Derivative Free King's-Type Family for Solving System of Nonlinear Equations

There has been very less work done on developing methods for solving systems of nonlinear equations using the idea of memorization. Methods without memory can be extended to methods with memory without using any additional function evaluations. Improvement in the convergence order of methods with memory gives an added benefit. The stability of such methods is another important concept which needs to be considered. In this work, we deal with the development of such methods. We first construct a sixth-order derivative free family without memory and then further extend it to methods with memory of order 7.40, 7.87 and 8.77. Numerical results show the superiority of these methods over the existing ones. Basins of attraction have also been presented to observe the dynamical behavior of the proposed methods and the comparison validates that the proposed methods are much more stable than the existing methods.