论文标题
Sor样迭代方法的最佳参数，用于求解绝对值方程的系统
Optimal parameter for the SOR-like iteration method for solving the system of absolute value equations
论文作者
论文摘要
求解的类似SOR的迭代方法用于查找向量$ x $的绝对值方程〜（ave），使得$ ax -| x | | -b = 0 $ with $ν= \ | a^{ - 1} \ | _2 <1 $。重新审视了KE和MA提出的类似SOR状迭代方法的收敛条件（[{\ emAppl。Math。Comput。}，311：195--202，2017]），并给出了一个新的证明，这在确定收敛区域和最佳迭代参数方面表现出了一些见解。沿着这一行，最佳参数将$ \ |t_ν（ω）\ | _2 $与$$t_ν（ω）= \ left（\ begin {array} {array} {cc} | 1-ω|＆ω|＆ω^2ν\ \ \ \\ | 1- | 1- | 1- | 1- | 1- | 1- | 1- |＆| 1- | | +e | +optiation探索了最小化$η_ν（ω）= \ max \ {| 1-Ω|，νΩ^2 \} $。最佳和近似的最佳参数是迭代无关的，$ν$的较大值是迭代参数$ω$的较小收敛区域。提出了数值结果，以证明具有最佳参数的SOR样迭代方法优于guo，wu和li提出的近似最佳参数（[[{\ emAppl。Math。Lett。}，97：107--113，2019]）。在某些情况下，与CPU时间相比，具有最佳参数的SOR样iTration方法比广义的Newton方法（Mangasarian，[{\ emOptim。Lett。}，3：101--108，2009]）用于求解AVE。
The SOR-like iteration method for solving the absolute value equations~(AVE) of finding a vector $x$ such that $Ax - |x| - b = 0$ with $ν= \|A^{-1}\|_2 < 1$ is investigated. The convergence conditions of the SOR-like iteration method proposed by Ke and Ma ([{\em Appl. Math. Comput.}, 311:195--202, 2017]) are revisited and a new proof is given, which exhibits some insights in determining the convergent region and the optimal iteration parameter. Along this line, the optimal parameter which minimizes $\|T_ν(ω)\|_2$ with $$T_ν(ω) = \left(\begin{array}{cc} |1-ω| & ω^2ν\\ |1-ω| & |1-ω| +ω^2ν\end{array}\right)$$ and the approximate optimal parameter which minimizes $η_ν(ω) =\max\{|1-ω|,νω^2\}$ are explored. The optimal and approximate optimal parameters are iteration-independent and the bigger value of $ν$ is, the smaller convergent region of the iteration parameter $ω$ is. Numerical results are presented to demonstrate that the SOR-like iteration method with the optimal parameter is superior to that with the approximate optimal parameter proposed by Guo, Wu and Li ([{\em Appl. Math. Lett.}, 97:107--113, 2019]). In some situation, the SOR-like itration method with the optimal parameter performs better, in terms of CPU time, than the generalized Newton method (Mangasarian, [{\em Optim. Lett.}, 3:101--108, 2009]) for solving the AVE.
