Third Order Methods for Multiple Roots of Nonlinear Equations  with Applications

Several new third-order iterative methods for finding multiple roots of nonlinear equations are developed and systematically compared with well-known existing schemes. These methods leverage knowledge of the root multiplicity and include generalizations of Newton, Halley, and Euler-Cauchy-type methods. Both qualitative and quantitative analyses are conducted: basins of attraction are employed to visualize convergence behavior in the complex plane, and metrics such as iteration count, divergence rate, and computational efficiency are systematically recorded. Among the 29 constructed methods, nine demonstrate competitive performance with classical approaches. One method consistently outperforms classical approaches, achieving the shortest average CPU time across all tested examples. The results show that careful parameter selection within the general iterative framework leads to robust and efficient methods for multiple roots, with potential applications in engineering problems such as supersonic flow and complex fluid modeling as demonstrated in the last two examples from chemistry. MSC Classification: 65H05 , 65B99 , 65Y20