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305199810JournalofNanjingUniversityofAeronautics&AstronauticsVol.30No.5Oct.1998Lanczos-QRX汪晓虹 周传荣(,210016)Kx=KMx,Lanczos-QR,v1,mLanczos:KVm=MVmTm+hmemTTmd,TmdQR,v1,v1,Kx-HMx0,,:QR;;Lanczos;;:O241.6;V214.2,:,X=(x1,,xn),:K(x)=X2M(x)p:K(x),M(x)Rnn,M(x),K(x),k:x(k)=x(k-1)+dk,K(x(k))=X2M(x(k))p,,,,,K=X2MLanczos,[1-2],Lanczos,Kx=KMxK,MRnn,K,M(1)X:1998-03-02;:1998-05-27,,195511(1)mLanczos:Tn=A1B1B1A2B2:Vn=[v1,,vn](VnTMVn=In),KVn=MVnTnKVm=MVmTm+MBmvm+1eTm,eTm=[0,,0,1]1m:hm=MBmvm+1KVm=MVmTm+hmeTmVTmhm=0(2)mLanczosHTm,y,Tmy=Hy(3)(H,x)(1)H(1)Ritz,x=Vmy(1)RitzKx-HMx=(KVm-MVmTm)y=hmeTmy(4)RitzH(1)m,:;VmM,Lanczos;v1(),1Lanczos-QR[3]A,AVm-VmTm=rmeTmAmLanczos,Tm:Tm=A1B1B1A2B2=1,,m-1rm=0:v1=XmS,:XmAXm=XmD,XTmXm=Im;D=diag(a1,,am),S=(s1,,sm)T(1),(1)M,MCholesky:M=LTL,(1)(L-1)TKL-1y=Ky(5)mLanczos(L-1)TKL-1qj=Bj-1qj-1+Ajqj+Bjqj+1,B0q0=0LT:KL-1qj=Bj-1LTL(L-1qj-1)+AjLTL(L-1qj)+BjLTL(L-1qj+1)v=L-1q,:Kvj=Bj-1Mvj-1+AjMvj+BjMvj+1KVm=MVmTm+MBmvm+1eTm,VTmMVm=Im,(1)mLanczos1(1)mLanczos,KVm-MVmTm=hmeTm50230:Tm=A1B1B1A2B2=1,,m-1hm=0:v1=XmS,XmKXm=MXmD,XTmMXm=Im,D=diag(d1,,dm),S=(s1,,sm)T1,KXm=MXmD(K,M)(m)Schur,D(1)1,Lanczos,Tm(1),(1)Vm,Vm,Lanczosv1(1)p,(3)H(1)QR,Lanczos,v1,(1)p,(1)pm=p+d(d=5+2.5p),v1,vT1Mv1=1mLanczos:KVm=MVmTm+hmeTm(mnn,VmM)Tmm:Hiyi(i=1,,m)TmdQR,:Lj=Hp+j,j=1,,d(Tmd):T(0)=Tm,QR:T(0)-L1I=Q1R1,Q1,R1KVm-MVmTm=hmeTm,KVmQ1-MVmQ1(R1Q1+L1I)=hmeTmQ1(6)T(1)=R1Q1+L1I(T(1)),QR:T(1)-L2I=Q2R2,KVmQ1Q2-MVmQ1Q2(R2Q2+L2I)=hmeTmQ1Q2T(2)=R2Q2+L2I,,,d,KVmQ1Qd-MVmQ1Qd(RdQd+LdI)=hmeTmQ1Qd(7)mLanczosv*1=VmQe1,:Q=Q1Qd,e1=(1,0,,0)Tm1(8)2TmmmH1HpL1LdT*=QTTmQ=a1b1b1a2b2=Q1Qd,Qj(j=1,,d)T(j-1)-LjI=QjRjQRbj0(1jp-1),bp=0;T*=T*pT*d,K(T*p)={H1,,Hp},K(T*d)={L1,,Ld},v*1=VmQe1=pj=1cjxjxj=Vmyj(1)RitzHjRitz,yj:Tmyj=Hjyj,1jp(1)mLanczos:KVm=MVmTm+hmeTm;Lj,Tmd5035:Lanczos-QRQR[4]T(j-1)-LjI=QjRjT(j)=RjQj+LjI(j=1,,d)T(0)=TmT*=QTTmQ,Q=Q1Qd;dj=1(Tm-LjI)=QR=Q1QdRdR1;q1=dj=1(Tm-LjI)e1=rQe1,r=eT1Re1q1=pi=1kjyi,yjTmyj=Hjyjbj0(1jp-1),1,bp=0,v*1=VmQe1=pj=1cjxjKx=KMxpLanczos-QR:p,d,m=p+d,E;v1,vT1Mv1=1;1mLanczos(2),Tm,Vm,hm;2Tm(Hj,yj),j=1,,m:Hp+j=Lj,j=1,,d3xj=Vmyj,j=1,,p,Kxj-HjMxjE,;,44Lj(j=1,,d),dQRT(j-1)-LjI=QjRj,j=1,,dT(0)=TmT(j)=RjQj+LjI5Q=Q1Qd,(8)Lanczosv*1=VmQe1;1[5],,:3tLanczosT*(t)p:H1(t),,Hp(t)j=1,,p,{Hj(t)}(t=1,2,),limtHj(t)=Kj,Kj(1)p21100(1),:E=7.01010N/m2,:Q=2770kg/m3450kg:si=5cm2(i=1,,100)511002200(2),:E=2.11011N/m2,:Q=7800kg/m30.23kg,si=64.5cm2(i=1,,200)5,p=5,d=14Lanczos-QR,1014,5(X=H/2P)x,68(812)15043022001X1/HzL-QRKx-HMxKx-HMx10.177610-50.177610-521.083710-51.083710-3132.915710-42.915710-143.589510-33.5895155.420410-25.4204112.876610-42.876610-8213.034910-413.034910-62313.889910-313.889910-3432.599210-232.599210-1538.213710-138.123711GolubGH,VanLoanCF.Matrixcomputations.Baltimore,MD:theJohnsHopkinsUni-versityPress,1983.322-3512PareettBN.Thesymmetriceigenvalueproblem.EnglewoodCliffs,NJ:PrenticeHall,1980.288-3013LehoucqRB,SorensenDC.Deflationtechniquesforanimplicitlyrestartedarnoldiitera-tion.SIAMMatrixAnalAppl,1996,17(4):789-8214,,..:,1979.163-1685SorensenDC.Implicitapplicationofpolynomialfiltersinak-steparnoldimethod.SIAMMatrixAnalAppl,1992,13(1):357-3855055:Lanczos-QRLanczos-QRMethodsforComputationofaLargeReal-SymmetricEigenvalueProblemsWangXiaohongZhouChuanrong(InstituteofVibrationEngineering,NUAANanjing,210016)AbstractTherecursionformulasoftheLanczos-QRmixedmethodareusedtodetermineafewsmallesteigenvaluesandassociatedeigenvectorsofalargesparsereal-symmetriceigenvalueproblems:Kx=KMx.AftermstepsoftheLanczosprocesswithinitialvectorv1,wehaveKVm=MVmTm+hmeTm.ChoosingdshiftsthelargesteigenvaluesofTm,wecarryoutdstepsshiftedQRfactorizationforTm,andinitialvectorv1isthenupdated.ThetruncateLanczosprocessisiterativelyrestartedtoforcetheinitialvectorcloserandclosertothedesiredinvariantsubspace,andtheresidualKx-HMx0.Numericalexamplesillustratethatthemethodcangiverapidconvergence,andtheresultisstable,effective.Keywords:QRfactorization;iterativemethods;Lanczosmethod;generaleigenvalueproblem;structuraldynamics50630