A Kantorovich-type analysis for a fast iterative method for solving nonlinear equations (Q881993)

The paper is concerned with the iterative solution of nonlinear equations in Banach space. Such equations appear in applied mathematics and in engineering (difference equations, differential equations, integral equations, nonlinear algebraic equations). The author is interested in numerical methods that avoid the expensive computation at each step of the Fréchet derivative of the operator from the left hand side of the equation. For this, an iterative method, using divided differences of order one, that generalizes the classical secant method, is considered. Under certain Kantorovich-type conditions, the quadratic convergence of the method is proved, in the local case as well as the semilocal case. Moreover, the proposed method is compared favorably with other iterative methods with order of convergence two. Numerical examples are discussed.