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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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