Higher-order derivative-free families of Chebyshev–Halley type methods with or without memory for solving nonlinear equations

In this paper, we present two new derivative-free families of Chebyshev–Halley type methods for solving nonlinear equations numerically. Both families require only three and four functional evaluations to achieve optimal fourth and eighth orders of convergence. Furthermore, accelerations of convergence speed are attained by suitable variation of a free parameter in each iterative step. The self-accelerating parameter is estimated from the current and previous iteration. This self-accelerating parameter is calculated using Newton’s interpolation polynomial of third and fourth degrees. Consequently, the R-orders of convergence are increased from 4 to 6 and 8 to 12, respectively, without any additional functional evaluation. The results require high-order derivatives reaching up to the eighth derivative. That is why we also present an alternative approach using only the first or at most the fourth derivative. We also obtain the radius of convergence and computable error bounds on the distances involved. Numerical experiments and the comparison of the existing robust methods are included to confirm the theoretical results and high computational efficiency. In particular, we consider a concrete variety of real life problems coming from different disciplines e.g., Kepler’s equation of motion, Planck’s radiation law problem, fractional conversion in a chemical reactor, the trajectory of an electron in the air gap between two parallel plates, Van der Waal’s equation which explains the behavior of a real gas by introducing in the ideal gas equations, in order to check the applicability and effectiveness of our proposed methods.