New optimal class of higher-order methods for multiple roots, permitting f′(xn) = 0

Finding multiple zeros of nonlinear functions pose many difficulties for many of the iterative methods. A major difficulty in the application of iterative methods is the selection of initial guess such that neither guess is far from zero nor the derivative is small in the vicinity of the required root, otherwise the methods would fail miserably. Finding a criterion for choosing initial guess is quite cumbersome and therefore, more effective globally convergent algorithms for multiple roots are still needed. Therefore, the aim of this paper is to present an improved optimal class of higher-order methods having quartic convergence, permitting f′(x) = 0 in the vicinity of the required root. The present approach of deriving this optimal class is based on weight function approach. All the methods considered here are found to be more effective and comparable to the similar robust methods available in literature.