Journal of the
Korean Mathematical Society JKMS

In this paper, we derive an upper bound for the distance from a point in the immediate basin of a root of a complex polynomial to the root itself. We establish that if $z$ is a point in the immediate basin of a root $\alpha$ of a polynomial $p$ of degree $d\geq 12$, then $|z-\alpha|\leq \frac{3}{\sqrt{d}}\left(6\sqrt{310}/35\right)^d|N_p(z)-z|$, where $N_p$ is the Newton map induced by $p$. This bound leads to a new bound of the expected total number of iterations of Newton's method required to reach all roots of every polynomial $p$ within a given precision, where $p$ is normalized so that its roots are in the unit disk.

Keywords: Root, polynomial, Newton's method

MSC numbers: Primary 26A18, 30C15

Supported by: This work was financially supported by Faculty of Science, Silpakorn University (Grant No. SRF-JRG-2558-02)