Efficient and Systematic Solution of Real and Complex Eigenvalue Problems Employing Simplex Chain Vertices Searching Procedure

This paper presents a novel method that is very efficient in solving multidimensional real and complex eigenvalue problems, commonly employed in electromagnetic analysis, which can be transformed into a nonlinear equation. The concept is realized as root tracing process of a real or complex function of N variables in the constrained space. We assume that the roots of the continuous function of N variables lie on the continuous (N-1) -dimensional hyperplane. The method uses regular N and (N-1)-Simplexes, at which vertices the considered function changes its sign. Based on (N-1)-Simplex, the function is evaluated at two new points that are vertices of new regular N-Simplexes for which (N-1)-Simplex is one of its (N-1)-faces. The algorithm, with the usage of stack, runs in an iterative mode tracing the roots inside the volume of the considered simplexes. As a result, the algorithm creates a chain of simplexes in the constrained region. The proposed algorithm is optimal in the sense of the number of function evaluations. The numerical results, real and complex dispersion characteristics of chosen microwave guides, have proven the versatility and efficiency of the proposed algorithm.