QR迭代法求矩阵特征值

QRA0098609610087120111023QRA:AAHAHQR()n1A(n),QRnAgr900900crg:mm:mm22.1Agr900900crg:mm:mmAA900¤900matlaBA=zeros(900,900)900¤900.mmijAA.matA900A900QR2.2QRQRnAA=Q1R1;A1=R1Q1A1=Q2R2;A2=R2Q2¢¢¢¢¢¢¢¢¢1Ak=Qk+1Rk+1;Ak+1=Rk+1Qk+1¢¢¢¢¢¢¢¢¢QiRiAi+1=QTi+1(Qi+1Ri+1)Qi+1=QTi+1AiQi+1Ai+1Aii!1AiAQR()AAHAH=2666666664a11a12a13:::a1n¡1a1na21a22a23:::a2n¡1a2n0a32a33a34::::::::::::::::::00::::::an;n¡1ann3777777775(1)AHQRQRAAH,AHQRAAHAHA()AHQRQRQRkjAi+1;ijjAi;ij+jAi+1;i+1jaccuracyi=1,2,¢¢¢¢¢¢n-1accuracyAHAQR(function[eigenvalue,°ag1]=QR(A,n))11AAH;2QR2.3AAHH=I¡2wwTwH2⫼䈾ᮃ㥋ᇨᖋবᤶᇚAবЎϞ⍋ỂԃḐⶽ䰉AH⫼ঢ়᭛ᮃবᤶᅲ⦄ϔ⃵ㄝᬜⱘQR䗁ҷˈे˖Ai=QiRiAi+1=RiQi䖒ࠄQR䗁ҷⱘ㊒ᑺ˛ᕫࠄ⒵䎇㊒ᑺⱘϞϝ㾦ⶽ䰉An䕧ߎAnⱘᇍ㾦ܗ㋴԰ЎAⶽ䰉ⱘ⡍ᕕؐAⶽ䰉NoYes㒧ᴳ⿟ᑣ⫼QR䗁ҷ⊩∖Aⱘ⡍ᕕؐ˄Џ⿟ᑣ˅1:(QR)AAHA1=H1AH1A2=H2A1H2¢¢¢¢¢¢An¡2=Hn¡2An¡3Hn¡2H1H2¢¢¢¢¢¢:H126666664a11a21a31...an137777775=26666664a011a0210...037777775(2)3H226666666664a012a022a032a042...a0n237777777775=26666666664a0012a0022a00320...037777777775(3)¢¢¢¢¢¢HiHiAi¡1iHi266666666666666664a01ia02ia03i¢¢¢a0iiai+1;iai+2;i...a0ni377777777777777775=266666666666666664a001ia002ia003i¢¢¢a00iia00i+1;i0...0377777777777777775(4)Ai¡1n¡i¡!x=[a0i+1;i;a0i+2;i;¢¢¢;a0ni]T®=k¡!xk2;n-i¡!e=[1;0;0;¢¢¢;0]T!=¡!x¡®¡!ek¡!x¡®¡!ek2(n¡i)¤(n¡i)H0i,Hi=Ii¤i00H0i#(5)Hi:Ai=HiAi¡1Hii=1i=n¡2n-2AH,A(functionAH=A-to-AHessenberg(A,n))4i=n-2?ᵘ䗴n-i㓈ⱘwi৥䞣߽⫼wiᵘ䗴(n-i)*(n-i)ⱘH(n-i)*(n-i)ᠽܙH(n-i)*(n-i)ᕫࠄHiAi=Hi*Ai-1*HiAⶽ䰉Ai-1Ai䕧ߎϞ⍋ỂԃḐⶽ䰉AH⫼䈾ᮃ㥋ᇨᖋবᤶᇚAⶽ䰉ⳌԐবᤶЎϞ⍋ỂԃḐⶽ䰉AH2:AAH2.4QRAAH.QRG(n¡1;n;µn¡1)G(n¡2;n¡1;µn¡1)¢¢¢¢¢¢G(1;2;µ1)AH=RG(i¡1;i;µi¡1)AH=QRQ=G(1;2;µ1)T¢¢¢¢¢¢G(n¡2;n¡1;µn¡1)TG(n¡1;n;µn¡1)TRAHnew=RQ=RG(1;2;µ1)T¢¢¢¢¢¢G(n¡2;n¡1;µn¡1)TG(n¡1;n;µn¡1)T5QAHii+1(n-1)AHi+1;i=0ii+1ci=AHi;iqAH2i;i+AH2i+1;isi=AHi+1;iqAH2i;i+AH2i+1;iii+1AH0i;j=ci¤AHi;j+si¤AHi+1;jAH0i+1;j=ci¤AHi+1;j¡si¤AHi;jj=1;2;¢¢¢¢¢¢;n.i1n-1,,QRRRQAHnew=RQ=RG(1;2;µ1)T¢¢¢¢¢¢G(n¡2;n¡1;µn¡1)TG(n¡1;n;µn¡1)TRii+1R0j;i=ci¤Rj;i+si¤Rj;i+1R0j;i+1=ci¤Rj;i+1¡si¤Rj;ij=1;2;¢¢¢¢¢¢;n.i1n-1,,AHnew=RQQR(AHnew=QR-iteration-once(AH,n))3function[eigenvalue;flag¡1]=QR(A;n)accuracy=10¡3,n=900,A2.1flag1=93449344AH0Aeigenvalue:mat9009006ᵘ䗴ঢ়᭛ᮃবᤶG(i,i+1)ᬍবAHⶽ䰉㄀i㸠੠㄀i+1㸠ⱘؐˈՓᕫAHi+1,i=0ˈㄝᬜഄᅲ⦄˖A=QR.ᣝ✻G(1,2)T,G(2,3)TĂĂG(n,n-1)Tᴹձ⃵ᬍবRⶽ䰉ⱘ㄀i߫੠㄀i+1߫ˈㄝᬜഄᅲ⦄˖Anew=RQDŽϞ⍋ỂԃḐⶽ䰉AHi=1:n-1ᕫࠄϞϝ㾦ⶽ䰉RNOYESi=1:n-1NOYES㒧ᴳᴀᄤ⿟ᑣˈᅲ⦄ձ⃵QR䗁ҷᕫࠄAnew⫼ঢ়᭛ᮃবᤶᅲ⦄AHⶽ䰉ⱘϔ⃵QR䗁ҷ3:QR1:11.959111.959111.928711.928711.878411.878311.867311.86680.06150.15320.15320.24400.30500.30500.39420.3942701002003004005006007008009000246810124:A9008