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In this paper, we firstly present a block robust structured multifrontal factorization method (in brief, BRSMF) using block diagonalonal structure of three temperature matrices, and then we propose a multi-core parallelization of BRSMF (in brief, MBRSMF) method based on the current mainstream parallel computer multi-core architecture. MBRSMF method parallelizes the nested dissection ordering, symbolic decomposition and numerical decomposition of BRSMF method, which aims to effectively solve three temperature linear systems on the multi-core computer. The multi-core parallelization of symbolic decomposition and numerical decomposition is based on the binary elimination tree. Theoretical analysis proves MBRSMF method has better load balancing capability. Numerical experiments show that the MBRSMF method is effective.

In this paper, based on global relaxed parallel multisplitting USAOR (GUSAOR) method presented by Zhang et al. (Appl Math Comput 20:121–132, 2008) and similar ideas used by Zhang and Li (Comput Math Appl 67:1954–195, 2014), we further analyze global relaxed parallel multisplitting USAOR (GUSAOR) method and obtain the weaker 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.

In 2013, Bai and Zhang (Numer Linear Algebra Appl, 20:425–439 2013) constructed modulus-based synchronous multisplitting iteration methods by an equivalent reformulation of the linear complementarity problem into a system of fixed-point equations and studied the convergence of them. In this paper, we generalize Bai and Zhang’s method and study relaxed modulus-based synchronous multisplitting multi-parameter methods for linear complementarity problems. Furthermore, when the system matrix is an H+-matrix, we give convergence results for our new methods under weaker conditions than those in Bai and Zhang’s. Finally, numerical experiments are presented to illustrate the efficiency of the proposed methods.

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.

In this paper, based on generalized conjugate residual squared (GCRS2) algorithm in Zhang et al. (2010 Third International Conference on Information and Computing, pp 326–329, 2010) and the ideas in Gu et al. (Appl Math Comput 186:1243–1253, 2007), we present an improved generalized conjugate residual squared (IGCRS2) algorithm, which is designed for distributed parallel environments. The improved algorithm reduces two global synchronization points to one by changing the computation sequence in the GCRS2 algorithm and all inner products per iteration are independent and communication time required for inner product can be overlapped with useful computation. Theoretical analysis and numerical comparison about isoefficiency analysis show that the IGCRS2 method has better parallelism and scalability than the GCRS2 method and the parallel performance can be improved by a factor of about 2.

In this paper, one new parallel version of GPBi-CG method (PGPBi-CG method, in brief) is proposed for solving large sparse linear systems with unsymmetrical coefficient matrices on distributed parallel environments. The method reduces three global synchronization points to one by reconstructing GPBi-CG method and the communication time required for the inner product can be efficiently overlapped with computation time of vector updates. It combines the elements of numerical stability with the characteristics of design of parallel algorithms. The cost is only slightly increased computation time, which can be ignored, compared with the reduction of communication time. Performance and isoefficiency analysis shows that the PGPBi-CG method has better parallelism and scalability than the GPBi-CG method. Numerical experiments show that the scalability can be improved by a factor 3 and the improvement in parallel communication performance approaches 66.7 %.