  Waveguide and resonant properties of diffractive structures are often
explained through the complex poles of their scattering matrices. Numerical
methods for calculating poles of the scattering matrix with applications in
grating theory are discussed. A new iterative method for computing the matrix
poles is proposed. The method takes account of the scattering matrix form in
the pole vicinity and relies upon solving matrix equations with use of matrix
decompositions. Using the same mathematical approach, we also describe a
Cauchy-integral-based method that allows all the poles in a specified domain to
be calculated. Calculation of the modes of a metal-dielectric diffraction
grating shows that the iterative method proposed has the high rate of
convergence and is numerically stable for large-dimension scattering matrices.
An important advantage of the proposed method is that it usually converges to
the nearest pole.
