On the Global Convergence of Broyden's Method

We consider Broyden's 1965 method for solving nonlinear equations. If the mapping is linear, then a simple modification of this method guarantees global and Q-superlinear convergence. For nonlinear mappings it is shown that the hybrid strategy for nonlinear equations due to Powell leads to R-superlinear convergence provided the search directions from a uniformly linearly indepenent sequence. We then explore this last concept and its connection with Broyden's method. Finally, we point out how the above results extend to Powell's symmetric version of Broyden's method