Generalized conformable fractional Newton-type method for solving nonlinear systems

In a recent paper, a conformable fractional Newton-type method was proposed for solving nonlinear equations. This method involves a lower computational cost compared to others fractional iterative methods. Indeed, the theoretical order of convergence is held in practice, and it presents a better numerical behaviour than fractional Newton-type methods formerly proposed, even compared to classical Newton-Raphson method. In this work, we design a generalization of this method for solving nonlinear systems by using a new conformable fractional Jacobian matrix, and a suitable conformable Taylor power series; and it is compared with classical Newton’s scheme. The necessary concepts and results are stated in order to design this method. Convergence analysis is made and a quadratic order of convergence is obtained, as in classical Newton's method. Numerical tests are made, and the Approximated Computational Order ofo Convergence (ACOC) supports the theory. Also, the proposed scheme shows good stability properties observed by means of convergence planes.