High-efficiency parametric iterative schemes for solving nonlinear equations with and without memory

Many practical problems, such as the Malthusian population growth model, eigenvalue computations for matrices, and solving the Van der Waals' ideal gas equation, inherently involve nonlinearities. This paper initially introduces a two-parameter iterative scheme with a convergence order of two. Building on this, a three-parameter scheme with a convergence order of four is proposed. Then we extend these schemes into higher-order schemes with memory using Newton's interpolation, achieving an upper bound for the efficiency index of ${7.88748}^{\frac{1}{3}}\approx 1.99057$. Finally, we validate the new schemes by solving various numerical and practical examples, demonstrating their superior efficiency in terms of computational cost, CPU time, and accuracy compared to existing methods.