In this paper, N-player linear quadratic differential games that are sign-indefinite for infinite horizon weakly coupled large-scale systems are discussed. After establishing the asymptotic structure and local uniqueness of the solution for cross-coupled sign-indefinite algebraic Riccati equations (CSARE), a new algorithm for solving CSARE is provided. It is shown that the proposed algorithm attains linear convergence. Moreover, in order to reduce the computational workspace, the recursive algorithm is combined. Finally, a high-order approximation strategy based on the proposed iterative solutions is described. As a result, it was recently proved that the numerical strategy achieves a high-order approximation of the equilibrium value. As another important feature, when the small parameters are unknown, a parameter-independent strategy is developed.