This paper describes a novel technique for the very efficient and accurate full-wave modal analysis of cylindrical waveguides with arbitrary cross-section. This new technique relies on solving the integral equations that provide the well-known Boundary Integral - Resonant Mode Expansion (BI-RME) method by the Nyström approach, instead of using the traditional Galerkin version of the Method of Moments (MoM), thus providing large savings on computational costs. Accuracy aspects of this simple and fast procedure, which are directly connected to the rigorous treatment of the singular behaviour of the integral equation kernels, are carefully considered for waveguides defined by straight, circular and/or elliptical arcs. Comparative benchmarks between the new technique and the original BI-RME method are successfully presented for single- and multi-ridged waveguides, elliptical waveguides and rectangular waveguides with rounded corners
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