Е.В. Чижонков - Конспект лекций по методам решения симметричных линейных систем (1162400), страница 4
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DZ®« £ ï ¢ (13) j = k 1 ¨ j = k , ¯®«ã稬 ¨§ (11) ¨(12) á¨á⥬ã ãà ¢¥¨©:(0 = (rk+1; rk 1 ) = hk+1k+1 (Ark ; rk 1) + (1i k+1 )(rk 1 ; rk 1 );0 = (rk+1; rk ) = k+1 (rk ; rk ) k+1 (rk ; Ark ) :kkr ;r )§ ¢â®à®£® ãà ¢¥¨ï ©¤¥¬ k+1 = ((Ark ; rk ) . ©¤¥¬ ¢¥«¨ç¨ã (Ark ; rk 1 ) . ¯¨è¥¬ ¢ëà ¥¨¥ ¤«ï rk ¨ ¨§ ¥£® ¢ëà §¨¬ Ark 1 :rk = k rk1k k Ark 1 + (1 k )rkâáî¤ ¨¬¥¥¬2! Ark 1 = 1 rkk1 k 1 k k 2r +r :k kk k11 k k 1 k k 2 k(r ; r ) +(r ; r ):k kk k ª ª ª (rk 1 ; rk ) = 0 ¨ (rk 2 ; rk ) = 0 , â® (rk ; Ark 1 ) = 1 (rk ; rk ) .
DZ®¤áâ ¢¨¬ íâ®k k(rk ; Ark 1 ) =1 k 1 k(r ; r )k¢ëà ¥¨¥ ¢ ¯¥à¢®¥ ãà ¢¥¨¥ á¨áâ¥¬ë ¤«ï ¯ à ¬¥â஢, ¯®«ã稬k+1 = 1k+1 (rk ; rk ) 1k (rk 1 ; rk 1 ) k 1:20楪 ¯®£à¥è®á⨠«£®à¨â¬ 楪 ¯®£à¥è®á⨠¬¥â®¤ ᮯàï¥ëå £à ¤¨¥â®¢ ãáâ ¢«¨¢ ¥âáï «®£¨ç®®æ¥ª¥ ¬¥â®¤ ¨áª®à¥©è¥£® £à ¤¨¥â®£® á¯ã᪠, ⮫쪮 ¤«ï áà ¢¥¨ï ¨á¯®«ì§ã¥âáï ¥ ®¯â¨¬ «ìë© ®¤®è £®¢ë© ¬¥â®¤, âà¥åá«®©ë© 祡ë襢᪨© ¯à®æ¥áá.¥©á⢨⥫ì®, ®¡ «£®à¨â¬ ¨¬¥îâ ®¤¨ ª®¢ãî âà¥åá«®©ãî ä®à¬ã, ® ¨â¥à æ¨®ë¥ ¯ à ¬¥âàë ¢ëç¨á«ïîâáï à §«¨ç®: ç¥à¥§ ᪠«ïàë¥ ¯à®¨§¢¥¤¥¨ï ¢ ¬¥â®¤¥ á¯àï¥ëå £à ¤¨¥â®¢ ¨«¨ ¯® ï¢ë¬ ä®à¬ã« ¬, § ¢¨áï騬 ®â £à ¨æ ᯥªâà ¬ âà¨æëA , ¢ âà¥åá«®©®¬ 祡ë襢᪮¬ ¬¥â®¤¥. §¨æ ¢ ᯮᮡ¥ ¢ëç¨á«¥¨ï ¯ à ¬¥â஢ ¯à¨¢®¤¨â ª ⮬ã, çâ® ¯à¨¡«¨¥¨ï ¯® ¬¥â®¤ã ᮯàï¥ëå £à ¤¨¥â®¢ ¬¨¨¬¨§¨àãîâ ª ¤®© ¨â¥à 樨 äãªæ¨® « F (x) .DZ®«®¨¬ y0 x0 ¨ ¯à®¢¥¤¥¬ n ¨â¥à 権 ¯® âà¥åá«®©®¬ã 祡ë襢᪮¬ã ¬¥â®¤ã á ç «ì®£® ¯à¨¡«¨¥¨ï y0 .
१ã«ìâ ⥠¯®«ã稬F0 (yn ) qn2 F0 (y0 ); qn = Tn 1p11221mp ; = M; 1 =:=q01 + 21n1+ ¯®¬¨¬, çâ® F (yn ) = (A(yn x); (yn x)) (Ax; x) , F0 (yn ) = (A(yn x); (yn x)) |ª¢ ¤à â A ®à¬ë ®è¨¡ª¨. ᫨ xn ¯®«ã祮 ¯® ¬¥â®¤ã ᮯàï¥ëå £à ¤¨¥â®¢, áâ àâãï á x0 , â® ¨¬¥¥â ¬¥á⮮楪 F (xn ) F (yn ); § ç¨â ¨ F0 (xn ) F0 (yn ): âáî¤ á«¥¤ã¥â á¯à ¢¥¤«¨¢®áâì¥à ¢¥á⢠jjxn xjjA qnjjx0 xjjA :ª®®¬¨ç ï ¢ëç¨á«¨â¥«ì ï ä®à¬ DZਠ¢¥¤¥ë¥ ¢ëè¥ âà¥åá«®©ë¥ á®®â®è¥¨ï ¨á¯®«ì§ãîâ ¤¢ 㬮¥¨ï ¬ âà¨æë ¢¥ªâ®à ª ¤®© ¨â¥à 樨 «£®à¨â¬ .
«ï ¬ âà¨æ ¡®«ì让 à §¬¥à®á⨠ç áâ® ¨á¯®«ì§ãîâ íª¢¨¢ «¥âãî (¢ ¯à¥¤¯®«®¥¨¨ ®âáãâáâ¢¨ï ¢ëç¨á«¨â¥«ì®© ¯®£à¥è®áâ¨)ä®à¬ã, âॡãîéãî ⮫쪮 ®¤®£® 㬮¥¨ï ¬ âà¨æë ¢¥ªâ®à. DZਢ¥¤¥¬ ¥¥ § ¯¨áì,á«¥¤ãï ¬®®£à 䨨 "Iterative Methods for Sparse Linear Systems" (Y.Saad, 2000).Algorithm (Conjugate Gradient) for A x = b1. Compute r0 := b Ax0 ; p0 := r02. For j = 0; 1; : : : until onvergene Do:3. j := (rj ; rj )=(Apj ; pj )4. xj +1 := xj + j pj5.
rj +1 := rj j Apj6. j := (rj +1 ; rj +1 )=(rj ; rj )7. pj +1 := rj +1 + j pj8. End Do 6DZ।®¡ãá«®¢«¨¢ ¨¥ë à áᬠâਢ ¥¬ à¥è¥¨¥ á¨áâ¥¬ë «¨¥©ëå ãà ¢¥¨© Ax = b , £¤¥ A = AT > 0 .DZ।¯®« £ ï, çâ® (A) 2 [m; M ℄ , ¬¥â®¤ ᮯàï¥ëå £à ¤¨¥â®¢ ¯®§¢®«ï¥â ¯®«ãç¨âì21ká«¥¤ãîéãî ®æ¥ªã ᪮à®á⨠á室¨¬®áâ¨: kxk xkA 6 qk kx0 xkA , £¤¥ qk = 212k ; 1 =1 + 1p pMmp p . ¤ ª® ¥á«¨ ®â®è¥¨¥ £à ¨æ ᯥªâà m=M ¬ «®, â® ¬¥â®¤ á室¨âáï ¬¥¤M+ m«¥®. «ï ¡®àì¡ë á í⨬ ¨á¯®«ì§ãîâ â ª §ë¢ ¥¬®¥ ¯à¥¤®¡ãá«®¢«¨¢ ¨¥ (¯¥à¥®¡ãá« ¢«¨¢ ¨¥, preondition).¬¥áâ® ç «ì®© á¨áâ¥¬ë ¡ã¤¥¬ à áᬠâਢ âì á¨á⥬㠢¨¤ B 1 A = B 1 b , £¤¥ B| ¥ª®â®à ï ¥¢ëத¥ ï ¬ âà¨æ . «ï ¯à®áâ®âë, ¯à®¢¥¤¥¬ ¨áá«¥¤®¢ ¨ï, ¨á¯®«ì§ã濫ãá«®©ë© 祡ë襢᪨© ¬¥â®¤, ¯®¨¬ ï, çâ® «®£¨ç®¥ ¨áá«¥¤®¢ ¨¥ ¬®® ¯à®¤¥« âì ¨ ¤«ï ¬¥â®¤ ᮯàï¥ëå £à ¤¨¥â®¢.
â ª,xk+1 xk+ B 1 Axk = B 1 b:k+1DZ।¯®«®¨¬, çâ® ¬ âà¨æ B , ª ª ¨ A , ᨬ¬¥âà¨ç ¨ ¯®«®¨â¥«ì® ®¯à¥¤¥«¥ .¡®§ ç ï ek = xk x , ¯®«ã稬ek+1 = (Ik+1 B 1 A)ek : ª ª ª1 B > 0 , â® áãé¥áâ¢ã¥â ¬ âà¨æ B 12 > 0 , çâ® B 12 B 21 = B . ¤¥« ¥¬ § ¬¥ãz k = B 2 ek . ®£¤ B 2 z k+1 = (B 2 B 211¨«¨1k+1 B 2 B 2 A)B 2 z k ;1z k+1 = (I11k+1C )z k ;£¤¥ C = B 12 AB 12 . 祢¨¤®, çâ® C ᨬ¬¥âà¨ç ¨1111(Cx; x) (B 2 AB 2 x; x) (AB 2 x; B 2 x)(Ay; y)(Ay; y)=== 1=1(x; x)(x; x)(x; x)(B 2 y; B 2 y) (By; y)¡®§ 稬(Ay; y)(Ay; y)M~ = sup; m~ = inf:y (By; y )y (By; y )~M«ï १ª®£® 㢥«¨ç¥¨ï ᪮à®á⨠á室¨¬®á⨠âॡã¥âáï, ç⮡ë M. ᫨ ¢§ïâìm~m~ mB = A , â® ¯®«ãç ¥¬ M=~ = 1 , ® è £¥ à¥è ¥¬ á¨á⥬ã íª¢¨¢ «¥âãî ç «ì®©.DZ®í⮬ã B ã® ¢ë¡¨à âì ãç¨âë¢ ï ¤¢ ¯à¨æ¨¯ : ¯à®áâ®âã ®¡à é¥¨ï ¨ ¬¨¨¬ «ì®áâì ®â®è¥¨ï M~ ª m~ .
«ï ¯à¨¬¥à à áᬮâਬ ¬®¤¥«ìãî § ¤ çã:(k(x)y0 )0 = f (x); 0 < kmin k(x) kmax;y(0) = y(1) = 0;kmaxkmin 1:k í⮬ á«ãç ¥ Mh 2 max . §ï¢ ¢ ª ç¥á⢥ B ¬ âà¨æã á¨áâ¥¬ë ¯à¨ k(x) 1 ,mkmin¯®«ã稬 (B 1 A) 2 [kmin ; kmax ℄ .22¡®¡é¥ë© ¬¥â®¤ ¯à®á⮩ ¨â¥à 樨¥â®¤ ¯à®á⮩ ¨â¥à 樨 ¢ë£«ï¤¨â á«¥¤ãî騬 ®¡à §®¬:xk+1 xk+ Axk = b;A = AT > 0; > 0:¡®¡é¥ë© ¬¥â®¤ ¯à®á⮩ ¨â¥à 樨 ®â«¨ç ¥âáï ¤®¡ ¢«¥¨¥¬ 㬮¥¨ï ¯¥à¢®£® á« £ ¥¬®£® ¥ª®â®àãî ¥¢ëத¥ãî ¬ âà¨æã B , â® ¥áâìBxk+1 xk+ Axk = b;A = AT > 0; > 0:DZãáâì ¢ë¯®«¥® ãá«®¢¨¥ B 2 A > 0 , ⮣¤ ®¡®¡é¥ë© ¬¥â®¤¯à®á⮩ ¨â¥à 樨 á室¨âáï ª x , â® ¥áâì ¤«ï «î¡®£® ç «ì®£® ¯à¨¡«¨¥¨ï ¬¥â®¤®¡®¡é¥®© ¨â¥à 樨 á室¨âáï ª à¥è¥¨î.®ª § ⥫ìá⢮.
¡®§ 稬⢥थ¨¥.ek = xk⮣¤ ek+1 = (I¨, á«¥¤®¢ ⥫ì®,x;B 1 A)ekAek+1 = (A AB 1 A)ek :âáî¤ (Aek+1 ; ek+1 ) = ((A AB 1 A)ek ; (I B 1 A)ek ) =(Aek ; ek ) (Aek ; B 1 Aek ) (AB 1 Aek ; ek ) + 2 (AB 1 Aek ; B 1 Aek ):¡®§ 稬 z k = B 1 Aek , ⮣¤ (Aek+1 ; ek+1 ) = (Aek ; ek ) 2 ( B ¤à㣮© áâ®à®ë, k kA z ; z ):2B (ek+1 ek ) + Aek = 0;á«¥¤®¢ ⥫ì®,zk = ª¨¬ ®¡à §®¬, ¯®«ãç ¥¬, çâ®ekek+1:kek+1k2A kek k2A + 2 ((B 2 A)(ek ek+1); ek ek+1) = 0: âà¨æã B ¬®® ¯à¥¤áâ ¢¨âì ¢ ¢¨¤¥ B = S + K , £¤¥ S = B+2BT , K = B 2BT .
ᨫã ᢮¥£® ¢¨¤ S ᨬ¬¥âà¨ç¥, K ª®á®á¨¬¬¥âà¨ç¥. DZ®í⮬ã (Bx; x) = (Sx; x) ¨(Sx; x) > 2 (Ax; x) , ®âáî¤ ¨§ ª®¥ç®¬¥à®á⨠¯à®áâà á⢠¢¥ªâ®à®¢ á«¥¤ã¥â, çâ®(Sx; x) > 2 (Ax; x) + "(x; x) , â® ¥áâì B 2 A > "I . ç¨â,kek+1k2A kek k2A + 2" (ek ek+1; ek ek+1) 6 0;23¨«¨kek+1 k2A kek k2A + 2" kek+1 ek k22 6 0§ í⮣® ¥à ¢¥á⢠᫥¤ã¥â, çâ® ¯®á«¥¤®¢ ⥫ì®áâì kek k2A ¬®®â®® ã¡ë¢ ¥â; á ¤à㣮© áâ®à®ë, ¯®á«¥¤®¢ ⥫ì®áâì ®£à ¨ç¥ ᨧã 0, á«¥¤®¢ ⥫ì®, áãé¥áâ¢ã¥â ¯à¥¤¥«¯®á«¥¤®¢ ⥫ì®áâ¨, §®¢¥¬ ¥£® ke1 k2A . ç¨â,1Xk=0kek+1 ek k22 6 2" (ke0 k2A ke1 k2A):«¥¤®¢ ⥫ì®, àï¤ á室¨âáï, § ç¨â, ¥£® ç«¥ë áâ६ïâáï ª 0:lim kek+1k!1§ á®®â®è¥¨ï¯®«ãç ¥¬, çâ®A 1Bek k2 = lim kxk+1k!1xk k2 = 0:xk+1 xk+ xk = A 1 b x1x xk = A 1 B (xk+1 xk ) ! 0;§ ç¨â, kx xk k2 ! 0 .
®ª § ⥫ìá⢮ § ¢¥à襮. ¯à ªâ¨ª¥ ¤«ï «î¡®© ¯®«®¨â¥«ì®®¯à¥¤¥«¥®© ¬ âà¨æë B ¬®®, ¢§ï¢ ¤®áâ â®ç® ¬ «®¥ , ¯®«ãç¨âì ®æ¥ªã B 2 A , , á«¥¤®¢ ⥫ì®, ¨ á室¨¬®áâì.¥â®¤ë ५ ªá æ¨¨à ¤¨æ¨®®, £®¢®àï ® ¬¥â®¤ å ५ ªá 樨, ¨¬¥îâ ¢ ¢¨¤ã á«¥¤ãî騥 ç¥âëॠ«£®à¨â¬ :¬¥â®¤ ª®¡¨¬¥â®¤ ãáá -¥©¤¥«ï¬¥â®¤ SOR (Suesive Overrelaxation)¬¥â®¤ SSOR (Symmetri SOR) ¤«ï A = AT > 0DZãáâì ã á ¥áâì á¨á⥬ «¨¥©ëå ãà ¢¥¨© ¢¨¤ Ax = b . §à¥§ ¥¬ ¬ âà¨æã A ç áâ¨: A = L + D + R , £¤¥ D { ¤¨ £® «ì ¬ âà¨æë A , L { ¨ïï âà¥ã£®«ì ï ç áâì,R { ¢¥àåïï âà¥ã£®«ì ï ç áâì.
®£¤ ª ¤ë© ¬¥â®¤ ५ ªá 樨 ¬®® ¯à¥¤áâ ¢¨âì ¢¢¨¤¥ ®¡®¡é¥®£® ¬¥â®¤ ¯à®á⮩ ¨â¥à 樨 ᮠ᢮¥© ¬ âà¨æ¥© B ¨ ¯ à ¬¥â஬ , ¨¬¥®: ¬¥â®¤ ª®¡¨: B = D; = 1; Dxk+1 + (L + R)xk = b; ¬¥â®¤ ãáá -¥©¤¥«ï: B = L + D; = 1; (D + L)xk+1 + Rxk = b; ¬¥â®¤ SOR: B = !L + D; = !; (D + !L)xk+1 + (D(! 1) + !R)xk = b;24 ¬¥â®¤ SSOR:¯¥à¢ë© è £: (D + !L)xk+1 + (D(! 1) + !R)xk = !b;¢â®à®© è £: (D + !R)xk+1 + (D(! 1) + !L)xk = !b:⢥थ¨¥. DZãáâì A = AT > 0 , ⮣¤ ¬¥â®¤ SOR á室¨âáï ¤«ï «î¡®£® ç «ì®£®¯à¨¡«¨¥¨ï , ! 2 (0; 2) .®ª § ⥫ìá⢮.
®áâ â®ç®áâì. DZãáâì ! 2 (0; 2) . DZ஢¥à¨¬ ¤«ï â ª¨å ! ¥à ¢¥á⢮ B !2 A > 0 :((Dx; x) + (!Lx; x))!((Lx; x) + (Dx; x) + (Rx; x)) =2!(Dx; x) = (1= (Dx; x)2!)(Dx; x) > 02 ª ª ª ¯à¨ ! = 1 ¬¥â®¤ SOR ¯à¥¢à é ¥âáï ¢ ¬¥â®¤ ãáá -¥©¤¥«ï, â® ª ª á«¥¤á⢨¥¯®«ãç ¥¬, çâ® ¬¥â®¤ ãáá -¥©¤¥«ï á室¨âáï.¥®¡å®¤¨¬®áâì. DZãáâì ¬¥â®¤ SOR á室¨âáï. DZ।áâ ¢¨¬ xk+1 = T xk + , £¤¥ T { ®¯¥à â®à ¯¥à¥å®¤ , (T ) < 1 . «ï í⮣® 㬮¨¬ á®®â®è¥¨¥ ¤«ï SOR D 1 , ¯®«ã稬(I + !D 1 L)xk+1 = ((1 !)I®âáî¤ !D 1 R)xk ) + !D 1 b;T = (I + !D 1 L) 1 ((1 !)I!D 1 R) : ª ª ª (T ) < 1 , â® ¬®¤ã«ì ¯à®¨§¢¥¤¥¨ï ᮡá⢥ëå ç¨á¥« ¬¥ìè¥ 1.
à ªâ¥à¨áâ¨ç¥áª¨© ¬®£®ç«¥ ¬ âà¨æë T ¨¬¥¥â ¢¨¤ d() = det(T I ) , ®âáî¤ ¯® ⥮६¥ ¨¥â nQá«¥¤ã¥â | ( 1)n d(0) = i (T ) . ç¨âi=1n Yi =i=1jdet(T )j = 1 j1 !jn = j1 !jn 6 n < 1:â® ¤ ¥â j1 !j < 1 . ®ª § ⥫ìá⢮ § ¢¥à襮. 7«®çë¥ ¢ ਠâë ५ ªá 樮ëå ¬¥â®¤®¢DZ।áâ ¢¨¬ ¢ á¨á⥬¥ Ax = b ¢¥ªâ®à x ¢ ¢¨¤¥ x = (u; p)T £¤¥ u 2 RNn p 2 RNp (ᮮ⢥âá⢥®,x 2 RNn +Np ). DZਠí⮬ ¬ âà¨æ A \à §®¡ì¥âáï" ç¥âëॠ¯®¤¬ âà¨æë11 A12A A(£¤¥ A11 ¨¬¥¥â à §¬¥àë Nn Nn A22 | Np Np ).A A2122 A11 00 A22DZ®«®¨¬ D =, L=, R = 00 A012 :«®çë© ¬¥â®¤ ª®¡¨ ®¯à¥¤¥«ï¥âáï á®®â®è¥¨¥¬ Dxn+1 + (L + R) xn = b .0 0A21 0DZà®æ¥¤ãà à¥è¥¨ï á¨áâ¥¬ë ¤ ë¬ ¬¥â®¤®¬ «®£¨ç ¯®â®ç¥ç®¬ã ¬¥â®¤ã ª®¡¨:à §¨æ § ª«îç ¥âáï ¢ ⮬, çâ® ¢¬¥áâ® ¤¥«¥¨ï ¤¨ £® «ìë¥ í«¥¬¥âë âॡã¥âáï 㬮¥¨¥ ®¡à âãî ¬ âà¨æã.
â®ç¥çëå ¬¥â®¤ å ¥®¡å®¤¨¬®¥ ãá«®¢¨¥ ¯à¨¬¥¨¬®á⨬¥â®¤ | aii 6= 0 . ¡«®çëå íâ® âॡ®¢ ¨¥ ¯¥à¥¯¨áë¢ ¥âáï ¢ ¢¨¤¥ det(Aii ) 6= 0 .25DZ®å®¨¬ ®¡à §®¬ ®¯à¥¤¥«ïîâáï ¡«®çë© ¬¥â®¤ ãáá { ¥©¤¥«ï (D + L)xn+1 + Rxn =xn+1 xnb ¨ ¡«®çë© ¬¥â®¤ SOR: (D + wL)+ Axn = b .w¥®à¥¬ë á室¨¬®á⨠¤«ï ¡«®çëå ¬¥â®¤®¢ «®£¨çë ⥮६ ¬ ® á室¨¬®á⨠¯®â®ç¥çëå ¬¥â®¤®¢. ¯à¨¬¥à, à áᬮâਬ ®á®¢ãî ⥮६㠮 á室¨¬®á⨠¬¥â®¤ SOR:⢥थ¨¥. DZãáâì A = AT > 0 , ⮣¤ ¡«®çë© ¬¥â®¤ SOR á室¨âáï ¤«ï «î¡®£® ç «ì®£® ¯à¨¡«¨¥¨ï , ! 2 (0; 2) .®ª § ⥫ìá⢮.
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