We present a new optimal class of Ostrowski's method for obtaining multiple zeros of univariate nonlinear functions. Several researchers tried to construct an optimal family of Ostrowski's method for multiple zeros, but they did not have success in this direction. The new strategy adopts a weight function approach. The design structure of new families of Ostrowski's technique is simpler than the existing classical families of the same order for multiple zeros. The classical Ostrowski's method of fourth-order can obtain a particular form for the simple root. Their efficiency is checked on a good number of relevant numerical examples. These results demonstrate the performance of our methods. We find that the new methods are just as competent as other existing robust techniques available in the literature.

Here, we suggest a high-order optimal variant/modification of Schröder’s method for obtaining the multiple zeros of nonlinear uni-variate functions. Based on quadratically convergent Schröder’s method, we derive the new family of fourth -order multi-point methods having optimal convergence order. Additionally, we discuss the theoretical convergence order and the properties of the new scheme. The main finding of the present work is that one can develop several new and some classical existing methods by adjusting one of the parameters. Numerical results are given to illustrate the execution of our multi-point methods. We observed that our schemes are equally competent to other existing methods.

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