Remote-Sensing Technologies and Data Processing Algorithms - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 206

where function G has components:

G i ðÞ ¼0 for i ¼ N þ 1 ; ...; N c ;

G i ðÞ ¼ X

N
k ; j¼1 ½a ijk n j ðÞn k ðÞþ b ij n j ðÞ for i ¼ 1

ð 2 : 41 Þ

; ...; N ;

where a ijk ðÞ ¼a ij k ; b ij ðÞ ¼b ij ; N c ¼ N þ N 2 þ N 2 N þ 1 2.

The Newton-Raphson method (Ram 2010) gives a possibility to solve system
( 2.40 ), ( 2.41 ). Let, introduce series of functions Y (1) (t),

ð Þ=

, Y (n) (t) such that Y (1) (t)is …

the first approximation of solution for system equations ( 2.40 ). Then, nth approx-

imation is solution of the following linear system:

h i þ X n h i o Y ð n Þ

j

h i

N c

dY ð n Þ
i

Y ð n 1 Þ
j

dt ¼ G i Y ð n 1 Þ ðÞ dG i Y ð n 1 Þ ðÞ ðÞ= = dY

j¼1

ð 2 : 42 Þ

As it was shown by Bellman and Dreyfus (1962), iterative process ( 2.42 )is
convergent by the quadratic law. Solution of Eq. ( 2.42 ) is described in the following
form:

Y ðÞ ðÞ ¼P ðÞþ X

N c

C k H ð k Þ ðÞ ð 2 : 43 Þ

k¼1

where P(t) is the particular solution of the Eq. ( 2.42 ), H (k) (t) is vector solution of
homogeneous system. To de ne function P(t) it is needed to solve the Eq. ( 2.42 )

under initial conditions: Yi(0) i (0) = 0 (i =1, , N c ). Functions H (k) (t) are determined as

…

solutions of the Cauchy problem:

n h i

=

o ½Y ð n Þ
j

dt ¼ X

N c

d Y ð n Þ
i

Y ð n 1 Þ
j

dG i Y ð n 1 Þ ðÞ ðÞ= dY ð i ¼ 1 ; ...; N c Þ ð 2 : 44 Þ

j¼1

1
0
.
0

0
1
.
0

0
0
.
1

H ð 1 Þ ð 0 Þ ¼ H ð 2 Þ ð 0 Þ ¼ H ð N c Þ ð 0 Þ ¼ ; ; ...; ; ð 2 : 45 Þ

( 2.45 ), constants C k are unknown initial
conditions of system of equations ( 2.41 ). Therefore, constants C k are de

As it is followed from Eqs. ( 2.42 )

-

ned from

condition: