Semilocal convergence of a continuation method in Banach spaces

Abstract

This paper is concerned with the semilocal convergence of a continuation method between two third-order iterative methods, namely, the Halley’s and the convex acceleration of Newton’s method, also known as the Super-Halley’s method. This convergence analysis is discussed using the recurrence relations approach. This approach simplifies the analysis and leads to improved results. The convergence analysis is established under the assumption that the second Frechet derivative satisfies Lipschitz continuity condition. An existence-uniqueness theorem is given. Also, a closed form of error bound is derived in terms of a real parameter α ∈ [0, 1]. Two numerical examples are worked out to demonstrate the efficacy of our approach. On comparing the existence and uniqueness region and error bounds for the solution obtained by our analysis with those obtained by using majorizing sequences [15], we observed that our analysis gives better results. Further, we have observed that for particular values of the α, our analysis reduces to those for the Halley’s method (α = 0) and the convex acceleration of Newton’s method (α = 1), respectively, with improved results.