On the Davidenko-Branin Method for Solving Simultaneous Nonlinear Equations

It has been conjectured that the Davidenko–Branin method for solving simultaneous nonlinear equations is globally convergent, provided that the surfaces on which each equation vanishes are homeomorphic to hyperplanes. We give an example to show that this conjecture is false. A more complicated example shows that the method may fail to converge to a zero of the gradient of a scalar function, so the associated method for function minimization is not globally convergent.