A Geometric Proof of Convergence for the QR Method |
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Page 31
... numbers . We now consider the itera- Wh + 1 + 1 R ' h + 1h + 1 = ( A - μ ( 1 ) ... numbers μ in such a way that for some fixed set of indices N ( h ) -μ ( h ) i e Nj N implies << 1 for ... real so that the computation is still performed in real.
... numbers . We now consider the itera- Wh + 1 + 1 R ' h + 1h + 1 = ( A - μ ( 1 ) ... numbers μ in such a way that for some fixed set of indices N ( h ) -μ ( h ) i e Nj N implies << 1 for ... real so that the computation is still performed in real.
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... numbers z that are attained by the expression Z Let F ( h ) 1 = + u Au where u runs over all vectors 0 . + u u + ( h ) ... real non - negative numbers creasing and therefore has a limit f ≥ 0 . ( 9.5 ) [ p ( r ) xp 18 de- is The case f ...
... numbers z that are attained by the expression Z Let F ( h ) 1 = + u Au where u runs over all vectors 0 . + u u + ( h ) ... real non - negative numbers creasing and therefore has a limit f ≥ 0 . ( 9.5 ) [ p ( r ) xp 18 de- is The case f ...
Page 48
... real numbers t with 0 ≤ t≤ 1 ( p ( r ) ( x + tz ) , T ( x + tz ) ) ( x + tz ) , T ( x + tz ) ) = ( p ( r ) x , Tx ) + t ( Therefore , ( r ) 0 = ( P t ( P t ( p ( r ) z , Tx ) + t ( p ( r ) x , Tz ) + ( P t2 ( p ( r ) z , Tz ) holds ...
... real numbers t with 0 ≤ t≤ 1 ( p ( r ) ( x + tz ) , T ( x + tz ) ) ( x + tz ) , T ( x + tz ) ) = ( p ( r ) x , Tx ) + t ( Therefore , ( r ) 0 = ( P t ( P t ( p ( r ) z , Tx ) + t ( p ( r ) x , Tz ) + ( P t2 ( p ( r ) z , Tz ) holds ...
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