A Decomposition Methodology and a Class of Algorithms for the Solution of Nonlinear Equations

In this paper we develop a decomposition methodology to find a solution of a system of nonlinear equations. The decomposition methodology is based on the matrix theoretic notion of nonzero transversal and the graph theoretic notion of minimal essential set. This decomposition methodology gives rise to a class of three iterative methods each of which uses a derivative matrix whose order is equal to the cardinality of the minimal essential set. We prove that the iterative methods are types of the Newton method. For the case of a system of linear equations, we show that the iterative methods are combinations of the generalized Gaussian elimination and the Newton method.