Image Processing Reference
In-Depth Information
This is equal to the
first element of the product 2Ax. Similarly, it can be shown that
q
f
(
x
)
q
x
i
is equal to the ith element of the product. Therefore,
q
x
T
Ax
q
x
¼
(
:
)
2Ax
3
213
Similarly, it can be shown that the second derivative of f
(
x
)
is given by
2
x
T
Ax
q
x
2
q
¼
2A
(
3
:
214
)
Example 3.53
2
Let x 2 R
3
and f (x)
13x
1
þ
8x
2
4x
3
6x
2
x
3
. Find
q
f
(
x
)
q
x
and
q
f (x)
q
x
2
.
¼
3x
1
x
2
þ
5x
1
x
3
þ
S
OLUTION
13x
1
þ
8x
2
4x
3
f (x)
¼
3x
1
x
2
þ
5x
1
x
3
þ
6x
2
x
3
2
4
3
5
2
4
3
5
¼ x
T
Ax
13
1
:
52
:
5
x
1
x
2
x
3
¼ x
1
½
x
2
x
3
1
:
583
2
:
53
4
The
first derivative of
f (x)is
2
4
3
5
2
4
3
5
¼
2
4
3
5
13
1
:
52
:
5
x
1
x
2
x
3
26x
1
3x
2
þ
5x
3
q
f (x)
q
x
¼
2Ax ¼
2
1
:
583
3x
1
þ
16x
2
þ
6x
3
2
:
53
4
5x
1
þ
6x
2
8x
3
The second derivative of f (x)is
2
4
3
5
26
35
2
f (x)
q
x
2
2Ax
q
x
¼
q
¼
q
2A ¼
3 66
56
8
Example 3.54
Let x 2 R
n
, A be an n n symmetric matrix, b be an n
1 column vector, and c a
2
f (x)
q
x
2
.
1
2
x
T
Ax þ b
T
x þ c. Find
q
f (x)
scalar. De
ne f (x)
¼
and
q
q
x
S
OLUTION
The
first derivative of
f (x)is
q
f (x)
q
x
¼ Ax þ b
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