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Chao Zhang

Yanfang Zhang

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Qiang Ye Show more... Subscribe SciFeed Related Info Link Google Scholar More by Authors Links on DOAJ Li, C. Wu, S. on Google Scholar Li, C. Wu, S. on PubMed Li, C. Wu, S. /ajax/scifeed/subscribe Table of Contents Altmetric share Share announcement Help format_quote Cite question_answer Discuss in SciProfiles thumb_up ... Endorse textsms ... Comment Need Help? Support

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Get Information clear JSmol Viewer clear first_page settings Order Article Reprints Font Type: Arial Georgia Verdana Font Size: Aa Aa Aa Line Spacing:    Column Width:    Background: Open AccessCommunication An Improved Convergence Condition of the MMS Iteration Method for Horizontal LCP of H+-Matrices by

Cuixia Li and

Shiliang Wu * School of Mathematics, Yunnan Normal University, Kunming 650500, China * Author to whom correspondence should be addressed. Mathematics 2023, 11(8), 1842; https://doi.org/10.3390/math11081842 Received: 15 March 2023 / Revised: 5 April 2023 / Accepted: 10 April 2023 / Published: 13 April 2023 (This article belongs to the Special Issue Optimization Theory, Method and Application) Download Download PDF Download PDF with Cover Download XML Versions Notes

Abstract: In this paper, inspired by the previous work in (Appl. Math. Comput., 369 (2020) 124890), we focus on the convergence condition of the modulus-based matrix splitting (MMS) iteration method for solving the horizontal linear complementarity problem (HLCP) with H + -matrices. An improved convergence condition of the MMS iteration method is given to improve the range of its applications, in a way which is better than that in the above published article. Keywords: horizontal linear complementarity problem; H+-matrix; the MMS iteration method MSC: 65F10; 90C33 1. IntroductionAs is known, the horizontal linear complementarity problem, for the given matrices A , B ∈ R n × n , is to find that two vectors z , w ∈ R n satisfy A z = B w + q ≥ 0 , z ≥ 0 , w ≥ 0 and z T w = 0 , where q ∈ R n is given, which is often abbreviated as HLCP. If A = I in (1), the HLCP (1) is no other than the classical linear complementarity problem (LCP) in [1], where I denotes the identity matrix. This implies that the HLCP (1) is a general form of the LCP.The HLCP (1), used as a useful tool, often arises in a diverse range of fields, including transportation science, telecommunication systems, structural mechanics, mechanical and electrical engineering, and so on, see [2,3,4,5,6,7]. In the past several years, some efficient algorithms have been designed to solve the HLCP (1), such as the interior point method [8], the neural network [9], and so on. Particularly, in [10], the modulus-based matrix splitting (MMS) iteration method in [11] was adopted to solve the HLCP (1). In addition, the partial motivation of the present paper is from complex systems with matrix formulation, see [12,13,14] for more details.Recently, Zheng and Vong [15] further discussed the MMS method, as described below.The MMS method [10,15]. Let Ω be a positive diagonal matrix and r > 0 , and let A = M A − N A and B = M B − N B be the splitting of matrices A and B, respectively. Assume that ( z ( 0 ) , w ( 0 ) ) is an arbitrary initial vector. For k = 0 , 1 , 2 , … until the iteration sequence ( z ( k ) , w ( k ) ) converges, compute ( z ( k + 1 ) , w ( k + 1 ) ) by z ( k + 1 ) = 1 r ( | x ( k + 1 ) | + x ( k + 1 ) ) , w ( k + 1 ) = 1 r Ω ( | x ( k + 1 ) | − x ( k + 1 ) ) , where x ( k + 1 ) is obtained by ( M A + M B Ω ) x ( k + 1 ) = ( N A + N B Ω ) x ( k ) + ( B Ω − A ) | x ( k ) | + r q . For the later discussion, some preliminaries are gone over. For a square matrix A = ( a i j ) ∈ R n × n , | A | = ( | a i j | ) , and 〈 A 〉 = ( 〈 a i j 〉 ) , where 〈 a i i 〉 = | a i i | and 〈 a i j 〉 = − | a i j | for i ≠ j . A matrix A = ( a i j ) ∈ R n × n is called a non-singular M-matrix if A − 1 ≥ 0 and a i j ≤ 0 for i ≠ j ; an H-matrix if its comparison matrix 〈 A 〉 is a non-singular M-matrix; an H + -matrix if it is an H-matrix with positive diagonals; and a strictly diagonally dominant (s.d.d.) matrix if | a i i | > ∑ j ≠ i | a i j | , i = 1 , 2 , … , n . In addition, A ≥ ( > ) B with A , B ∈ R n × n , means a i j ≥ ( > ) b i j for i , j = 1 , 2 , … , n .For the MMS method with H + -matrix, two new convergence conditions are obtained in [15], which are weaker than the corresponding convergence conditions in [10]. One of these is given below.Theorem 1 ([15]). Assume that A , B ∈ R n × n are two H + -matrices and Ω = d i a g ( ω j j ) ∈ R n × n with ω j j > 0 , i , 2 , … , n , | b i j | ω j j ≤ | a i j | ( i ≠ j ) a n d s i g n ( b i j ) = s i g n ( a i j ) , b i j ≠ 0 . Let A = M A − N A be an H-splitting of A, B = M B − N B be an H-compatible splitting of B, and M A + M B Ω be an H + -matrix. Then the MMS method is convergent, provided one of the following conditions holds:(a) Ω ≥ D A D B − 1 ;(b) Ω

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