PDF-Bindel,Fall2012MatrixComputations(CS6210)
Author : yoshiko-marsland | Published Date : 2016-04-24
xkxkykxkyxFigure1Constructionofare ectortotransformxintokxkykyk1transformationstointroducesubdiagonalzerosinGaussianeliminationThisleadsustothefollowingalgorithmtocomputetheQRdecompositionfunct
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Bindel,Fall2012MatrixComputations(CS6210): Transcript
xkxkykxkyxFigure1Constructionofare ectortotransformxintokxkykyk1transformationstointroducesubdiagonalzerosinGaussianeliminationThisleadsustothefollowingalgorithmtocomputetheQRdecompositionfunct. 118 brPage 2br Moti ation olynomial oot 64257nding standard algor ithm QR iter ation on companion matr ix Rob ust softw are xists It nor mwise bac kw ard stab le It used in Matlab But it tak es time and stor age Use str ucture in QR iter ates to get anysuch xedpointiterationviaasplittingofthematrixA,i.e.bywritingA=M KwithMnonsingular.ThenwecanrewriteAx=bintheformMx=Kx+b;or,equivalently,x=x+M 1(b Ax):The xedpointiterationisthenx(k+1)=M 1(Kx(k)+b)= 1.Row1:y1=d12.Row2:2y1+y2=d2,ory2=d2 2y13.Row3:3y1+2y2+y3=d3,ory3=d3 3y1 2y2Moregenerally,theforwardsubstitutionalgorithmforsolvingunitlowertri-angularlinearsystemsLy=dlookslikey=d;fori=2:ny(i)=d(i) L theelementsinL.Forexample,considerwhathappenswhenwefactorthefollowingmatrixwithoutpivoting:A=111=10 11101 1:Ifweroundu22to 1,thenwehave10 1110 1=1106=A;thatis,aroundingerrorint
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