# TR-2010-10

## The end-game for Newton iteration

Todd Dupont; L. Ridgway Scott. 30 December, 2010.

Communicated by L. Ridgway Scott.

### Abstract

We consider the angular orientations of convergent iterates generated by
Newton's method in multiple space dimensions.
We show that the limiting behavior of the iterates can be described by
a tensor eigenproblem.
We give an extensive computational analysis of this tensor eigenproblem
in two dimensions.
In a large fraction of cases, the tensor eigenproblem has a discrete number of
solutions to which the Newton directions converge quickly, but there is also a
large fraction of cases in which the behavior is more complicated.
We contrast the angular orientations of iterates generated by
Newton's method with the corresponding directions of the
continuous Newton algorithm.

### Original Document

The original document is available in PDF (uploaded 30 December, 2010 by
L. Ridgway Scott).