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Averaging the behavior of the weight vector after the first critical point has
been reached (
t
≥
t
0
), the following formula can be written:
w(
t
)
=
w(
t
)
2
z
n
∀
t
≥
t
0
(2.104)
where
z
n
is the unit eigenvector associated with the smallest eigenvalue of
R
.
From eq. (2.96) the discrete law can easily be deduced for updating the weight
modulus. This discrete law can be regarded as the discretization of the following
ODE:
w)
E
2
T
y
2
w
d
w
w
=
s
2
T
(w
yx
−
w
w
(2.105)
dt
T
2
Without loss of generality, the input data are considered as Gaussian; after some
matrix algebra, which can be found in [181], the following equation is derived:
(
p
)
−
λ
+
λ
n
tr
(
R
)
p
dp
dt
=
s
2
2
n
(2.106)
where
λ
n
is its smallest eigenvalue of
R
, the input data autocorrelation matrix.
Solving this differential equation for
s
(
p
)
=
1 (OJAn), with the initial condition
p
(
0
)
=
1 for the sake of simplicity, yields
p
(
t
)
=
e
−
λ
n
+
λ
n
tr
(
R
)
t
(2.107)
and considering that the quantity in brackets is never negative, it follows that
p
(
t
)
→∞
when
t
→∞
exponentially
(2.108)
Hence, the norm of
w(
t
)
diverges very fast. This divergence also arises when
s
(
p
)
=
p
(LUO); indeed,
1
p
(
t
)
=
1
(2.109)
2
−
λ
)
t
−
n
+
λ
n
tr
(
R
In this case the divergence happens in a
finite
time (
sudden divergence
); that is,
p
(
t
)
→∞
when
t
→
t
∞
(2.110)
where
1
t
∞
=
2
−
λ
)
(2.111)
n
+
λ
n
tr
(
R
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