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with (4.19), the following is obtained while taking into account matrix general
property (
GB
)
+
=
B
+
G
+
(and with adding the weights):
=
B
0
+
B
(
B
+
G
+
WGB
)
−1
B
+
G
+
W
(
Y
−
G
0
−
GB
0
) .
X
(4.20)
Equations system (3.7) has been solved in Sect. 3.2 just using relation (4.20)
with account of restrictions to the parameters (3.8). We should mention, that
the matrix of system (4.20) being a subject to inversion is still symmetric and
positive defined, with concern to all similar matrices, which will be presented
below. To compute the product of several matrices effectively is to use the
above-described approach of multiplying amatrix from the right-hand side by
a vector with consequent choosing of the last matrix columns of the product
as such vectors.
Now consider the general case of nonlinear relationship (4.7) between
observations
Y
and parameters
X
. Take certain initial values of parame-
ters
X
0
≡{
x
0,
k
}
and expand (4.7) into a Taylor series accounting for only
the linear item:
K
∂
g
i
(
x
1
,...,
x
K
)
∂
=
=
y
i
g
i
(
x
0,1
,...,
x
0,
K
)+
(
x
k
−
x
0,
k
),
i
1,...,
N
.
(4.21)
x
k
=
k
1
=
Difference
y
i
(
x
1
,...,
x
K
)−
g
i
(
x
0,1
,...,
x
0,
K
)
y
i
(
x
1
,...,
x
K
)−
y
i
(
x
0,1
,...,
x
0,
K
)is
a linear function of parameter difference
x
k
−
x
0
k
(in the considered approxi-
mation). It allows the constructing of the iteration algorithm for the nonlinear
dependence using the above-obtained solution for the case of the linear one.
This standard approach of reducing the nonlinear problems to the linear ones
is known as
alinearization
. System (4.21) is converted to the matrix form as:
Y
(
X
)
(
X
−
X
0
)+
Y
(
X
0
) ,
=
G
0
·
(4.22)
=
1,...,
K
,calculatedinpoint
X
0
. This specification of the matrix of derivatives
is convenient because in the linear case of (4.9) the matrix
G
evidently has
the same meaning; hence, the successiveness of the specifications is kept.
The operator of the direct problem solution also keeps initial specification
G
(
X
,
U
∂
|∂
=
where
G
0
is thematrixof partial derivative (
g
i
(
x
1
,...,
x
K
)
x
k
),
i
1,...,
N
,
k
X
), but for a brevity we will further write just
G
(
X
). Iterationally
applying the above-considered solution with LST as per (4.15) to (4.22) we
obtain:
\
=
X
n
+(
G
n
WG
n
)
−1
G
n
W
(
Y
−
G
(
X
n
)) ,
X
n
+1
(4.23)
=
where
n
0,1,2,...isanumberoftheiteration.
There are three difficulties of the practical application of (4.23):
- indeterminacy of the zeroth approximation choice;
- necessity of elaborating the criteria for the iteration interruption;
- possible large spread in the desired values during the consequent itera-
tions.
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