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). 
Mario M. Gra\c{c}a and Pedro M. Lima
math.NA