B now try the single shift qr iteration on h c finally

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Unformatted text preview: 1)n , and this is also upper Hessenberg (constructing a 12 23 ( 4 × 4 example will convince you). Since multiplication by an upper-triangular matrix can’t alter the upper-Hessenberg structure, the matrix R1 Q1 = H2 at the second step of the QR iteration is again upper Hessenberg, and so on for each successive step. Being able to iterate with Hessenberg matrices results in a significant reduction of arithmetic. Note that if A = AT , then Hk = HT for each k, which means that each Hk is tridiagonal in structure. k • D E Convergence. When the Hk ’s converge, the entries at the bottom of the first subdiagonal tend to die first—i.e., a typical pattern might be ⎛ ∗ ⎜∗ Hk = ⎝ 0 0 ∗ ∗ ∗ 0 ⎞ ∗∗ ∗ ∗⎟ ⎠. ∗∗ T H When is satisfactorily small, take (the (n, n)-entry) to be an eigenvalue, and deflate the problem. An even nicer state of affairs is to have a zero (or a satisfactorily small) entry in row n − 1 and column 2 (illustrated below for n = 4) ⎛ ⎞ ∗∗∗∗ ⎜∗ ∗ ∗ ∗⎟ Hk = ⎝ (7.3.19) ⎠ 0 00 IG R Y P because the trailing 2 × 2 block will yield two eigenvalues by the quadratic formula, and thus complex eigenvalues can be revealed. • O C Shifts. Instead of factoring Hk at the k th step, factor a shifted matrix Hk − αk I = Qk Rk , and set Hk+1 = Rk Qk + αk I, where αk is an approximate real eigenvalue—a good candidate is αk = [Hk ]nn . Notice that σ (Hk+1 ) = σ (Hk ) because Hk+1 = QT Hk Qk . The inverse power method k is now at work. To see how, drop the subscripts, and write H − αI = QR as QT = R(H − αI)−1 . If α ≈ λ ∈ σ (H) = σ (A) (say, |λ − α| = with α, λ ∈ ), then the discussion concerning the inverse power method in Example 7.3.8 insures that the rows in QT are close to being left-hand eigenvectors of H associated with λ. In particular, if qT is the last row in QT , then n rnn eT = eT R = qT QR = qT (H − αI) = qT H − αqT ≈ (λ − α)qT , n n n n n n n so rnn = |rnn | ≈ (λ − α)qT n Copyright c 2000 SIAM 2 = and qT ≈ ±eT . The significance of this n n Buy...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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