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

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