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Many optimal order multiple root techniques, which use derivatives in the algorithm, have been proposed in literature. But contrarily, derivative free optimal order techniques for multiple root are almost nonexistent. By this as an inspirational factor, here we present a family of optimal fourth order derivative-free techniques for computing multiple roots of nonlinear equations. At the beginning the convergence analysis is executed for particular values of multiplicity afterwards it concludes in general form. Behl et. al derivative-free method is seen as special case of the family. Moreover, the applicability and comparison is demonstrated on different nonlinear problems that certifies the efficient convergent nature of the new methods. Finally, we conclude that our new methods consume the lowest CPU time as compared to the existing ones. This illuminates the theoretical outcomes to a great extent of this study. [ABSTRACT FROM AUTHOR]

We propose a novel family of seventh-order iterative methods for computing multiple zeros of a nonlinear function. The algorithm consists of three steps, of which the first two are the steps of recently developed Liu–Zhou fourth-order method, whereas the third step is based on a Newton-like step. The efficiency index of the proposed scheme is 1.627, which is better than the efficiency index 1.587 of the basic Liu–Zhou fourth-order method. In this sense, the proposed iteration is the modification over the Liu–Zhou iteration. Theoretical results are fully studied including the main theorem of local convergence analysis. Moreover, convergence domains are also assessed using the graphical tool, namely, basins of attraction which are symmetrical through the fractal like boundaries. Accuracy and computational efficiency are demonstrated by implementing the algorithms on different numerical problems. Comparison of numerical experiments shows that the new methods have an edge over the existing methods. [ABSTRACT FROM AUTHOR]