A robust iterative family for multiple roots of nonlinear equations: Enhancing accuracy and handling critical points.

Abstract

[EN]This research article introduces an iterative method that exhibits an optimal fourth-order convergence rate, ensuring rapid and accurate approximation of the roots. Unlike conventional methods, the proposed algorithm can successfully converge even when the derivative is zero or approaches zero in the vicinity of the desired root. This remarkable feature enhances the applicability of the method, allowing it to handle situations where conventional methods fail due to the presence of critical points such as the roots of . The convergence analysis of the proposed method is presented, showing its superior performance compared to other methods. Extensive numerical experiments are conducted to validate the efficiency and accuracy of the algorithm. The results indicate that the proposed iterative method not only achieves fast convergence but also exhibits robustness in handling various types of nonlinear equations. Its ability to converge even in the presence of zero or near-zero derivatives significantly expands the scope of applications, making it a valuable tool for solving complex problems in science and engineering.