Stable Homogeneous Parameter, Homogeneous Geometric Newton Method with Complex Root Solutions

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In a past thesis, we proposed a homogeneous parameter, homogeneous geometric Newton-Raphson method for dealing with a rational polynomial curve and we concluded that the algorithm is robust and locally unique. After more experiments, we noticed that we have to get over not only divergence but also oscillation problems. Oscillation happens only when the equation has complex roots. So we propose a new geometric Newton-Raphson method, which can cope with complex root solutions. The algorithm has the features that the conventional homogeneous parameter, homogeneous Newton-Raphson method has and further more can find complex roots. The results of our experiments show that the algorithm is robust with respect to both divergence and oscillation, and also has local uniqueness property.
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