A broad range of scientific simulations involve solving large-scale computationally expensive linear systems of equations. Iterative solvers are typically preferred over direct methods when it comes to large systems due to their lower memory requirements and shorter execution times. However, selecting the appropriate iterative solver is problem-specific and dependent on the type and symmetry of the coefficient matrix. Gauss-Seidel (GS) is an iterative method for solving linear systems that are either strictly diagonally dominant or symmetric positive definite. This technique is an improved version of Jacobi and typically converges in fewer iterations. However, the sequential nature of this algorithm complicates the parallel extraction. In fact, most parallel derivatives of GS rely on the sparsity pattern of the coefficient matrix and require matrix reordering or domain decomposition. In this article, we introduce a new algorithm that exploits the convergence property of GS and adapts the parallel structure of Jacobi. The proposed method works for both dense and sparse systems and is straightforward to implement. We have examined the performance of our method on multicore and many-core architectures. Experimental results demonstrate the superior performance of the proposed algorithm compared with GS and Jacobi. Additionally, performance comparison with built-in Krylov solvers in MATLAB showed that in terms of time per iteration, Krylov methods perform faster on CPUs, but our approach is significantly better when executed on GPUs. Lastly, we apply our method to solve the power flow problem, and the results indicate a significant improvement in runtime, reaching up to 87 times faster speed compared with GS.

中文翻译：

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并行雅可比嵌入高斯-赛德尔方法


广泛的科学模拟涉及求解大规模计算成本昂贵的线性方程组。对于大型系统，迭代求解器通常比直接方法更受青睐，因为迭代求解器的内存需求较低且执行时间较短。然而，选择适当的迭代求解器是针对特定问题的，并且取决于系数矩阵的类型和对称性。 Gauss-Seidel (GS) 是一种迭代方法，用于求解严格对角占优或对称正定的线性系统。该技术是雅可比的改进版本，通常在更少的迭代中收敛。然而，该算法的顺序性质使并行提取变得复杂。事实上，GS 的大多数并行导数依赖于系数矩阵的稀疏模式，并且需要矩阵重新排序或域分解。在本文中，我们介绍了一种新算法，该算法利用 GS​​ 的收敛特性并采用 Jacobi 的并行结构。所提出的方法适用于密集和稀疏系统，并且易于实现。我们检查了我们的方法在多核和众核架构上的性能。实验结果表明，与GS和Jacobi相比，该算法具有优越的性能。此外，与 MATLAB 中内置 Krylov 求解器的性能比较表明，就每次迭代的时间而言，Krylov 方法在 CPU 上执行速度更快，但我们的方法在 GPU 上执行时明显更好。最后，我们应用我们的方法来解决潮流问题，结果表明运行时间显着改善，速度比 GS 快了 87 倍。