In this paper, we consider numerical methods for the linear complementarity problem (LCP). By introducing a positive diagonal parameter matrix, the LCP is transformed into an equivalent fixed-point equation and the equivalence is proved. Based on such equation, the general fixed-point (GFP) method with two cases are proposed and analyzed when the system matrix is a $ P $-matrix. In addition, we provide several concrete sufficient conditions for the proposed method when the system matrix is a symmetric positive definite matrix or an $ H_{+} $-matrix. Meanwhile, we discuss the optimal case for the proposed method. The numerical experiments show that the GFP method is effective and practical.

For the large sparse linear complementarity problem, by reformulating it as an implicit fixed point equation problem, Bai propose a class of modulus-based matrix splitting iteration methods in [12]. In this paper, we discuss one form of these methods-the general modulus-based Jacobi iteration method, proved the convergence, and derive the domain and the optimum value of the parameter for one special situation. Numerical results show that this method is superior to some modulus-related methods in computing efficiency and feasible aspects in some situations.