An Efficient Two-Point Iteration Scheme for Robust Approximation of Multiple Roots With Applications

In recent years, several optimal fourth-order iterative methods have been designed for finding multiple roots of nonlinear equations, assuming that the multiplicity m of the root is known in advance. These methods are advantageous in providing faster convergence when the multiplicity is specified but often rely on existing techniques that may limit flexibility or performance across diverse types of nonlinear problems. In contrast to these existing approaches, this paper introduces a novel iteration scheme for approximation of multiple roots based on a variant of Newton’s scheme and an analytic weight function. Specifically, it requires only one evaluation of the function f and two evaluations of the first derivative f′ per iteration, achieving an optimal order of convergence of four while keeping computational costs low. The theoretical advantages of this new family lie in its adaptability and streamlined structure, making it potentially robust across varying conditions and equations. Numerical tests and comparisons are provided on a variety of application-based problems, including isentropic supersonic flow, continuous stirred tank reactor, blood rheology model, osteoporosis in Chinese women, and eigenvalue problem, highlighting the competitive performance of the proposed scheme relative to some existing fourth-order methods. Compared to several existing fourth-order algorithms, the proposed method provides better results in terms of both computational efficiency and minimizing residual errors. Moreover, the numerical and theoretical results are validated by plotting the convergence regions of our iteration scheme in terms of phase portraits of attractor basins for the above-stated nonlinear problems. The phase portraits of attractor basins generated by the proposed iteration approach exhibit wider convergence regions in less time as compared to many existing techniques, which confirm its robustness and superiority.