A general family of derivative free with and without memory root finding methods

In this manuscript, we construct a general family of optimal derivative free iterative methods by using rational interpolation. This family is further extended to a family of with-memory methods with increased order of convergence by employing two free parameters. At each iterative step, we use a suitable variation of the free parameters. These parameters are computed by using the information from current and previous iterations so that the convergence order of the existing family is increased from \(2^{n}\) to \(2^{n}+2^{n-1}+2^{n-2}\) without using any additional function evaluations. To check the performance of newly developed iterative schemes with and without memory, an extensive comparison with the existing with- and without memory methods is done by taking some real world problems and standard nonlinear functions. Numerical experiments illustrate that the proposed family of methods with-memory retain better computational efficiency and fast convergence speed as compared to existing with- and without memory methods. The performance of the methods is also analyzed visually by using complex plane. Numerical and dynamical comparisons confirm that the proposed families of with and without memory methods have better efficiency, convergence regions and speed in contrast with the existing methods of the same kind.