On the Downhill Method The downhill method is a numerical method for solving complex equations f(z) = 0 on which the only restriction is that the function w = f(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. CACM December, 1969 Bach, H. downhill method, complex relaxation method, complex iteration, complex equation, transcendental complex equation, algebraic complex equation 5.15 CA691206 JB February 15, 1978 2:54 PM 1806 4 1806 1806 5 1806 1806 5 1806 1806 5 1806 1803 5 1806 535 5 1806 1806 6 1806