Enclosing the solutions of nonlinear systems of equations

The quadratically convergent Newton-type methods and its variants are generally used for solving the nonlinear systems of equations. Most of these methods use the convexity conditions [7] of the involved bounded linear operators for their convergence. Alefeld and Herzberger [1] have proposed a quadratically convergent iterative method for enclosing the solutions of the special type of nonlinear system of equations arising from the discretization of nonlinear boundary value problems which do not require the convexity conditions but uses the subinverses of the bounded linear operators. In this paper, we have proposed a modification of this method which takes it further faster. The proposed method uses both the superinverses and subinverses of bounded linear operators. At the expense of slightly more computations than used in [1], the rate of convergence of our method enhances from quadratic to cubic. Finally, the method is tested on a numerical example.