HouseholderQRAlgorithmically, this method is very close to G.E. with no pivoting. There, a GausstransformMk=I+mketkwas used to introduce zeros below the (k, k) element ofA(k-1), givingA(k)G=MkA(k-1)G. Here, a Householder reflectorHk=I-βukutkreplaces the Gauss transform as the operator that introduces zeros below the (k, k)element:A(k)H=HkA(k-1)H.LetA∈Rm×n, withm≥n, and letp= min (n, m-1). The HouseholderQRfactorization ofAcan be coarsely described asA(0) =AFor k = 1:pComputeuso that (I-βuut)A(k-1)has zeros below its (k, k) entryComputeA(k)=HkA(k-1)EndThere are some important details to consider yet, but it is essentially this simple.We know that ifuis a Householder vector forx, then it is a multiple ofx±x2e1,and thatHx= (I-βuut)x=±x2e1, withβ= 2/(utu). As with G.E. we canview thekthstep asA(k)=R(k)X(k)0˜A(k)=I00˜HkR(k-1)X(k-1)0˜A(k-1)=HkA(k-1),whereR(k)isk×kupper triangular, and˜A(k)is (m-k)×(n-k). Hereutk= (0t,˜utk), where ˜u
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