The following table shows performance of Newtons method for the problem
f1=x**2-y-2=0 and f2=x*y+1=0 with initial guess -4,4
x1
x2
f1
f2
||inv(jacob)||
-4.0000
4.0000
10.000
-15.000
.33333
-2.4722
1.7778
2.3341
-3.3951
.48011
-1.8176
.87522
.42851
-.59082
.60279
-1.6346
.63831
.33506E-01
-.43366E-01
.65321
-1.6182
.61819
.26912E-03
-.33013E-03
.65832
See also the picture. The solutions of the system occur
where the three colors corresponding to the surfaces z=f1(x,y), z=f2(x,y)
and z=0
(red , dark blue and light blue) meet. The iterates are represented by
vertical bars.
Continuation method for systems
of algebraic
equations
f(x,tau)=0. Optional computation of turning and (simple) bifurcation
points
Optional automatic construction of
complete
bifurcation diagrams.
Rootfinders based on Newtons method or its modification will run into
considerable trouble when presented with a problem where singularities of
the Jacobian occur. In this case the use of unconstrained or
constrained minimization might help,
since there exist minimization methods which are rather robust if the
constraint gradients are linearly dependent.
To use an unconstrained minimizer, choose e.g. f(x)=||F(X)||^2 (the
Euclidean length squared) and minimize this with some code proposed in
the unconstrained minimization section.
To use a constrained minimizer, choose simply
f==0 as an objective function and minimize this under the constraint
F(x)=0.
The following code works often successfully in this situation:
