A note on the use of the Neumann expansion in calculating the scatter from rough surfaces

The Neumann expansion has been used to compute the solutions of the magnetic-field integral equation (MFIE) for two-dimensional, perfectly conducting, Gaussian, rough surfaces. For surfaces whose roughness is of a similar order to the incident wavelength, it is shown that the expansion may diverge rapidly. The rate of convergence is compared with the conjugate-gradient (CG) method, whose convergence is sure. When it converges, the Neumann expansion convergence is more rapid. It is concluded that the Neumann expansion is not suitable without qualification as a numerical solution to the rough surface MFIE. Moreover, the failure of the Neumann expansion of the solution of the discrete representation of the MFIE provides strong evidence that the use of the Neumann expansion as a formal solution to the MFIE is open to doubt.>