A New Higher-Order Scheme for Multiple Roots with Unknown Multiplicity

In this work, we propose a new efficient iterative method to find multiple roots of nonlinear equations with unknown multiplicity n. The new scheme is free from the second derivative and consists of three steps derived from Enhanced Halley’s method introduced by the contributors and a Newton step. To increase efficiency, the first derivatives were approximated using forward difference, central difference, and Hermite interpolation techniques. It is demonstrated that the implemented method achieves sixth order of convergence. As an application, we apply the new method to chemical engineering problem (volume from van der Waals), biomedical engineering (blood rheology model) and ten academic problems. Comparisons and examples clarify that the new method outperforms existing methods with the same order of convergence.