Improved modulus-based matrix splitting iteration method for horizontal implicit complementarity problems

In this paper, based on the implicit fixed-point equation of the horizontal implicit complementarity problem (HICP), we modify several steps in the framework of the modulus-based matrix splitting (MMS) iteration method (Linear and Multilinear Algebra, 71: 2392–2408, 2023) for the HICP, and construct an improved modulus-based matrix splitting (IMMS) iteration method for the HICP. Compared with the MMS method proposed recently, it is not needed to solve a linear system with coefficient matrix A at each iteration of the IMMS method, which leads to less computation time and faster convergence. Then the proposed IMMS method may exhibit better numerical performance than the MMS in practical calculations. We analyze the convergence conditions of the IMMS iteration method when the system matrices are positive definite or \(H_{+}\) $H_{+}$ -matrices. Also, we deduce the choice for the parameter matrix \(\Omega \) $\mathrm{\Omega }$ in the IMMS method, which is vital to its implementation. In addition, the algebra properties of the IM accelerated overrelaxation (IMAOR) method are established. Finally, some numerical examples are given to validate the efficiency and advantage of the IMMS method for the HICP.