求解绝对值方程组的广义SOR型方法

Generalized SOR-like Methods for Solving Absolute Value Equations

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。

For solving the absolute value equations Ax-|x|=b,a generalized SOR-like(GSOR)method is pro⁃posed by introducing the preconditioning matrix and using the relaxation parameter matrix instead of a single relaxa⁃tion 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 solva⁃bility 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 conver⁃gence 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.