Information Technology Reference
In-Depth Information
Consequently, it is natural to define
p.x
0
/
p
0
.x
0
/
;
x
1
D
x
0
(4.153)
and generally we have
p.x
k
/
p
0
.x
k
/
:
x
kC1
D
x
k
(4.154)
Let us now turn our attention to system (
4.148
). The key point in deriving New-
ton's method for this system is the multi-dimensional version of the Taylor series.
Consult e.g. Apostol [4] to see that for a smooth function F.x;y/,wehave
6
F.x
C
x; y
C
y/
D
F.x;y/
C
x
@F
@x
.x; y/
C
y
@F
@y
.x; y/
.x
2
;xy;y
2
/:
C
(4.155)
O
Now, we can proceed as in the scalar case. Suppose that .x
0
;y
0
/ are approximations
of .x
;y
/ that solve the system
f.x;y/
D
0;
g.x; y/
D
0:
(4.156)
Using (
4.155
), we get
f.x
0
C
x; y
0
C
y/
D
f.x
0
;y
0
/
C
x
@f
@x
.x
0
;y
0
/
C
y
@f
@y
.x
0
;y
0
/
.x
2
;xy;y
2
/;
C
(4.157)
O
and
g.x
0
C
x; y
0
C
y/
D
g.x
0
;y
0
/
C
x
@g
@x
.x
0
;y
0
/
C
y
@g
@y
.x
0
;y
0
/
.x
2
;xy;y
2
/:
C
(4.158)
O
Since we want x and y to be such that
f.x
0
C
x; y
0
C
y/
0;
g.x
0
C
x; y
0
C
y/
0;
(4.159)
it is reasonable to define x and y to be the solution of the linear system
6
Note that
O
."; ı/ is shorthand for
O
."/
C
O
.ı/.