Numerical solution of separable nonlinear equations with a singular matrix at the solution

We present a numerical method for solving the separable nonlinear equation A(y)z + b(y) = 0, where A(y) is an m x N matrix and b(y) is a vector, with y is an element of R-n and z is an element of R-N. We assume that the equation has an exact solution (y*, z*). We permit the matrix A(y) to be singular at the solution y* and also possibly in a neighborhood of y*, while the rank of the matrix A(y) near y* may differ from the rank of A(y*) itself. We previously developed a method for this problem for the case m = n + N, that is, when the number of equations equals the number of variables. That method, based on bordering the matrix A(y) and finding a solution of the corresponding extended system of equations, could produce a solution of the extended system that does not correspond to a solution of the original problem. Here, we develop a new quadratically convergent method that applies to the more general case m >= n + N and produces all of the solutions of the original system without introducing any extraneous solutions.