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تسجيل مستخدم جديدOptimal parameter for the SOR-like iteration method for solving the system of absolute value equations
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The SOR-like iteration method for solving the absolute value equations~(AVE) of finding a vector $x$ such that $Ax - |x| - b = 0$ with $ u = |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 $|T_ u(omega)|_2$ with $$T_ u(omega) = left(begin{array}{cc} |1-omega| & omega^2 u |1-omega| & |1-omega| +omega^2 u end{array}right)$$ and the approximate optimal parameter which minimizes $eta_{ u}(omega) =max{|1-omega|, uomega^2}$ are explored. The optimal and approximate optimal parameters are iteration-independent and the bigger value of $ u$ is, the smaller convergent region of the iteration parameter $omega$ 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.
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