Second-derivative free methods of third and fourth order for solving nonlinear equations

Abstract

We present here a one-parameter family of iterative methods for solving nonlinear equations. All the methods of the family have third-order convergence, except the one which has the fourth-order convergence. Per iteration, all these methods require two evaluations of the function and one evaluation of the first derivative. Therefore, the fourth-order method is more economic than the third-order methods. Numerical examples are given to support that the methods thus obtained are competitive with other similar robust methods. Moreover, it is shown by way of illustration that the fourth-order method is very useful in the applications requiring high precision in computations.