A Geometric Proof of Convergence for the QR Method |
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Page 16
... assume that T has the + property TT = I. For if not , we decompose T into T = T.R with T. having this property and R ... assumed situation . -1 Let the orthogonal complement of T be spanned by the orthogonal n * ( n - p ) matrix S , so ...
... assume that T has the + property TT = I. For if not , we decompose T into T = T.R with T. having this property and R ... assumed situation . -1 Let the orthogonal complement of T be spanned by the orthogonal n * ( n - p ) matrix S , so ...
Page 20
... assume that there is an index q for which the strict inequality || > | +1 | holds . q Let the invariant subspaces S and S Soq 1 + + զո ( Hypothesis H12 ) . be spanned by the columns of the matrices G1 and G2 , which we assume to be ...
... assume that there is an index q for which the strict inequality || > | +1 | holds . q Let the invariant subspaces S and S Soq 1 + + զո ( Hypothesis H12 ) . be spanned by the columns of the matrices G1 and G2 , which we assume to be ...
Page 21
... assume that there are two indices p and q for which 1 ^ , | ≥ ... ≥ 1 ^ q | > | ^ q + 11 2 ... ≥ 1201 Let the invariant subspaces S Soq ' Sap ' Σλ 12 + 12 ... 21120 holds ( H12 ) . n S respectively be spanned by pn the columns of the ...
... assume that there are two indices p and q for which 1 ^ , | ≥ ... ≥ 1 ^ q | > | ^ q + 11 2 ... ≥ 1201 Let the invariant subspaces S Soq ' Sap ' Σλ 12 + 12 ... 21120 holds ( H12 ) . n S respectively be spanned by pn the columns of the ...
Common terms and phrases
block of order BUUREMA Cauchy-Schwarz inequality columns complex number computation is performed compute a unitary denote F.L. BAUER follows functie G₁ G₂ h 1h h large h lim h h Xn+1 h+1 h+1 H₁₂ Hence Hessenberg form Hessenberg matrices Hessenberg matrix hh+1 infinite subset initial matrix invariant subspace inverse iteration iteration process Krylov sequences lemma lim p(r limit f limiting polynomial linear subspaces lower diagonal block mass-center method with shifts minimizing polynomial monic non-singular matrix normal matrix orthogonal complement p-dimensional subspace P₁ P₁o performed in real polynomial of degree PROOF OF CONVERGENCE QR algorithm QR method quence real arithmetic real numbers section 9 sequence of linear sequence of matrices SP2han strict inequality subspaces converges tary matrix tend to zero unitary matrix vectors x+z W₂ wiskunde X₁₂ X₂ zijn