CENTER FOR COMPUTATIONAL MATHEMATICS COLLOQUIUM
                JOINT WITH THE OPTIMIZATION SEMINAR

                  UNIVERSITY OF COLORADO AT DENVER



TITLE:   Qualitative analysis of Newton's method
 

SPEAKER: Vladimir Janovsky,  Charles University, Prague, Czech Republic
         

DATE:    Monday, September 22, 1997  


PLACE:   Math Conference Room 656             (PLEASE NOTE UNUSUAL ROOM)
         UCD Building, 1250 14th St., Denver


TIME:    noon (Refreshments served at 11:45 am) 



ABSTRACT

A global convergence of the Newton method 
is studied by techniques of the Singularity Theory. 
It is assumed that the objects of the computation
(roots of a mapping) are "organized" by a particular 
singular point of the mapping being subjected to small perturbations.

It will be shown that the Liapunov-Schmidt reduction process yields 
an invariant, exponentially attracting manifold of the Newton flow
(i.e., the continuous variant of the Newton method).
If the singular point is a fold or a cusp then normal forms
of the reduced 2-D Newton flow will be presented. Moreover, experimental
evidence suggests that discrete versions of the normal forms
may explain the performance of the ordinary Newton method
in a neighbourhood of the mentioned singular points.