Modulus-based matrix splitting iteration methods with new splitting scheme for horizontal implicit complementarity problems

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In this paper, we propose the horizontal implicit complementarity problems, which cover many complementarity problems that have wide applications. After reformulating the horizontal implicit complementarity problem as an implicit fixed-point equation, the modulus-based matrix splitting iteration methods are applied to solve the problems. Besides, this paper introduces a new kind of matrix splitting, called modulus-based Hermitian and skew-Hermitian matrix relaxation splitting. The corresponding modulus-based matrix splitting iteration method is shown superior to the nonsmooth Newton's method and the existing matrix splitting schemes by some numerical experiments. The convergence analysis is given and validates that our methods are more practical in applications.