The method of simple iterations with correction of convergence and the minimal discrepancy method for plasmonic problems

The problem of searching for complex roots of the dispersion equations of plasmon-polaritons along the boundaries of the layered structure-vacuum interface is considered. Such problems arise when determining proper waves along the interface of structures supporting surface and leakage waves, including plasmons and polaritons along metal, dielectric and other surfaces. For the numerical solution of the problem, we consider a modification of the method of simple iterations with a variable iteration parameter leading to a zero derivative of the right side of the equation at each step, i.e. convergent iterations, as well as a modification of the minimum residuals method. It is shown that the method of minimal residuals with linearization coincides with the method of simple iterations with the specified correction. Convergent methods of higher orders are considered. The results are demonstrated by examples, including complex solutions of dispersion equations for plasmon-polaritons. The advantage of the method over other methods of searching for complex roots in electrodynamics problems is the possibility of ordering the roots and constructing dispersion branches without discontinuities. This allows you to classify modes.