Third- and fifth-order Newton–Gauss methods for solving nonlinear equations with n variables

Based on the mean-value theorem of multivariable vectors function F(x), two new iterative schemes with third-order and fifth-order convergence are constructed respectively by using Gauss quadrature formula for solving systems of nonlinear equations. Their error equations and asymptotic numerical convergence constants are obtained. The two suggested methods are compared with the related methods for solving systems of nonlinear equations and boundary-value problems of nonlinear ODEs in the numerical examples to demonstrate the efficiency and practicality.