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538
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540
is revealed by looking at a generic 4 × 4 pattern for
Hk+1 = RQ + αI
⎛
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0
0 ∗
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0 ∗
⎝∗
0
0 ≈ ∗
⎝0
0
0 ∗
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0 ∗
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∗ ⎛ It is illegal to print, duplicate, or distribute this material
Please report violations to meyer@ncsu.edu = ⎞⎛ ∗
∗
∗⎠⎝∗
∗
0
0
∗
∗
∗
α± ⎞ ∗
∗
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0 ⎠≈ ∗
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∗ ⎛ ∗
⎝∗
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0 ⎞ ⎛ 0
α
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+⎝
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±1
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α± ⎞
⎠ α
α ⎞ α ⎠. D
E The strength of the last approximation rests not only on the size of , but
it is also reinforced by the fact that ≈ 0 because the 2-norm of the last
row of Q must be 1. This indicates why this technique (called the single
shifted QR iteration) can provide rapid convergence to a real eigenvalue. To
extract complex eigenvalues, a double shift strategy is employed in which the
eigenvalues αk and βk of the lower 2 × 2 block of Hk are used as shifts
as indicated below:
Factor: Hk − αk I = Qk Rk ,
Set: T
H IG
R (so Hk+1 = QT Hk Qk ),
k Hk+1 = Rk Qk + αk I Factor: Hk+1 − βk I = Qk+1 Rk+1 ,
Set: Hk+2 = Rk+1 Qk+1 + βk I
.
.
. (so Hk+2 = QT+1 QT Hk Qk Qk+1 ),
k
k Y
P The nice thing about the double shift strategy is that even when αk is
complex (so that βk = αk ) the matrix Qk Qk+1 (and hence Hk+2 ) is
real, and there are eﬃcient ways to form Qk Qk+1 by computing only the
ﬁrst column of the product. The double shift method typically requires very
few iterations (using only real arithmetic) to produce a small entry in the
(n − 2, 2)-position as depicted in (7.3.19) for a generic 4 × 4 pattern. O
C Exercises for section 7.3 7.3.1. Determine cos A for A = −π/2
π/2 π/2
−π/2 . 7.3.2. For the matrix A in Example 7.3.3, verify with direct computation that
λ1 t
0
eλ1 t G1 + eλ2 t G2 = P e 0 eλ2 t P−1 = eAt .
7.3.3. Explain why si...

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