Abstract
In this paper, we present a generalized SOR-like iteration method to solve the non-Hermitian positive definite linear complementarity problem (LCP), which is obtained by reformulating equivalently the implicit fixed-point equation of the LCP as a two-by-two block nonlinear equation. The convergence properties of the generalized SOR-like iteration method are discussed under certain conditions. Numerical experiments show that the generalized SOR-like method is efficient, compared with the SOR-like method and the modulus-based SOR method.
Highlights
Based on the implicit fixed-point equation of the linear complementarity problem (LCP)(q, M), a class of modulus iteration method in [7] and its various versions have been presented in the literature. e goal of modulus iteration method is to take z |x| + x and w |x| − x such that the LCP(q, M) can be equivalently transformed into a system of fixed-point equations: (I + M)x (I − M)|x| − q
For other forms of iteration methods, one can see [18,19,20,21,22]. We focus on this situation where the involved matrix M of the LCP(q, M) in (1) is non-Hermitian positive definite
By reformulating equivalently the implicit fixed-point equation of the LCP(q, M) as a two-by-two block nonlinear equation, based on the GSOR iteration method in [23], we extend the GSOR iteration method for the LCP(q, M) in (1) with its two-by-two block form. at is to say, we present a generalized SOR-like iteration method to solve the LCP(q, M). e convergence conditions of the generalized SOR-like iteration method are discussed under suitable choices of the involved parameter
Summary
The generalized SOR-like iteration method is established. To this end, we take z |x| + x and w Ω(|x| − x), where Ω is a nonnegative diagonal matrix, and the LCP(q, M) can be equivalently transformed into the following fixed-point equations:. When ω τ in (11), the generalized SOR-like iteration method reduces to the SOR-like method [24]. En, we give the following main result with respect to generalized SOR-like iteration method (11). exk, eyk e xk 2 + e yk 2 This implies that the generalized SOR-like method is convergent. Let M ∈ Rn×n be non-Hermitian positive definite and δ (Ω + M)− 1(Ω − M) This implies that the SOR-like method is convergent
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