3.8 Method Based on Jacobian Estimation
87
y
1
y
2
y
3
j
A
z
1
z
2
z
3
j+m
a 1
a 1
a 2
a 2
Fig. 3.1 Transformation of a sphere of small radius into a counterpart ellipsoid
vector ζ k i into ζ k i+m . Consequently, the vectors y i are transformed into
y i+m = ζ k i+m − ζ j+m .
Assuming that the radius ε is sufficiently small, one can introduce the operator
A j as follows
y i+m = A j y i .
The operator A j describes the system in variations. To estimate the operator A,
the least-square method can be employed:
min S
A j
= min
1
N
A j
N
i=0
(y i+m − A j y i )
2
.
This yields the following system of equations of the dimension n × n:
A j V = C, (V ) kl =
1
N
N
i=1
y
k
i y
l
i ,
(C) kl =
1
N
N
i=1
y
k
i+m y
l
i ,
87
y
1
y
2
y
3
j
A
z
1
z
2
z
3
j+m
a 1
a 1
a 2
a 2
Fig. 3.1 Transformation of a sphere of small radius into a counterpart ellipsoid
vector ζ k i into ζ k i+m . Consequently, the vectors y i are transformed into
y i+m = ζ k i+m − ζ j+m .
Assuming that the radius ε is sufficiently small, one can introduce the operator
A j as follows
y i+m = A j y i .
The operator A j describes the system in variations. To estimate the operator A,
the least-square method can be employed:
min S
A j
= min
1
N
A j
N
i=0
(y i+m − A j y i )
2
.
This yields the following system of equations of the dimension n × n:
A j V = C, (V ) kl =
1
N
N
i=1
y
k
i y
l
i ,
(C) kl =
1
N
N
i=1
y
k
i+m y
l
i ,