Two-Point Iterative Methods for Solving Quadratic Equations and its Applications

Kung-Traub conjecture states that an iterative method without memory for locating the zero of a scalar equation could achieve convergence order 2d−1, where d is the total number of function evaluations, but proposed algorithm produces convergence order of r+2, where r is a positive integer with three function evaluations for solving quadratic equations, which is better than expected maximum convergence order. Therefore, we show that the conjecture fails for quadratic equations. Also, we extend proposed algorithm to solving systems which involving quadratic equations. We test our methods with some numerical experiments including application to one dimensional and two dimensional Bratu problems.