Author： Sahari Mohamed Lamine Djellit Illhem
Publisher： Taylor & Francis Ltd
ISSN： 0020-7160
Source： International Journal of Computer Mathematics, Vol.86, Iss.3, 2009-03, pp. : 407-422
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Abstract
Conjugate gradient method is a root-finding algorithm to non-linear equations. In this paper, we suggest extending this method for a polynomial to the complex plane. Through the experimental and theoretical mathematics method, we drew the following conclusions: (1) the conjugate gradient is a dynamical system with two complex parameters; (2) locally conditions for convergence to any roots of complex functions is given; (3) the conjugate gradient method may fail to converge to all roots for cubic with three simple roots; (4) the boundary of conjugate gradient basins are fractals in some cases, and depends on the parameters; (5) the algorithm is then improved by introducing a method to determine the optimal parameters.
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