CS205  Class 7
Readings
: Heath 5.6
Systems of Nonlinear Equations
1.
Let’s turn our attention back to systems of nonlinear equations, i.e. A(x)=b or F(x)=0.
a.
Here the
Jacobian
matrix, J(x), is rather useful as a linearization of the nonlinear problem.
b.
Here we define
/
ij
i
j
J
F
x
where each equation of F(x)=0 is written individually as
( )
0
i
F x
, and
each
j
x
is the jth component of the x vector.
c.
For example, consider
1
1
2
( )
sin
4
0
F x
x
x
and
2
2
1
2
( )
0
F
x
x
x
which can be written in
matrix form as
1
2
2
1
2
sin
4
( )
0
x
x
F x
x
x
. Then the Jacobian matrix is
2
1
1
cos
( )
2
1
x
J x
x
.
d.
Note that J(x), that is J, generally depends on the x vector.
e.
In general, we write J(x)=F’(x) and note that the Jacobian is the multidimensional generalization of
f’(x).
f.
Thus conditioning in multiple dimensions depends on the Jacobian matrix just as conditioning for
scalars depends on f’(x).
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 Spring '07
 Nonlinear system, xk, Systems of Nonlinear Equations, Jacobian matrix, xk yk

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