Convergence ball of a modified secant method with convergence order 1.839…

A local convergence theorem on a modified secant method with convergence order 1.839… for solving nonlinear equations is established under the hypotheses that the second-order and third-order derivative of the function involved are bounded. An estimate of the radius of the convergence ball of this method is obtained. Moreover, an error estimate is provided which matches the convergence order of the method.