On second derivative-free zero finding methods

High order root-finding algorithms are constructed from formulas for approximating higher order logarithmic and standard derivatives. These formulas are free of derivatives of second order or higher and use only function evaluation and/or first derivatives at multiple points. Richardson extrapolation technique is applied to obtain better approximations of these derivatives. The proposed approaches resulted in deriving a family of root-finding methods of any desired order. The first member of this family is the square root iteration or Ostrowski iteration. Additionally, higher order derivatives are approximated using multi-point function evaluations. We also derived a procedure for fourth order methods that are dependents only on the function and its first derivative evaluated at multiple points.