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On iterates of Möbius transformations on fields

Author: Sam Northshield
Journal: Math. Comp. 70 (2001), 1305-1310
MSC (2000): Primary 41A20; Secondary 65B99, 11B39, 12J20
Published electronically: March 6, 2000
MathSciNet review: 1710202
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Abstract: Let $p$ be a quadratic polynomial over a splitting field $K$, and $S$ be the set of zeros of $p$. We define an associative and commutative binary relation on $G\equiv K\cup \{\infty \}-S$ so that every Möbius transformation with fixed point set $S$ is of the form $x$ ``plus'' $c$ for some $c$. This permits an easy proof of Aitken acceleration as well as generalizations of known results concerning Newton's method, the secant method, Halley's method, and higher order methods. If $K$ is equipped with a norm, then we give necessary and sufficient conditions for the iterates of a Möbius transformation $m$ to converge (necessarily to one of its fixed points) in the norm topology. Finally, we show that if the fixed points of $m$ are distinct and the iterates of $m$ converge, then Newton's method converges with order 2, and higher order generalizations converge accordingly.

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Additional Information

Sam Northshield
Affiliation: Plattsburgh State University, Plattsburgh, New York 12901

Keywords: Fibonacci numbers, Newton's method, secant method, Aitken acceleration, M\"obius transformation, field, order of convergence
Received by editor(s): October 3, 1997
Received by editor(s) in revised form: January 18, 1999, and July 30, 1999
Published electronically: March 6, 2000
Additional Notes: This paper was written while the author was on sabbatical leave at the University of Minnesota. The author appreciated the hospitality of the Mathematics Department during that time.
Article copyright: © Copyright 2000 American Mathematical Society