Double newtonisation of fixed point sequences
Mario M. Graca
TL;DR
This work tackles the challenge of computing fixed points when the derivative at the fixed point is neutral ($|u'(x^*)|=1$) by constructing a universal accelerator. It introduces a double Newtonisation: first transform $u$ into a hyperbolic $v$ via $v(x)=\frac{u(x)-x\,u'(x)}{1-u'(x)}$, then form the standard accelerator $w$ by $w(x)=\frac{v(x)-x\,v'(x)}{1-v'(x)}$, turning $x^*$ into a super attracting fixed point. The framework encompasses logarithmic and hyperbolic fixed-point sequences, proves kernel characterizations, and demonstrates broad applicability through several concrete examples, including real and complex cases. The results yield a unified, practical method for accelerating fixed-point iterations and computing multiple zeros with high efficiency, even in challenging neutral or near-neutral settings.
Abstract
A neutral fixed point of a real iteration map $u$ becomes a super attracting fixed point using a suitable double newtonisation. The map $u$ is so transformed into a map $w$ which is here called the standard accelerator of $u$. The map $w$ provides a unifying process to deal with a large set of fixed point sequences which are not convergent or converge slowly. Several examples illustrate the main results obtained.