Self-Adaptive Mesh Generator for Global Complex Roots and Poles Finding Algorithm

In any global method of searching for roots and poles, increasing the number of samples increases the chances of finding them precisely in a given area. However, the global complex roots and poles finding algorithm (GRPF) (as one of the few) has direct control over the accuracy of the results. In addition, this algorithm has a simple condition for finding all roots and poles in a given area: it only requires a sufficiently dense initial grid. However, in practice, this requirement may turn out to be very difficult to implement. For a complex and sophisticated function, the use of a regular high-density mesh may be ineffective or even impossible due to limited computational resources. In this article, a method for creating a self-adaptive initial mesh for this algorithm is presented. The proposed solution uses gradient calculation to identify areas that require mesh refinement, including areas where a zero and a pole are in close proximity. The adaptive mesh allows for faster and more accurate analysis of functions with a much smaller number of samples. As shown in the numerical examples, this approach reduces the number of function calls by several orders of magnitude, and thus significantly reduces the computational time.