This paper considers the problem of solving a system of n nonlinear equations in n variables, when the underlying functions are continuously differentiable and their derivative satisfies a Lipschitz condition. We restrict our attention to methods which are based on complementary pivoting, also known as fixed point algorithms. We show that certain realizations of these methods achieve quadratic convergence when they reach sufficiently close to the solution. And in these cases, without using the derivatives of the mappings, the computational work involved is comparable to that of Newton's method.