Abstract
Abstract In this paper, based on the relationship between the linear complementarity problem and its reformulated fixed-point equation, we discuss the conditions of the modulus-based type iteration methods. Moreover, we present some convergence results on the two-step modulus-based matrix splitting iteration method with an H + {H}_{+} -matrix. Finally, we give the numerical experiments.
Highlights
The linear complementarity problem is to solve z ∈ Rn such that zTr = 0 with z ≥ 0, r = Az + q ≥ 0, (1)where A = ∈ Rn×n, q ∈ Rn are given and the symbol “T” denotes the transpose operation
In order to compute the numerical solution of the LCP(A, q) with a large and sparse system matrix A, many kinds of modulus-based type iteration methods are presented recently, such as the modulus-based iteration method [8], the nonstationary extrapolated modulus method [9], the modified modulus-based iteration method [10], the modulus-based matrix splitting iteration method [3], the general modulus-based matrix splitting method [6], the two-step modulus-based matrix splitting iteration method [11], the twosweep modulus-based matrix splitting iteration method [12], the preconditioned general modulus-based matrix splitting method [13], the accelerated modulus-based matrix splitting iteration methods [14,15], and so on
By studying the connection between the LCP(A, q) and the fixed-point equation, we propose the conditions of the modulus-based type iteration method, which extends the application scope of this method
Summary
We further study the modulus-based type iteration methods for solving the LCP(A, q) in this paper. By studying the connection between the LCP(A, q) and the fixed-point equation, we propose the conditions of the modulus-based type iteration method, which extends the application scope of this method. For the two-step modulus-based matrix splitting iteration method [11], the convergence has been further studied in [16] Since this method is very effective, it has been applied to solve other complementarity problems. For the recent works on the two-step modulus-based methods for solving other complementarity problems, readers can refer to [29,30,31,32,33,34,35]. We provide the numerical experiments to verify the proposed results and illustrate the special cases of this method
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