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1、EE236C(Spring2011-12)3.Conjugategradientmethod?conjugategradientmethodforlinearequations?convergenceanalysis?conjugategradientmethodasiterativemethod?nonlinearconjugategradientmethod3-1Unconstrainedquadraticminimization1TTminimizef(x)=xAx?bx2nwithA∈S++?equivalenttosolvingAx=b?r
2、esidualr=b?Axisnegativegradientatx:r=??f(x)conjugategradientmethod?inventedbyHestenesandStiefelaround1951?themostwidelyusediterativemethodforsolvingAx=b,withA?0?canbeextendedtonon-quadraticunconstrainedminimizationConjugategradientmethod3-2Krylovsubspacesde?nition:asequenceofne
3、stedsubspaces(K0?K1?K2?···)k?1K0={0},Kk=span{b,Ab,...,Ab}fork≥1ifKk+1=Kk,thenKi=Kkforalli≥kkeyproperty:A?1b∈K(evenwhenK6=Rn)nnCayley-Hamiltontheorem:p(A)=An+aAn?1+···+aI=0where1nnn?1p(λ)=det(λI?A)=λ+a1λ+···+an?1λ+an1
