Computer solution of the scattering problem for a groove in a metallic plane using the modal method

The roots of the complex transcendental equations that result from the application of the modal method to the scattering problem for a metallic groove are obtained iteratively as fixed points of entire functions of the form Fc(z), where c, z C. Iterations are performed with Fc(z) or an appropriate branch of its multiple-valued inverse function, that is, zj+1 = Fc(zj) or zj+1 = F−1c(zj), respectively. Since convergence fails near double roots, an insightful study of the problem is made and high-precision solutions near double roots are obtained by interpolation. Examples are given to illustrate the behaviour of the methods in different situations, with a connection to fractal theory.