Expanding the Applicability of Stirling’s Method under Weaker Conditions and Restricted Convergence Regions

In this paper we have provided sufficient conditions to study semilocal and local convergence of the Stirling’s method. The method is used to find fixed points of nonlinear operator equation. We assume Lipschtiz continuity type conditions on the first Fréchet derivative of the operator but no contractive conditions as in earlier works. This way expand the applicability of this method. Here we introduce a new type of majorizing sequences instead of usual majorizing sequences and recurrence relations. Finally the paper will be concluded with numerical examples and a favorable comparison with known results.

A CONTINUATION METHOD AND ITS CONVERGENCE FOR SOLVING NONLINEAR EQUATIONS IN BANACH SPACES

A continuation method is a parameter based iterative method establishing a continuous connection between two given functions/operators and used for solving nonlinear equations in Banach spaces. The semilocal convergence of a continuation method combining Chebyshev's method and Convex acceleration of Newton's method for solving nonlinear equations in Banach spaces is established in [J. A. Ezquerro, J. M. Gutiérrez and M. A. Hernández [1997] J. Appl. Math. Comput.85: 181–199] using majorizing sequences under the assumption that the second Frechet derivative satisfies the Lipschitz continuity condition. The aim of this paper is to use recurrence relations instead of majorizing sequences to establish the convergence analysis of such a method. This leads to a simpler approach with improved results. An existence–uniqueness theorem is given. Also, a closed form of error bounds is derived in terms of a real parameter α ∈ [0, 1]. Four numerical examples are worked out to demonstrate the efficacy of our convergence analysis. On comparing the existence and uniqueness region and error bounds for the solution obtained by our analysis with those obtained by using majorizing sequences, it is found that our analysis gives better results in three examples, whereas in one example it gives the same results. Further, we have observed that for particular values of the α, our analysis reduces to those for Chebyshev's method (α = 0) and Convex acceleration of Newton's method (α = 1) respectively with improved results.