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 7.8874813≈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.