A family of third-order methods to solve nonlinear equations by quadratic curves approximation

A one-parameter family of iteration functions for finding the simple roots of nonlinear equations is presented. The iteration process is based on one-point approximation by the quadratic equation x2 + ay2 + bx + cy + d = 0, where the unknowns b, c and d are determined in terms of a. Different choices of a correspond to different approximating quadratic curves, viz. parabola, circle, ellipse and hyperbola. Euler, Chebyshev, Halley, super-Halley methods and, as an exceptional case, Newton method are seen as the special cases of the family. All the methods of the family are cubically convergent except Newton's which is quadratically convergent.