New higher order methods for solving nonlinear equations with multiple roots

For a nonlinear equation f ( x ) = 0 having a multiple root we consider Steffensen’s transformation, T . Using the transformation, say, F q ( x ) = T q f ( x ) for integer q ≥ 2 , repeatedly, we develop higher order iterative methods which require neither derivatives of f ( x ) nor the multiplicity of the root. It is proved that the convergence order of the proposed iterative method is 1 + 2 q − 2 for any equation having a multiple root of multiplicity m ≥ 2 . The efficiency of the new method is shown by the results for some numerical examples.