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The Inverse Power Method
Assume that
A
∈
C
n
×
n
has a
n
linearly independent eigenvectors
v
1
, v
2
, . . . , v
n
, and
associated eigenvalues
λ
1
, λ
2
, . . . , λ
n
, with

λ
1

>

λ
2
 ≥ 
λ
i

, i
= 3
,
4
, . . . , n
. Then (
λ
1
, v
1
)
is a dominant eigenpair of
A
, and for almost all
x
∈
C
n
,
x
k
≡
A
k
x
→
v
1
as
k
→ ∞
.
How fast does such an iteration converge? Write
x
=
n
X
i
=1
c
i
v
i
. Then
A
k
x
λ
k
1
=
c
1
v
1
+
c
2
(
λ
2
λ
1
)
k
v
2
+
n
X
i
=3
c
i
(
λ
i
λ
1
)
k
v
i
(1)
and it is clear (yes?) that the error gets multiplied by about

λ
2
/λ
1

at each step (we say
that the convergence is linear with asymptotic error constant

λ
2
/λ
1

). So the smaller the
ratio

λ
2
/λ
1

, the better, and if

λ
1
 ≈ 
λ
2

then we expect very slow convergence.
Let
B
∈
C
n
×
n
have eigenvalues
μ
i
labeled so that

μ
i
 ≥ 
μ
i
+1

. The power method
applied
to
B
converges to the dominant eigenvector of
B
(if one exists) with a speed that depends
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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
 Power

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