Graphics Programs Reference
In-Depth Information
The methodworks like this:
1.
Let
v
be an approximation to
x
1
(a randomvector of unit magnitude will do).
2.
Solve
Az
=
v
(9.28)
for the vector
z
.
3. Compute
|
z
|
.
4. Let
v
=
z
/
|
z
|
and repeat steps 2-4 until the change in
v
is negligible.
At the conclusion of the procedure,
|
z
|=±
1
/λ
1
and
v
=
x
1
. The sign of
λ
1
is de-
terminedas follows:if
z
changes sign between successive iterations,
λ
1
is negative;
otherwise,
λ
1
is positive.
Let us now investigate why the methodworks.Since the eigenvectors
x
i
of
Eq. (9.27) areorthonormal, they canbe usedas the basisfor any
n
-dimensional vector.
Thus
v
and
z
admit the unique representations
n
n
v
=
v
i
x
i
z
=
z
i
x
i
(a)
i
=
1
i
=
1
Note that
v
i
and
z
i
are not the elements of
v
and
z
, but the components with respect
to the eigenvectors
x
i
.Substitutioninto Eq. (9.28) yields
n
n
A
z
i
x
i
−
v
i
x
i
=
0
i
=
1
i
=
1
But
Ax
i
=
λ
i
x
i
,sothat
n
(
z
i
λ
i
−
v
i
)
x
i
=
0
i
=
1
Hence
v
i
λ
i
z
i
=
It followsfromEq. (a)that
n
n
v
i
λ
1
λ
i
v
i
λ
i
x
i
=
1
λ
1
z
=
x
i
i
=
1
i
=
1
v
1
x
1
+
1
λ
1
v
2
λ
v
3
λ
1
λ
2
x
2
+
1
λ
3
x
3
+···
=
(9.29)
|
λ
1
/λ
i
|
<
=
Since
1), weobservethat the coefficientof
x
1
has becomemore promi-
nent in
z
than it was in
v
; hence
z
is abetter approximation to
x
1
. Thiscompletes the
first iterativecycle.
1 (
i
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