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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/.
 
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