Adaptive Parallel Methods for Polynomial Equations with Unknown Multiplicity

New two-step simultaneous iterative techniques are proposed for solving polynomial equations with multiple roots of unknown multiplicity. The developed schemes achieve a local convergence order of ten and address key limitations of existing solvers, namely their dependence on prior multiplicity information and their reduced efficiency when dealing with clustered or repeated roots. Root multiplicities are adaptively estimated within the iterative process, avoiding additional function evaluations beyond those required for parallel updates. The robustness and stability of the proposed methods are assessed using both random and distant initial guesses and validated on benchmark polynomials as well as nonlinear models from biomedical engineering. The numerical results show notable improvements in residual error, iteration count, CPU time, memory usage, and overall convergence rate compared with established classical techniques. These findings demonstrate that the proposed schemes provide reliable, high-order, and computationally efficient tools for solving challenging nonlinear problems in science and engineering.