On the downhill method

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.