Relationship theorem between nonlinear polynomial equations and the corresponding Jacobian matrix

This paper provides a proof of a relationship theorem between nonlinear analogue polynomial equations and the corresponding Jacobian matrix. This theorem is also verified generally effective for all nonlinear polynomial algebraic system of equations. By using this relationship theorem, we give a Newton formula without requiring the evaluation of nonlinear function vector as well as a simple formula to estimate the relative error of the approximate Jacobian matrix. The presented theorem can easily reduce numerical analogue equations of nonlinear initial value problems to the simple linear ones without any linearization procedures. Therefore, stability analysis of nonlinear initial value problems can be carried out based on the well-known results for linear problems. Finally, some possible applications of this theorem in nonlinear system analysis are also discussed.