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The downhill method is a numerical method for solving complex equations ƒ( z ) = 0 on which the only restriction is that the function w = ƒ( z ) must be analytical. An introduction to this method is given and a critical review of relating literature is presented. Although in theory the method always converges, it is shown that a fundamental dilemma exists which may cause a breakdown in practical applications. To avoid this difficulty and to improve the rate of convergence toward a root, some modifications of the original method are proposed and a program (FORTRAN) based on the modified method is given in Algorithm 365. Some numerical examples are included.

Three analysis methods for reflector antennas, the spherical near-field geometric theory of diffraction (SNFGTD), the moment method (MM) and physical optics (PO) are compared. In the SNFGTD method, the far field from the antenna is found by a spherical near-field transformation (SNF) of the tangential electric near field on a sphere surrounding the antenna once this has been found using the geometrical theory of diffraction (GTD). SNFGTD and MM agree very well, whereas PO differs with respect to the cross-polar pattern. Some discrepancies between SNFGTD and MM are shown to be related to inaccuracies in the GTD diffraction coefficients close to the reflection boundary.

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