Graphics Reference
In-Depth Information
for
x
. We shall work out the case where n is 2 as an example.
4.7.1. Example.
To find a root to the system of equations
(
)
=
fxy
gxy
,
,
0
0
(
)
=
.
Solution.
We define a sequence of points (x
k
,y
k
) that converge to the root. Suppose
that we already have defined the kth point (x
k
,y
k
). The Taylor expansions for f and g
around (x
k
,y
k
) are
(
)
=
(
)
+
(
)
(
)
+
(
)
(
)
+
fxy fxy
,
,
fxy xx
,
-
fxy yy
,
-
...
k k
xk k
k
yk k
k
(
)
=
(
)
+
(
)
(
)
+
(
)
(
)
+
gxy gxy
,
,
gxy xx
,
-
gxy yy
,
-
...
k k
xk k
k
yk k
k
Truncating higher order terms means that we want to solve
(
)
(
)
+
(
)
(
)
=-
(
)
fx y x x
,
-
fx y y y
,
-
fx y
,
xk k
k
yk k
k
k k
(
)
(
)
+
(
)
(
)
=-
(
)
gxy x x
,
-
gxy y y
,
-
gxy
,
.
xk k
k
yk k
k
k k
Cramer's rule implies that
=+
-+
fg
gf
y
y
x
x
k
+
1
k
J
=+
-+
gf
fg
x
x
y
y
,
k
+
1
k
J
where J = f
x
g
y
- g
x
f
y
and all the partials are evaluated at (x
k
,y
k
). If (x
0
,y
0
) is an approx-
imation to a root (a,b), then one can show that convergence to (a,b), if any, is quad-
ratic. Conditions that are sufficient to guarantee such convergence assuming that
(x
0
,y
0
) is “close enough” to (a,b) are of the type:
(1) The derivatives of f and g up to order 2 are continuous and bounded in a
neighborhood
U
of (a,b).
(2) The Jacobian J does not vanish in
U
.
See [ConD72].
Now methods for finding zeros of functions have a much wider application than
simply that specific problem. Many other problems can be rephrased in those terms.
Specifically, problems that involve finding the inverse of a function are often rephrased
in terms of finding solutions to an equation of the form (4.14). As an example of that
type of problem, suppose that one is given a function p :
R
2
Æ
R
3
, a point
q
Œ
R
3
, and
one wants to find values x and y so that p(x,y) =
q
. This problem can be expressed in
terms of finding a zero of the function
(
)
=-
(
)
fxy
,
q
pxy
,
.
(4.16)
