Two-step iterative methods for multiple roots and their applications for solving several physical and chemical problems

In this manuscript, we introduce a two-step convergent iterative scheme to compute the roots with multiplicity m$$ m $$ of nonlinear equations. Most of the schemes in literature have flexibility at either first substep or second step, while our scheme has the flexibility of choice at both substeps so that many existing methods are obtained as special cases. Our scheme has another advantage over earlier studies: It not only works for m >= 2$$ m\ge 2 $$ but also m >= 1$$ m\ge 1 $$. It is shown that the methods obtained by our scheme are optimal as they satisfy Kung-Traub conjecture. Several physical and chemical examples are considered to check the performance of the methods and to analyze the theoretical results, and it is found that our methods show preferred outcomes over the existing schemes. Likewise, the basins of attraction of the schemes also confirm the same results for the performance of the suggested techniques over the compared ones.