Global relaxed non-stationary multisplitting multi-parameter methods

Abstract

Based on the local relaxed method and the system relaxed method of parallel multisplitting described by Frommer and Mayer, the paper presents a global relaxed non-stationary multisplitting multi-parameter method by introducing some relaxed parameters. Convergence of these methods when coefficient matrices are H -matrices is investigated. Efficiency of the global relaxed non-stationary multisplitting multi-parameter method is shown by numerical experiments. These indicate that, when choosing the approximately optimal relaxed parameters, the methods have a faster convergent rate than previous methods.

Global Modulus-Based Synchronous Multisplitting Multi-Parameters TOR Methods for Linear Complementarity Problems

“…Obviously, one can find that the conditions of Theorem 4 in this paper are wider than those of Theorem 2.3 in [28]. Furthermore, we have more choices for the splitting A = B − C which makes multisplitting iterative methods converge.…”

“…In 2008, based on global relaxed non-stationary multisplitting multi-parameter TOR algorithm (GRNMMTOR) for the large sparse linear system [26], Zhang, Huang and Gu [28] got the corresponding theorem: Theorem 3. Let A be an H-matrix, and for k = 1, 2, ..., l, L k and F k be strictly lower triangular matrices [26].…”

A Note on the Inner-Outer Iterative Method for Solving the Linear Equation Ax = b

“…Among them, Prof. Bai [2][3][4][5][6][7][8][9] did a mountain of great work and constructed the parallel nonlinear AOR method about matrix multisplitting, the parallel chaotic multisplitting method, the two-stage multisplitting method under suitable constraints about two-stage multisplitting, some new hybrid algebraic multilevel preconditioning algorithms, nonstationary multisplitting iterative algorithms, and the nonstationary multisplitting two-stage iterative algorithms. Apart from these methods, Gu et al [13,14], Cao et al [16][17][18], Wang et al [15,24,26,27,29,30], and Zhang et al [28,29,31] also constructed relaxed nonstationary two-stage multisplitting algorithms, nested stationary iterative algorithms, relaxed parallel multisplitting AOR, USAOR, and SSOR algorithms on an H-matrix, two relaxed multisplitting algorithms for different weighting types when A is a monotone matrix or M-matrix, the parallel multisplitting TOR algorithm, and the global relaxed parallel multisplitting USAOR (GUSAOR) algorithm. Recently, Tian et al [22] studied the inner-outer iterative method for the linear equation Ax � b and deduced the corresponding convergence of the inner-outer algorithm.…”

A note on parallel multisplitting TOR method forH-matrices

“…The concept of multisplitting for the parallel solution of linear system was introduced by O'Leary and White [9] and further studied by many authors [3][4][5][6][7][8]10,[12][13][14]. The multisplitting method can be considered an extension and parallel generalization of the classical block Jacobi method [2].…”