# Graduate School in Nonlinear Science

### Sponsored by the Danish Research Academy

**
MIDIT OFD CATS
Modelling, Nonlinear Dynamics Optics and Fluid Dynamics Chaos and Turbulence Studies
and Irreversible Thermodynamics Risø National Laboratory Niels Bohr Institute and
Technical University of Denmark Building 128 Department of Chemistry
Building 321 P.O. Box 49 University of Copenhagen
DK-2800 Lyngby DK-4000 Roskilde DK-2100 Copenhagen Ø
Denmark Denmark Denmark
**

**THE HALLEY METHOD**

by **Xavier Buff**

Laboratoire de Mathematiques Emile Picard

Universite Paul Sabatier,

Toulouse, France

MIDIT-seminar 472

**Wednesday March 01, 2000, 11.00 h**

at MIDIT, MAT, Bldg. 303, Room 026, DTU,

**Abstract**:
A well known algorithm to find the roots of a polynomial is the
Newton's method. I will present a variant due to the astronomer
that discovered the comet. Instead of approximating P by its
tangent, one can approximate P by the Moebius transformation
that best oscullates P.

One then has to iterate the rational map

H_P(z) = z - P(z)/ ( P'(z) - P(z)P''(z)/(2P'(z)) )