Continuous Newton method for star-like functions

• Main Author: Yakov Lutsky
• Format: Article
• Language: English
• Published: Texas State University 2005-04-01
• Series: Electronic Journal of Differential Equations
• Subjects: Newton method; star-like functions; continuous semigroup.
• Online Access: http://ejde.math.txstate.edu/conf-proc/12/l1/abstr.html

We study a continuous analogue of Newton method for solving the nonlinear equation $$ varphi (z) =0, $$ where $varphi(z)$ holomorphic function and $0inoverline{varphi ( D)}$. It is proved that this method converges, to the solution for each initial data $zin D$, if and only if $varphi(z)$ is a star-like function with respect to either an interior or a boundary point. Our study is based on the theory of one parameter continuous semigroups. It enables us to consider convergence in the case of an interior as well as a boundary location of the solution by the same approach.