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http://dx.doi.org/10.4134/JKMS.j190457

ON THE DISTANCE TO A ROOT OF COMPLEX POLYNOMIALS UNDER NEWTON'S METHOD  

Chaiya, Malinee (Department of Mathematics Faculty of Science Silpakorn University)
Chaiya, Somjate (Department of Mathematics Faculty of Science Silpakorn University)
Publication Information
Journal of the Korean Mathematical Society / v.57, no.5, 2020 , pp. 1119-1133 More about this Journal
Abstract
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 α of a polynomial p of degree d ≥ 12, then ${\mid}z-{\alpha}{\mid}{\leq}{\frac{3}{\sqrt{d}}\(6{\sqrt{310}}/35\)^d{\mid}N_p(z)-z{\mid}$, where Np 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;
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