A family of methods for solving nonlinearequations using quadratic interpolation

A two-parameter derivative-free family of methods for finding the simple and real roots of nonlinear equations is presented. The approximation process is carried out by using interpolation on three successive points (χκ, Yκ) to determine the coefficients c, d, e in the general quadratic equation aχ2 + by2 + cχ + dy + e = 0 in the terms of the coefficients a, b. Different choices of a, b correspond to different quadratic forms. Muller and inverse parabolic interpolation methods are seen as special cases of the family. Geometrical relationships with other methods are established. It is shown that the order of convergence is 1.84. Some numerical examples are given