Digital Signal Processing Reference
In-Depth Information
The dimension of the update equation can be further decreased as
2
3
@f
@w
@f
@w
¼
2
m
4
5
)Dw ¼
2
m
@f
Dw
Dw
@w
:
Complex Newton Updates
Proposition 2 Given function f
(
.
)
defined in Proposition 1, Newton update in
R
2
N
given by
2
f
@ w
R
@ w
R
@
D w
R
¼
@
f
@ w
R
(1
:
30)
is equivalent to
@f
@w
H
1
H
1
@f
@w
Dw ¼
(
H
2
H
1
H
2
H
1
)
1
(1
:
31)
2
N
, where
in
C
2
f
@w@w
T
2
f
@w@w
H
:
@
@
H
1
W
and
H
2
W
(1
:
32)
Proof 2 By using (1.25) and (1.26), the real domain Newton updates given in (1.30)
can be written as
2
f
@ w
C
@ w
C
@
D w
C
¼
@
f
@ w
C
which can then put into the form
2
3
@f
@w
@f
@w
Dw
Dw
¼
4
5
H
2
H
1
H
1
H
2
where H
1
and H
2
are defined in (1.32).
We can use the formula for the inverse of a partitioned positive definite matrix ([49],
2
f
@ w
C
@ w
C
@
p. 472) when the nonnegative definite matrix
is positive definite, to write
2
3
@f
@w
@f
@w
¼
4
5
T
1
H
2
H
1
T
Dw
Dw
T
H
1
H
T
2
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