Continuous Newton method for star-like functions
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-...
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| Format: | Article |
| Language: | English |
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Texas State University
2005-04-01
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| Series: | Electronic Journal of Differential Equations |
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| Online Access: | http://ejde.math.txstate.edu/conf-proc/12/l1/abstr.html |
| Summary: | 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. |
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| ISSN: | 1072-6691 |