Local convergence of efficient secant-type methods for solving nonlinear equations (Q433325)

The paper deals with the problem of approximating a locally unique solution of a nonlinear equation \(F(x)= 0\), where \(F\) is defined on an open convex subset of a Banach space \(X\) with values in a Banach space \(Y\). Well-known methods are these of Newton and Chebyshev. In [J. Comput. Appl. Math. 235, No. 10, 3195--3206 (2011; Zbl 1215.65102)], the second author et. al. introduced a new family of iterative methods free from derivatives. They replaced the first derivative \(F\) by means of a divided difference of order one.NEWLINENEWLINE In the present paper, modifications of this method (the so-called secant-type-method) not examined earlier are proposed. These family of methods converges with order two. A unified estimate of the radius of convergence ball as well as an error analysis are given. Numerical examples (2 simple one-dimensional equations, an integral equation and the Chandrasekhar equation) validate the theoretical results.