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In 2013, Bai and Zhang (Numer Linear Algebra Appl 20:425-439, 2013) constructed modulus-based synchronous multisplitting methods by an equivalent reformulation of the linear complementarity problems into a system of fixed-point equations and studied their convergence. In 2014, Zhang and Li (Comput Math Appl 67:1954-1959, 2014) analyzed and obtained the weaker convergence results for linear complementarity problems. In this paper, based on their ideas, we further study modulus-based synchronous block multisplitting multi-parameter methods for linear complementarity problems. Furthermore, the convergence results of our new method in this paper are wider than those in literature when the system matrix is a block $$H_{+}$$ -matrix. Therefore, new results provide a guarantee for the optimal relaxation parameters. [ABSTRACT FROM AUTHOR]

In 2001, Chang studied the convergence of parallel multisplitting TOR method for H-matrices [D.W. Chang, The parallel multisplitting TOR(MTOR) method for linear systems, Comput. Math. Appl. 41 (2001), pp. 215-227]. In this paper, we point out some gaps in the proof of Chang's main results solving them. Moreover, we improve some of Chang's convergence results. A numerical example is presented in order to illustrate the improvement of Chang's convergence region. [ABSTRACT FROM AUTHOR]

Abstract: An improved generalized product-type bi-conjugate gradient (GPBi-CG) method (IGPBi-CG method, in brief) for solving large sparse linear systems with unsymmetrical coefficient matrices is proposed for distributed parallel environments. The method reduces three global synchronization points to two by reconstructing GPBi-CG method and the communication time required for the inner product can be efficiently overlapped with useful computation. The cost is only slightly increased computation time, which can be ignored compared with the reduction of communication time. Performance and isoefficiency analysis show that the IGPBi-CG method has better parallelism and scalability than the GPBi-CG method. Numerical experiments show that the scalability can be improved by a factor greater than 1.5 and the improvement in parallel communication performance approaches 33.3˙%. [Copyright &y& Elsevier]

Some preconditioners for accelerating the classical iterative methods are given in Zhang et al. [Y. Zhang and T.Z. Huang, A class of optimal preconditioners and their applications, Proceedings of the Seventh International Conference on Matrix Theory and Its Applications in China, 2006. Y. Zhang, T.Z. Huang, and X.P. Liu, Modified iterative methods for nonnegative matrices and M-matrices linear systems, Comput. Math. Appl. 50 (2005), pp. 1587-1602. Y. Zhang, T.Z. Huang, X.P. Liu, A class of preconditioners based on the (I+S(α))-type preconditioning matrices for solving linear systems, Appl. Math. Comp. 189 (2007), pp. 1737-1748]. Another kind of preconditioners approximating the inverse of a symmetric positive definite matrix was given in Simons and Yao [G. Simons, Y. Yao, Approximating the inverse of a symmetric positive definite matrix, Linear Algebra Appl. 281 (1998), pp. 97-103]. Zhang et al. 's preconditioners and Simons and Yao's are generalized in this paper. These preconditioners are all of low construction cost, which all could be taken as approximate inverse of M-matrices. Numerical experiments of these preconditioners applied with Krylov subspace methods show the effectiveness and performance, which also show that the preconditioners proposed in this paper are better approximate inverse for M-matrices than Simons'. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, based on GCRS algorithm in Zhang and Zhao (2010) and the ideas in Gu et al. (2007), we present an improved generalized conjugate residual squared (IGCRS) algorithm that is designed for distributed parallel environments. The new improved algorithm reduces two global synchronization points to one by changing the computation sequence in the GCRS algorithm in such a way that all inner products per iteration are independent so that communication time required for inner products can be overlapped with useful computation. Theoretical analysis and numerical comparison of isoefficiency analysis show that the IGCRS method has better parallelism and scalability than the GCRS method, and the parallel performance can be improved by a factor of about 2. Finally, some numerical experiments clearly show that the IGCRS method can achieve better parallel performance with a higher scalability than the GCRS method and the improvement percentage of communication is up to 52.19% averagely, which meets our theoretical analysis. [Copyright &y& Elsevier]

In this paper, based on local relaxed parallel multisplitting USAOR (LUSAOR) method presented by Zhang et al. (2008) and similar ideas used by Zhang and Li (2014), we further analyze local relaxed parallel multisplitting USAOR (LUSAOR) method and obtain the better convergence results compared to Zhang et al.’s when the system matrix is an H -matrix. Moreover, convergence graph and numerical examples clearly show that our new convergence domain is wider. [ABSTRACT FROM AUTHOR]

Abstract: Recently, a Newton’s iterative method is attracting more and more attention from various fields of science and engineering. This method is generally quadratically convergent. In this paper, some Chebyshev-type methods with the third order convergence are analyzed in detail and used to compute approximate inverse preconditioners for solving the linear system Ax = b. Theoretic analysis and numerical experiments show that Chebyshev’s method is more effective than Newton’s one in the case of constructing approximate inverse preconditioners. [Copyright &y& Elsevier]