On the geometry of Householder’s and Halley’s methods

In this paper, some modified Newton’s methods for solving nonlinear equations are proposed. Householder and Halley have suggested a variant of New- ton’s method in which they take an osculating parabola respectively an hyperbola instead of the tangent line. The proposed methods go faster to approximated solution, but it requires more calculation efforts. The order of convergence of both methods is three. By some interesting numerical problems, the theoretical results are illustrated. In the appendix, we give a foretaste of the proposed methods in the complex plane.