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Optimal Derivative-Free One-Point Algorithms for Computing Multiple Zeros of Nonlinear Equations

In this paper, we describe iterative derivative-free algorithms for multiple roots of a nonlinear equation. Many researchers have evaluated the multiple roots of a nonlinear equation using the first- or second-order derivative of functions. However, calculating the function’s derivative at each iteration is laborious. So, taking this as motivation, we develop second-order algorithms without using the derivatives. The convergence analysis is first carried out for particular values of multiple roots before coming to a general conclusion. According to the Kung–Traub hypothesis, the new algorithms will have optimal convergence since only two functions need to be evaluated at every step. The order of convergence is investigated using Taylor’s series expansion. Moreover, the applicability and comparisons with existing methods are demonstrated on three real-life problems (e.g., Kepler’s, Van der Waals, and continuous-stirred tank reactor problems) and three standard academic problems that contain the root clustering and complex root problems. Finally, we see from the computational outcomes that our approaches use the least amount of processing time compared with the ones already in use. This effectively displays the theoretical conclusions of this study.