Solving systems of nonlinear equations when the nonlinearity is expensive

Construction of multi-step iterative method for solving system of nonlinear equations is considered, when the nonlinearity is expensive. The proposed method is divided into a base method and multi-step part. The convergence order of the base method is five, and each step of multi-step part adds additive-factor of five in the convergence order of the base method. The general formula of convergence order is 5(m−2)5(m−2) where m(≥3)m(≥3) is the step number. For a single instance of the iterative method we only compute two Jacobian and inversion of one Jacobian is required. The direct inversion of Jacobian is avoided by computing LU factors. The computed LU factors are used in the multi-step part for solving five systems of linear equations that make the method computational efficient. The distinctive feature of the underlying multi-step iterative method is the single call to the computationally expensive nonlinear function and thus offers an increment of additive-factor of five in the convergence order per single call. The numerical simulations reveal that our proposed iterative method clearly shows better performance, where the computational cost of the involved nonlinear function is higher than the computational cost for solving five lower and upper triangular systems.