A modification of Newton's method for analytic mappings having multiple zeros
A modification of Newton's method for iteratively computing singular zero points of analytic functions is studied. Under mild assumptions the method converges quadratically, without a priori information concerning the nature of the singularity. The method generalizes to systems \textit{H. Van de Vel}'s iterative method for finding zero points of an analytic map in a single variable [Computing 14, 167-171 (1975; Zbl 0301.65028)]. The latter method determines the zero point \(z^*\) superlinearly and also the multiplicity \(\mu_{z^*}(f)\) of \(z^*\). The present generalization gives approximations of zero points \(z^*\in\mathbb{C}^n\) and also of the orders \(k_1,\cdots,k_n\) of the homogeneous principal parts of \(f\) at \(z^*\). That is, \(f_j(z)=\sum_{|\alpha|\geq k_j}c_{j,\alpha}(z-z^*)^{\alpha}\) where \(\alpha\) is a multi-index. Several numerical examples are given which illustrate the effectiveness of the approach.
scientific article; zbMATH DE number 1309795