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In this paper, the linear quadratic N-players Nash games for infinite horizon large scale interconnected systems are discussed. The main contribution in this paper is that a new algorithm for solving the cross-coupled algebraic Riccati equations (CARE) is proposed. In order to improve the convergence rate and reduce the computing workspace, Newton's method and the fixed point algorithm are combined. As a result, it is newly proved that Nash equilibrium strategies obtained achieve a high-order approximation of the exact equilibrium. Furthermore, when the weak coupling parameter is unknown, it is shown that the proposed parameter independent Nash strategies are equivalent to the classical linear quadratic approximate controllers.