This paper presents a novel multi-step iterative scheme to solve system of nonlinear equa- tions. Because the cost of calculating the Fréchet derivative evaluation and its inversion is significant, in order to achieve its high computational efficiency, we have tried to calculate Fréchet derivative and its inverse less per cycle by using the proposed multi-step iterative schemes. The basic iterative scheme has a convergence order of four; therefore, repetition of the second step can achieve higher convergence order. In fact, adding a new step to the base iterative scheme each time increases the convergence order by two units. The multi-step iterative schemes have convergence-order 2 m ,where m is the number of steps of the multi- step iterative schemes. Also, the computational efficiency of the iterative scheme is compared with other available methods. The numerical results presented confirm the theoretical results. A number of nonlinear system of equations associated with the numerical approximation of ordinary, partial, and fractional differential equations are made up and solved.
