Abstract
Root-finding is a common problem in various fields of physics and mathematics. The basic version of this problem can be formulated as an equation f(z)=0 where z∈ℂ and f is a meromorphic function on a complex plane ℂ. In some cases, a conjugated problem has to be solved as well to investigate the properties of the inspected system: f(z)−1=0. Iterative Householder-like methods show high convergence rate, however, the convergence itself is strongly dependent on the input parameters such as an initial guess. If the function f is complicated enough, there can be no prior information about its zeros and poles, and thus, no way to correctly pick the initial guess. The aim of the algorithm proposed in this article is to find all zeros and poles inside a selected region without any information about the local behavior of the function f. Additionally, the method is generalized to a parametric problem with continuous solutions: f(z)→f(z;p) where p∈Rd, d∈N∪{0}. The basis of the algorithm consists of a triangulation process and an implementation of the Cauchy’s argument principle to inspect the constructed subregions. The refinement process is accompanied by an absolute value gradient ∇ln|f| analysis that significantly decreases the number of points needed to correctly identify the locations of all zeros and poles. The algorithm is presented as a hybrid class for d∈{0,1} that can be directly used to solve corresponding problems.
Published Version
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