A Novel Geometric Modification to the Newton-Secant Method to Achieve Convergence of Order and Its Dynamics

Figure 1

Geometric modification of Newton-Secant method. is the point given by Newton’s method. In the next step, using condition (4), we have that . Thus the line passing through the points and gave us the better approximation to the root . This approximation is better than given by Newton’s method. The concave case is analogous to convex case.

(a) Convex case

(b) Concave case