Higher order multi-step iterative method for computing the numerical solution of systems of nonlinear equations: Application to nonlinear PDEs and ODEs

In the present study, we consider multi-step iterative method to solve systems of nonlinear equations. Since the Jacobian evaluation and its inversion are expensive, in order to achieve a better computational efficiency, we compute Jacobian and its inverse only once in a single cycle of the proposed multi-step iterative method. Actually the involved systems of linear equations are solved by employing the LU-decomposition, rather than inversion. The primitive iterative method (termed base method) has convergence-order (CO) five and then we describe a matrix polynomial of degree two to design a multi-step method. Each inclusion of single step in the base method will increase the convergence-order by three. The general expression for CO is 3s-1, where s is the number of steps of the multi-step iterative method. Computational efficiency is also addressed in comparison with other existing methods. The claimed convergence-rates proofs are established. The new contribution in this article relies essentially in the increment of CO by three for each added step, with a comparable computational cost in comparison with existing multi-steps iterative methods. Numerical assessments are made which justify the theoretical results: in particular, some systems of nonlinear equations associated with the numerical approximation of partial differential equations (PDEs) and ordinary differential equations (ODEs) are built up and solved