Complex Border Tracking Algorithm for Determining of Complex Zeros and Poles and its Applications

This paper proposes a new algorithm, very efficient numerically, used for locating zeros and poles of complex function and its exemplary applications in microwave technique problems. The algorithm starts its run from two arbitrary regions on the complex plane, where the searched function yields values with different signs for either or both real and imaginary parts. The algorithm runs along the path created by the chain of equilateral triangles, at whose vertices the function is evaluated. When analyzing the signs of the function, the tracking directions are chosen along the borders of complex regions, where zeros/poles are further expected. Calculations of complex propagation coefficients for a slab dielectric waveguide, complex dispersion coefficients of graphene transmission lines, and a zero/pole analysis of microwave filters are given to show some of the possible applications of the proposed algorithm. A comparison to other methods is also included.