Quantitative Behavior and Probabilistic Convergence of Iterative Methods for Solving Multiple Roots: A Numerical Exploration

To address the convergence rate degradation of standard Newton iteration methods for nonlinear equations with multiple roots, this study systematically investigates the dynamic behavior and convergence properties of iterative methods for solving multiple roots. First, under the condition of a countable state space, we analyze, based on existing Markov chain theory, the convergence conditions and rates for nine iterative formats, including the modified Newton method and Halley method. Next, extending the research to general state spaces, we discuss a potential probabilistic analysis framework for probability convergence, first arrival time expectation, and distribution convergence rate using Markov chain and drift analysis tools. Numerical experiments demonstrate that, as the root multiplicity increases from 1 to 7, the convergence probability of the standard Newton method decreases from 0.98 to 0.35, while the average first arrival time increases from 6.2 to 190.3 iterations. The results indicate that the performance of iterative methods for multiple roots strongly depends on explicit utilization of multiple roots information or possession of high-order convergence properties, thereby improving both convergence probability and first arrival time performance. This study provides quantitative evidence and novel insights for theoretical analysis and efficient algorithm design of iterative methods in complex scenarios. In addition, the sensitivity of the iterative method to the initial value is discussed. It is pointed out that the adaptive estimation strategy provides a good compromise between robustness and efficiency compared with high-order methods, such as the Halley method, when the prior information of root weight is not available.