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Showing papers by "Zhong-Zhi Bai published in 1996"


Journal ArticleDOI
TL;DR: A unified framework for the construction of various synchronous and asynchronous parallel matrix multisplitting iterative methods, suitable to the SIMD and MIMD multiprocessor systems, is presented, and its convergence theory is established under rather weak conditions.
Abstract: A unified framework for the construction of various synchronous and asynchronous parallel matrix multisplitting iterative methods, suitable to the SIMD and MIMD multiprocessor systems, respectively, is presented, and its convergence theory is established under rather weak conditions. These afford general method models and systematical convergence criterions for studying the parallel iterations in the sense of matrix multisplitting. In addition, how the known parallel matrix multisplitting iterative methods can be classified into this new framework, and what novel ones can be generated by it are shown in detail.

95 citations


Journal ArticleDOI
TL;DR: This new algorithm, when its relaxation parameters are suitably chosen, can not only afford extensive choices for parallely solving the linear complementarity problems, but also can greatly improve the convergence property of itself.
Abstract: For the linear complementarity problem, we set up a class of parallel matrix multisplitting accelerated overrelaxation (AOR) algorithm suitable to multiprocessor systems (SIMD-systems). This new algorithm, when its relaxation parameters are suitably chosen, can not only afford extensive choices for parallely solving the linear complementarity problems, but also can greatly improve the convergence property of itself. When the system matrices of the problems are either H -matrices with positive diagonal elements or symmetric positive definite matrices, we establish convergence theories of the new algorithm in a detailed manner.

58 citations


Journal ArticleDOI
TL;DR: In this article, a class of parallel nonlinear AOR methods for solving the large scale system of nonlinear equations with continuous diagonal matrix splitting is presented, and the global and monotone convergence of the method is proved.
Abstract: We set up a class of parallel nonlinear AOR method in the sense of matrix multi-splitting for solving the large scale system of nonlinear equations Ax + ϕ(x) = b with A ∈ L(Rn) nonsingular, b ∈ Rn and ϕ : Rn → Rn being continuously diagonal. The global as well as the monotone convergence of this method is proved.

41 citations


Journal ArticleDOI
TL;DR: In this article, a comparison theorem about the parallel nonlinear AOR method is presented, which describes the influence of either the multisplitting of the coefficient matrix or the pair of the relaxation parameters on the convergence rate of this method.
Abstract: A new comparison theorem about the parallel nonlinear AOR method [1] is set up, which describes in detail the influence of either the multisplitting of the coefficient matrix or the pair of the relaxation parameters on the convergence rate of this method.

21 citations


Journal ArticleDOI
TL;DR: The two-sided approximation properties and the influences on the convergence rates from the relaxation parameters about the new methods about the authors' new methods are shown, and sufficient conditions guaranteeing the methods to converge globally are discussed.
Abstract: We set up a class of parallel nonlinear multisplitting AOR methods by directly multisplitting the nonlinear mapping involved in the nonlinear complementarity problems. The different choices of the relaxation parameters can yield all the known and a lot of new relaxation methods, as well as a lot of new relaxed parallel nonlinear multisplitting methods for solving the nonlinear complementarity problems. The two-sided approximation properties and the influences on the convergence rates from the relaxation parameters about our new methods are shown, and sufficient conditions guaranteeing the methods to converge globally are discussed. Finally, a lot of numerical results show that our new methods are feasible and efficient.

17 citations


Journal ArticleDOI
TL;DR: A class of parallel nonlinear multisplitting successive overrelaxation methods for solving the large sparse nonlinear complementarity problems on the modern high-speed multiprocessors is presented.
Abstract: By making use of the nonlinear multisplitting and the nonlinear relaxation techniques, we present, in this paper, a class of parallel nonlinear multisplitting successive overrelaxation methods for solving the large sparse nonlinear complementarity problems on the modern high-speed multiprocessors. These new methods particularly include the so-called nonlinear multisplitting SOR-Newton method. Under suitable conditions, we establish the local convergence theories of the new methods, and investigate their asymptotic convergence rates. A lot of numerical results show that our new methods are feasible and efficient for parallel solving the nonlinear complementarity problems.

16 citations


Journal ArticleDOI
TL;DR: In this article, a class of hybrid algebraic multilevel preconditioning methods for solving systems of linear equations with symmetric positive-definite matrices resulting from the discretization of many second-order elliptic boundary value problems by the finite element method is presented.

13 citations


Journal ArticleDOI
TL;DR: In this article, a comparison theorem on the monotone convergence rates of the parallel nonlinear multisplitting accelerated overrelaxation (AOR) method for solving the large scale nonlinear complementarity problem is established.
Abstract: A new comparison theorem on the monotone convergence rates of the parallel nonlinear multisplitting accelerated overrelaxation (AOR) method for solving the large scale nonlinear complementarity problem is established. Thus, the monotone convergence theory of this class of method is completed.

10 citations


Journal ArticleDOI
TL;DR: The localQ-superlinear convergence of the algorithm is proved without introducing anm-step refactorization and the numerical results of the new algorithm are compared with those of the known algorithms, implying that the new algorithms is satisfactory.
Abstract: In this paper, we establish a class of sparse update algorithm based on matrix triangular factorizations for solving a system of sparse equations. The localQ-superlinear convergence of the algorithm is proved without introducing anm-step refactorization. We compare the numerical results of the new algorithm with those of the known algorithms, The comparison implies that the new algorithm is satisfactory.

6 citations


Journal ArticleDOI
TL;DR: In this paper, a comparison theorem is given for the nonlinear multisplitting relaxation method, and an important modification is proposed for it, and a modification for it is presented.
Abstract: A new comparison theorem is given for the nonlinear multisplitting relaxation method [1], and an important modification is proposed for it, too

4 citations