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

Abstract

We present a numerical method for solving the separable nonlinear equation A(y)z + b(y) = 0, where A(y) is an m × N matrix and b(y) is a vector, with y ∈Rn and z ∈RN. 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.