A convergence analysis of an iterative algorithm of order \(1.839\ldots\) under weak assumptions

Abstract

We provide new and weaker sufficient local and semilocal conditions for the convergence of a certain iterative method of order 1.839\(\ldots\) to a solution of an equation in a Banach space. The new idea is to use center-Lipschitz/Lipschitz conditions instead of just Lipschitz conditions on the divided differences of the operator involved. This way we obtain finer error bounds and a better information on the location of the solution under weaker assumptions than before.

Keywords

• Banach space
• majorizing sequence
• Halley method
• Euler-Chebyshev method
• divided differences of order one and two
• Fréchet-derivative