彭小飞, 余文松, 陈饶杰. 求解绝对值方程组的广义SOR型方法[J]. 华南师范大学学报(自然科学版), 2024, 56(1): 104-111. doi: 10.6054/j.jscnun.2024012
引用本文: 彭小飞, 余文松, 陈饶杰. 求解绝对值方程组的广义SOR型方法[J]. 华南师范大学学报(自然科学版), 2024, 56(1): 104-111. doi: 10.6054/j.jscnun.2024012
PENG Xiaofei, YU Wensong, CHEN Raojie. Generalized SOR-like Methods for Solving Absolute Value Equations[J]. Journal of South China Normal University (Natural Science Edition), 2024, 56(1): 104-111. doi: 10.6054/j.jscnun.2024012
Citation: PENG Xiaofei, YU Wensong, CHEN Raojie. Generalized SOR-like Methods for Solving Absolute Value Equations[J]. Journal of South China Normal University (Natural Science Edition), 2024, 56(1): 104-111. doi: 10.6054/j.jscnun.2024012

求解绝对值方程组的广义SOR型方法

Generalized SOR-like Methods for Solving Absolute Value Equations

  • 摘要: 为了求解大规模的绝对值方程Ax-|x|=b, 利用预处理技术及参数矩阵取代单参数的策略, 文章提出了一类广义SOR型(GSOR) 方法。通过选取适当的预处理矩阵或参数, GSOR方法能简化为已有的一种SOR型(NSOR)方法或导出更有效的SOR型方法。而且, 基于Ax-|x|=b方程解的唯一性条件, 建立了GSOR方法的收敛性定理并给出了该方法的拟最优参数。特别地, 利用截断的Neumann展开构建了一个新的预处理矩阵, 由此导出了一种特殊的GSOR方法,记为GSOR-1方法。文章进一步证明: GSOR-1方法具有比NSOR方法更小的拟最优收敛因子。数值测试进一步揭示: GSOR-1方法比NSOR方法具有更快的收敛速度且耗费更少的计算时间。

     

    Abstract: For solving the absolute value equations Ax-|x|=b, a generalized SOR-like (GSOR) method is proposed by introducing the preconditioning matrix and using the relaxation parameter matrix instead of a single relaxation parameter. With the appropriate preconditioning matrices or parameters, the GSOR method can reduce to an existing SOR-like (NSOR) method or lead to some new SOR-like methods. Moreover, based on the unique solvability of Ax-|x|=b, the convergence theory of the GSOR method is established and its quasi-optimal parameters are given. In particular, by using truncated Neumann expansion, a new preconditioning matrix is constructed and a special GSOR method(GSOR-1) is derived. It has been proved that the GSOR-1 method has the smaller convergence factor than the NSOR method. Numerical tests further reveal that the GSOR-1 method has faster convergence rate and costs less computational times than the NSOR method.

     

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