Local convergence study of tenth-order iterative method in Banach spaces with basin of attraction

Many applications from computational mathematics can be identified for a system of non-linear equations in more generalized Banach spaces. Analytical methods do not exist for solving these type of equations, and so we solve these equations using iterative methods. We introduced a new numerical technique for finding the roots of non-linear equations in Banach space. The method is tenth-order and it is an extension of the fifth-order method which is developed by Arroyo et.al. ^[1]. We provided a convergence analysis to demonstrate that the method exhibits tenth-order convergence. Also, we discussed the local convergence properties of the suggested method which depends on the fundamental supposition that the first-order Fréchet derivative of the involved function $\mathrm{{\rm Y}}$ satisfies the Lipschitz conditions. This new approach is not only an extension of prior research, but also establishes a theoretical concept of the radius of convergence. We validated the efficacy of our method through various numerical examples. Our method is comparable with the methods of Tao Y et al. ^[2]. We also compared it with higher-order iterative methods, and we observed that it either performs similarly or better for the numerical examples. We also gave the basin of attraction to demonstrate the behaviour in the complex plane.