Transformation methods for finding multiple roots of nonlinear equations

In this paper, to estimate a multiple root p of an equation f(x) = 0, we transform the function f(x) to a hyper tangent function combined with a simple difference formula whose value changes from −1 to 1 as x passes through the root p. Then we apply the so-called numerical integration method to the transformed equation, which may result in a specious approximate root. Furthermore, in order to enhance the accuracy of the approximation we propose a Steffensen-type iterative method, which does not require any derivatives of f(x) nor is quite affected by an initial approximation. It is shown that the convergence order of the proposed method becomes cubic by simultaneous approximation to the root and its multiplicity. Results for some numerical examples show the efficiency of the new method.