# The reason why the Newton's method complains eventually

Here we are trying to find solutions to the system: *F=0*. That is
*
f*_{x}=0,

f_{y}=0,

f_{z}=0.

Newton's method leads to the following iterations:
*
x*_{n+1}= x_{n}-(DF_{xn})^{-1}.
F(x_{n}).

The iteration process breaks down when * DF*_{xn}
,which is the same as the hessian matrix of *f* at * x*_{n}
, has rank less
than three. Recall that we are approching the singular curves on which
the hessian matrices have rank deficiency. So the Newton's
method complains eventually.

Chia-Hsing Nien <nien@geom.umn.edu>
Last modified: Tue May 28 13:25:43 1996