Biomedical Engineering Reference
In-Depth Information
C.4
Definitions of Several Vector Norms
T
The
p
th order norm (
p
≥
0) of an
N
-dimensional vector,
x
=[
x
1
,...,
x
N
]
is defined
such that
p
=
|
p
1
/
p
p
p
x
x
1
|
+|
x
2
|
+···+|
x
N
|
.
(C.62)
When
p
2, the norm is called the
L
2
-norm, which is equal to the conventional
Euclidean norm,
=
x
1
+
x
2
+···+
x
N
.
x
2
=
(C.63)
=
The
L
2
-norm is usually denoted
x
. When
p
1, the norm is expressed as
x
1
=|
x
1
|+|
x
2
|+···+|
x
N
|
,
(C.64)
which is called the
L
1
-norm. When
p
=
0, we can define
T (
x
)
, which is called the
indicator function, such that
0
x
=
0
T (
x
)
=
(C.65)
1
x
=
0
,
with the norm expressed as
N
x
0
=
1
T (
x
i
).
(C.66)
i
=
Namely,
x
0
is equal to the number of non-zero components of
x
. When
p
=∞
,
the norm becomes
x
∞
=
max
{
x
1
,
x
2
,...,
x
N
}
.
(C.67)
Namely, the
x
∞
is equal to the maximum component of
x
.
C.5
Derivative of Functionals
This section provides a brief explanation of the derivative of functionals. A conven-
tional function relates input variables to output variables. A functional relates input
functions to output variables. A simple example of a functional is:
ʲ
I
=
f
(
x
)
dx
,
(C.68)
ʱ
where
f
(
x
)
is defined in
[
ʱ, ʲ
]
.This
I
is a functional of the function
f
(
x
)
.The
functional is often denoted as
I
[
f
(
x
)
]
.
