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_{x_n})^-1. F(x_n).

The iteration process breaks down when DF_{x_n} ,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