Root finding by high order iterative methods based on quadratures

We discuss a recursive family of iterative methods for the numerical approximation of roots of nonlinear functions in one variable. These methods are based on Newton–Cotes closed quadrature rules. We prove that when a quadrature rule with n + 1 nodes is used the resulting iterative method has convergence order at least n + 2, starting with the case n = 0 (which corresponds to the Newton’s method).