¯®á«¥¤¥¥ ¢ëà ¥¨¥ ¯®«®¨â¥«ì® 8x 6= 0 , § ç¨âQ1A > 0:2 ª¨¬ ®¡à §®¬, ¬ë ¤®ª § «¨, çâ® ¬¥â®¤ SSOR ï¥âáï ®¡®¡é¥ë¬ ¬¥â®¤®¬ ¯à®á⮩¨â¥à 樨 á ¬ âà¨æ¥© B = Q ¨ ¯à¨ w 2 (0; 2) ¢ë¯®«ï¥âáï ãá«®¢¨¥11Q= B> A,B2A > 0;2 § ç¨â ¬¥â®¤ SSOR á室¨âáï.
⢥थ¨¥ ¤®ª § ®. 9ᨬ¯â®â¨ç¥áª ï ®¯â¨¬¨§ æ¨ï ¡«®ç®£® ¬¥â®¤ SSOR áᬮâਬ § ¤ çã ®¯â¨¬¨§ 樨 !minmax (T(!; )) . ¯¥à â®à ¯¥à¥å®¤ ¨¬¥2(0;2) (Tï ) SSOR¥â ¢¨¤ TSSOR= !D 1 R + I 1 (1 !) I !D 1 L!D 1 L + I 1 (1 !) I !D 1 R . ëà §¨¬ à¥è¥¨¥ § ¤ ç¨ á®¡áâ¢¥ë¥ § 票ïTSSOR y = y ç¥à¥§ à¥è¥¨¥ ᯥªâà «ì®© § ¤ ç¨ ¤«ï ¬¥â®¤ ª®¡¨ Tï x = x . «ïí⮣® ¯¥à¥¯¨è¥¬!D1R + I 1!DTSSOR ==I !A111 A120I1L + I 1= 1=!A111 A12II0I01!A22 A21 I:;)(1 !)2 I + !2 (1 !)(2 !)A111 A12 A221 A21!(1 !)(2 !)A221 A21!(1 !)(2 !)A111 A12 !3 (2 !)A111 A12 A221 A21 A111 A12(1 !)2 I + !2 (2 !)A221 A21 A111 A12:33®áâ â®ç® ¨áª âì § ¢¨á¨¬®áâì ¬¥¤ã ᮡá⢥묨 ¢¥ªâ®à ¬¨ ¢ ¢¨¤¥y1 = x1 ;y2 = x2(18)£¤¥ { ¥¨§¢¥áâë© ¯ à ¬¥âà. ®£¤ , á ãç¥â®¬ á®®â®è¥¨ïTï x = xA111 A12 x2 = x1 ;A221 A21 x1 = x2(19)¬®® § ¯¨á âì ᯥªâà «ìãî § ¤ çã ¤«ï ¬ âà¨æë TSSOR :(1 !)2 + !2 (1 !)(2 !)2 + !(1 !)(2 !) + !3 (2 !)3 = !(1 !)(2 !) + (1 !)2 + !2 (2 !)2 = : ᫨ ¯à¥®¡à §®¢ âì á¨á⥬ã á«¥¤ãî騬 ®¡à §®¬!(2 !)(1 ! + !2 2 ) = (1 !)2 !2 2 (1 !)(2 !)!(1 !)(2 !) = [ (1 !)2 !2 2 (1 !)(2 !)℄;¨ ¨áª«îç¨âì , â® ¯®«ãç¨âáï ª¢ ¤à ⮥ ãà ¢¥¨¥ ¤«ï 室¥¨ï ¥¨§¢¥á⮣® :2 (2(1 !)2 + !2 2 (2 !)2 ) + (1 !)4 = 0 .!2 2 (2 !)21;2 = (1 !)2 +2r4 44 (1 !)2 (2 !)2 !22 + ! (24 !) ¬¥â¨¬, çâ® ¯ à ¬¥â஬ ¤«ï ª®à¥© 1;2 ¢ëáâ㯠¥â 2 , ¯®í⮬㠪®«¨ç¥á⢮ ᮡá⢥ëå § 票© ¤«ï Tï ¨ TSSOR ᮢ¯ ¤ ¥â..ª.
1 > 2 > 0 , ⮠ᯥªâà «ìë© à ¤¨ãá ¬ âà¨æë TSSOR ¡ã¤¥â ®¯à¥¤¥«ïâì ¢¥«¨ç¨ 1 . ëà ¥¨¥ ¤«ï 1 ¬®®â®® § ¢¨á¨â ®â 2 , ¯®í⮬㠬 ªá¨¬ã¬ ¯® ¤®á⨣ ¥âáï¯à¨ = (Tï ) < 1: )r!4 4 (2 !)4!2 2 (2 !)2(TSSOR ) = (1 !)2 ++ (1 !)2 (2 !)2 !2 2 +:24«ï 室¥¨ï ¬¨¨¬ã¬ ¯® ! ¢ëç¨á«¨¬ ¯à®¨§¢®¤ãî0! (TSSOR ) = 2(! 1) + 22 (1 !)+2 !(1 p!)(2 !)(2 6! + 3!2 ) + 4 (2! !2 )3 (1 !): )2 !2 (1 !)2 (2 !)2 + 1=44 !4 (2 !)40! (TSSOR ) = 0 ¯à¨ ! = 1 , ¯®í⮬㠮¯â = (TSSOR )!=1 = 2 . DZ®«ã稫®áì â ª®¥ ¥+á®®â®è¥¨¥ ¤«ï ®¯â¨¬ «ì®£® § 票ï ᪮à®á⨠á室¨¬®áâ¨, ª ª ¨ ¤«ï ¬¥â®¤ ãáá ¥©¤¥«ï: (Tz ) = 2 .
.ª. ®¤¨ è £ ¬¥â®¤ SSOR § ª«îç ¥âáï ¢ ¢ë¯®«¥¨¨ ¯®á«¥¤®¢ â¥«ì® è £®¢ ¢¥à奩 ¨ ¨¥© ५ ªá 樨, ¨ ¯à¨ ! = 1 ¬¥â®¤ SOR ᮢ¯ ¤ ¥âá ¬¥â®¤®¬ ãáá -¥©¤¥«ï, â® ¬®® ᤥ« âì ¢ë¢®¤, çâ® ¡«®çë© 2 2 ¬¥â®¤ SSOR¥ ¤ ¥â ã᪮२ï á室¨¬®á⨠¯® áà ¢¥¨î á ¬¥â®¤®¬ ãáá -¥©¤¥«ï, ¯®í⮬㠮 ¥¨á¯®«ì§ã¥âáï.¤ ª® ¯®«ã祮¥ ¯à¨ ¨áá«¥¤®¢ ¨¨ ¬¥â®¤ SSOR ¯à¥¤áâ ¢«¥¨¥ ¤«ï ®¯¥à â®à ¯¥à¥å®¤ TSSOR = I Q 1 A ¯à¨¬¥ïîâ á«¥¤ãî騬 ®¡à §®¬: 䨪á¨àãîâ ¥ª®â®à®¥ !34¨ ¨á¯®«ì§ãîâ Q ª ª ¯à¥¤®¡ãá« ¢«¨¢ ⥫ì, â.ª.
SSOR Q(xk+1 xk ) + Axk = b ¨ Q =QT > 0 , ¯®á«¥ 祣® ¯à¨¬¥ïîâ ¬¥â®¤ ᮯàï¥ëå £à ¤¨¥â®¢. ª¨¬ ®¡à §®¬, ¬®® § ª«îç¨âì, çâ® ¬¥â®¤ SOR ï¥âáï ¨«ãç訬 ¯® ᪮à®áâ¨á室¨¬®á⨠á।¨ à áᬮâà¥ëå ५ ªá 樮ëå ¬¥â®¤®¢, ¯®â®¬ã ª ª 2 > (TSOR )¯à¨ ®¯â¨¬ «ìëå § 票ïå ¯ à ¬¥â஢. áᬮâਬ ¤ «¥¥ ¬®¤¨ä¨æ¨à®¢ ë© ¬¥â®¤SOR ¨ ¨áá«¥¤ã¥¬ ¯®§¢®«ï¥â «¨ ¢¢¥¤¥¨¥ ¤®¯®«¨â¥«ì®£® ¯ à ¬¥âà ¯®«ãç¨âì ¢ë¨£àëè¢ á室¨¬®áâ¨.®¤¨ä¨æ¨à®¢ ë© ¬¥â®¤ SOR (MSOR)8><uk+1 ukA11+ A11 uk + A12 pk = b1!k+1 pk>: A p+ A21 uk+1 + A22 pk = b2 :22!0 ᫨ ¯®«®¨âì ! = !0 , â® ¯®«ã稬 ®¡ëçë© ¬¥â®¤ SOR . ¯¥à â®à ¯¥à¥å®¤ TMSOR =I!0 A221 A210I 1(1 !)I0!A111 A12 :(1 !0 )I ᫨ A = AT > 0 , Tï x = x; TMSOR y = y , â® ¢ë¯®«¥®( + !1) = !0 !2 . ¬¥â¨¬, çâ® ¯à¨ ¯®¤áâ ®¢ª¥ ! = !0 ᮢ ¯®«ãç ¥¬ ¢ â®ç®á⨠ᮮâ®è¥¨ï ¤«ïTSOR .®ª § ⥫ìá⢮: 㤥¬ ¨áª âì ¢§ ¨¬®á¢ï§ì ¬¥¤ã ᮡá⢥묨 ¢¥ªâ®à ¬¨ à áᬠâਢ ¥¬ëåᯥªâà «ìëå § ¤ ç ¢ ¢¨¤¥ (18).
¬®¨¬ ®¡¥ ç á⨠TMSOR y = y I0¬ âà¨æã !0 A 1 A I , ¯®«ã稬⢥थ¨¥ 1.1)( + !02221(1 !)y1 !A111 A12 y2 = y1(1 !0 )y2 = [!0 A221 A21 y1 + y2 ℄:DZ®¤áâ ¢¨¬ ¢ëà ¥¨ï (18) ¨ (19), ⮣¤ á¨á⥬ ¯à¥®¡à §ã¥âáï ª ¢¨¤ã: + ! 1 = ! + !0 1 = 1 !0 ;¯®á«¥ ¯¥à¥¬®¥¨ï ãà ¢¥¨© ¯®«ã稬 â®, çâ® âॡ®¢ «®áì ¤®ª § âì. p®à¨ ¯®«ã祮£® ¢ ã⢥थ¨¨ 1 ª¢ ¤à ⮣® ãà ¢¥¨ï 1;2 = 1 2 !!0 (2 + 1) ,0 !!0 2£¤¥ = ! + ! +.2⢥थ¨¥ 2.
DZਠA = AT > 0 ¬¥â®¤ MSOR á室¨âáï ¤«ï ¯à®¨§¢®«ì®£® ç «ì®£® ¯à¨¡«¨¥¨ï , !; !0 2 (0; 2) .⢥थ¨¥ 3. ¥è¥¨¥ § ¤ ç¨ á¨¬¯â®â¨ç¥áª®© ®¯â¨¬¨§ 樨min (TMSOR ) = (TSOR )!=!0 =! = ! 1;!;!0 2(0;2)£¤¥ ! = !0 = ! = p 2 2.1 + 1 (Tï )§ ã⢥थ¨ï 3 á«¥¤ã¥â, çâ® ¬®¤¨ä¨ª æ¨ï ¬¥â®¤ SOR ¥ ¯à¨¢®¤¨â ª ã᪮२îá室¨¬®áâ¨, â® ¥áâì á।¨ ५ ªá 樮ëå ¬¥â®¤®¢ SOR ï¥âáï ¨«ãç訬.¥¬ 1ᯮ¬®£ ⥫ìë¥ à¥§ã«ìâ âë à §¤¥«¥ ¯à¨¢®¤ïâáï ®á®¢ë¥ ®¡®§ ç¥¨ï ¨ ¯®áâ ®¢ª § ¤ ç¨ ¨ ¤®ª §ë¢ îâáï¢á¯®¬®£ ⥫ìë¥ à¥§ã«ìâ âë, ¨á¯®«ì§ã¥¬ë¥ ¢ ¤ «ì¥©è¥¬.1.1.
á®¢ë¥ ®¡®§ ç¥¨ï¨ ¯®áâ ®¢ª § ¤ 稡®§ 稬 ç¥à¥§ U ¨ P ¥¢ª«¨¤®¢ë ¯à®áâà á⢠¢¥ªâ®à®¢ à §¬¥à®á⥩ Nu ¨ Npᮮ⢥âá⢥®, Z = U P: ¯¨áì ¢¥ªâ®à z ¢ ¢¨¤¥ z = fu; pg 2 Z ®§ ç ¥â, çâ®® á®á⮨⠨§ ¤¢ãå ª®¬¯®¥â: u 2 U; p 2 P . «ï á¨áâ¥¬ë «¨¥©ëå «£¥¡à ¨ç¥áª¨å ãà ¢¥¨© L z = F í⮠ᮮ⢥âáâ¢ã¥â à §¡¨¥¨î ª¢ ¤à ⮩ ¬ âà¨æë L ¡«®ª¨Lij (1 i; j 2) :11 L12 ;L= LL21 L22à §¬¥à®á⨠ª®â®àëå ®¯à¥¤¥«ïîâáï à §¬¥à®áâﬨ ª®¬¯®¥â ¢¥ªâ®à z : L11 ï¥âáïNu Nu ¬ âà¨æ¥©, L12 | Nu Np , L21 | Np Nu , L22 | Np Np . DZà ¢ ï ç áâìá¨á⥬ë F ¨¬¥¥â ¯à¥¤áâ ¢«¥¨¥, «®£¨ç®¥ z : F = ff; 'g 2 Z . ᫨ L11 ¥¢ëத¥ , â® ¬®® ®¯à¥¤¥«¨âì ¬ âà¨æã S :S = L=L11 (L22L21 L111 L12 ); §ë¢ ¥¬ãî ¤®¯®«¥¨¥¬ ãà ¤«ï ¬ âà¨æë L ®â®á¨â¥«ì® L11 (¤«ï 㤮¡á⢠¢§ïâë¬ á® § ª®¬ ¬¨ãá).
¥ § 稬®áâì ®¯à¥¤¥«ï¥âáï á«¥¤ãî騬 ä ªâ®à¨§®¢ ë¬ ¯à¥¤áâ ¢«¥¨¥¬ L :1L I0 L11 0IL12 ;11L= L L 1 I(1.1)0S 0I21 11£¤¥ I ¨¬¥¥â á¬ëá« ¥¤¨¨ç®© ¬ âà¨æë ᮮ⢥âáâ¢ãî饣® à §¬¥à . ®à¬ã« (1.1) ᢮¤¨â à¥è¥¨¥ á¨á⥬ë Lz = F ä®à¬ «ì® ª ®¡à é¥¨î ¤¢ãå ¯®¤¬ âà¨æ L11 ¨ S , ä ªâ¨ç¥áª¨ | ⮫쪮 S , â ª ª ª ¢ ¥¥ ®¯à¥¤¥«¥¨¥ 㥠¢å®¤¨â L111 . ª¨¬ ®¡à §®¬,íâ®â ¯à¨¥¬ ¯®§¢®«ï¥â ¯®¨§¨âì à §¬¥à®áâì à¥è ¥¬®© § ¤ ç¨ ¯ã⥬ ᢥ¤¥¨ï ¥¥ ª à ¢®á¨«ì®© á ¬ âà¨æ¥© á¯¥æ¨ «ì®© áâàãªâãàë. «¥¥ ¬ë ¡ã¤¥¬ ¨¬¥âì ¤¥«® á ¢¥é¥á⢥®© á¨á⥬®© «¨¥©ëå «£¥¡à ¨ç¥áª¨å ãà ¢¥¨© L" z = F á ¯ à ¬¥â஬ " 0 á¯¥æ¨ «ì®£® ¢¨¤ :L"z ABTB"D u = f'pF;(1.2)36¥¬ 1 ᯮ¬®£ ⥫ìë¥ à¥§ã«ìâ â룤¥ A = AT > 0; D = DT > 0 | ª¢ ¤à âë¥ ¬ âà¨æë à §¬¥à®¢ Nu Nu ¨ Np Np , B | ¯àאַ㣮«ì ï, ¢ ®¡é¥¬ á«ãç ¥, ¬ âà¨æ à §¬¥à Nu Np .㤥¬ ¯à¥¤¯®« £ âì, çâ® ¬ âà¨æ L" ¥¢ëத¥ ¯à¨ «î¡®¬ " 0 . â® ãá«®¢¨¥,¢ ᨫã ä ªâ®à¨§ 樨 L" ¢¨¤ (1.1), ®§ ç ¥â ¥¢ëத¥®áâì ¤®¯®«¥¨ï ãà S" =BT A 1B++" D .
DZ® ¯®áâ஥¨î S0 = S0T ¨ ®¡« ¤ ¥â ᢮©á⢮¬ ¯®«®¨â¥«ì®© ¯®«ã®¯à¥¤¥«¥®áâ¨, â.¥. (S0 p; p) 0 ¤«ï ¯à®¨§¢®«ì®£® p 2 P . DZ®í⮬㠯ਠ" > 0 ¬ âà¨æ S" (¨,á«¥¤®¢ ⥫ì®, L" ) ¥¢ëத¥ ¢á¥£¤ , ¯à¨ " = 0 ãá«®¢¨ï ¥¢ëத¥®á⨠¬ âà¨æL0 ¨ S0 ®áïâ íª¢¨¢ «¥âë© å à ªâ¥à. â® ®§ ç ¥â ¨áª«îç¨â¥«ì®áâì á¨âã æ¨¨ á" = 0 , ¯®í⮬㠮ᮢ®¥ ¨§«®¥¨¥ ¡ã¤¥â ¯®á¢ï饮 ¨¬¥® í⮬ã á«ãç î, ®¡®¡é¥¨¥à¥§ã«ìâ ⮢ ¤«ï " > 0 ¡ã¤¥â ¯à®¢®¤¨âìáï ¯® ¬¥à¥ ¥®¡å®¤¨¬®áâ¨.¡à ⨬ ¢¨¬ ¨¥ á®®â®è¥¨¥ ¬¥¤ã à §¬¥à®áâﬨ ¯à®áâà á⢠U ¨ P , á«¥-¤ãî饥 ¨§ ãá«®¢¨ï ¥¢ëத¥®á⨠¨á室®© § ¤ ç¨. ® ¢á¥£¤ ¨¬¥¥â ¢¨¤Nu Np :¥©á⢨⥫ì®, ¢ ¯à®â¨¢®¬ á«ãç ¥ à áᬮâਬ à¥è¥¨¥ ®¤®à®¤®© á¨á⥬ë L0 z = 0¢¨¤ z = f0; pg .