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The Francis Algorithm
Recall that we have a shifted QR iteration that converges quickly to a
reduced
Hessenberg matrix if the
s
i
are close to an eigenvalue of
H
(in fact, only one
iteration does it if
s
i
is an eigenvalue of
H
(this is called an
ultimate shift
)). If
H
is
reduced, we can decouple (or deﬂate) and continue with a strictly smaller problem
than before. Here is the iteration:
Q
i
R
i
=
H
i

s
i
I
H
i
+1
=
R
i
Q
i
+
s
i
I
=
Q
t
i
H
i
Q
i
Nonsymmetric real matrices may have complex eigenvalues, which must occur in
conjugate pairs:
u
+
iv
and
u

iv
. If we apply a complex shift
s
i
=
u
+
iv
to the
iteration, then
Q
i
,
R
i
and
H
i
+1
will be complex. This requires more storage, and
more computation (one complex multiplication requires 4 real multiplications and 2
real additions).
Now if we immediately follow the
s
i
=
u
+
iv
shift iteration with an iteration with
shift
s
i
+1
=
u

iv
, then everything becomes again real. We can understand this by
noting that two iterations above gives the same
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This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.
 Fall '10
 MARK

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