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In this work, we construct a family of seventh order iterative methods for finding multiple roots of a nonlinear function. The scheme consists of three steps, of which the first is Newton's step and last two are the weighted-Newton steps. Hence, the name of the scheme is 'weighted-Newton methods'. Theoretical results are studied exhaustively along with the main theorem describing convergence analysis. Stability and convergence domain of the proposed class are also demonstrated by means of using a graphical technique, namely, basins of attraction. Boundaries of these basins are fractal like shapes through which basins are symmetric. Efficacy is demonstrated through numerical experimentation on variety of different functions that illustrates good convergence behavior. Moreover, the theoretical result concerning computational efficiency is verified by computing the elapsed CPU time. The overall comparison of numerical results including accuracy and CPU-time shows that the new methods are strong competitors for the existing methods. [ABSTRACT FROM AUTHOR]

Many higher order multiple-root solvers that require derivative evaluations are available in literature. Contrary to this, higher order multiple-root solvers without derivatives are difficult to obtain, and therefore, such techniques are yet to be achieved. Motivated by this fact, we focus on developing a new family of higher order derivative-free solvers for computing multiple zeros by using a simple approach. The stability of the techniques is checked through complex geometry shown by drawing basins of attraction. Applicability is demonstrated on practical problems, which illustrates the efficient convergence behavior. Moreover, the comparison of numerical results shows that the proposed derivative-free techniques are good competitors of the existing techniques that require derivative evaluations in the iteration. [ABSTRACT FROM AUTHOR]