**4. Conclusions**

Locating the range or interval of the required root that provides sure convergence of an iterative method is one of the difficult problems in computational analysis. This paper addressed this problem and expanded the applicability of Methods (2) and (3) using hypotheses only on the functions appearing in these techniques. Further, we provided the radii of ball convergence and error bounds using Lipschitz conditions. This type of study was not addressed in the earlier work. With the help of the radius of convergence, we can find the range of initial estimate *p*∗ that tells us how close it must be for granting the convergence of Methods (2) and (3). Finally, the applicability of new approach was illustrated with several numerical examples.

**Author Contributions:** All co-authors contributed to the conceptualization, methodology, validation, formal analysis, writing the original draft preparation, and editing.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
