Optimal parameter for the SOR-like iteration method for solving the system of absolute value equations

The SOR-like iteration method for solving the absolute value equations~(AVE) of finding a vector $x$ such that $Ax - |x| - b = 0$ with $\nu = \|A^{-1}\|_2 < 1$ 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 $\|T_\nu(\omega)\|_2$ with $$T_\nu(\omega) = \left(\begin{array}{cc} |1-\omega| & \omega^2\nu \\ |1-\omega| & |1-\omega| +\omega^2\nu \end{array}\right)$$ and the approximate optimal parameter which minimizes $\eta_{\nu}(\omega) =\max\{|1-\omega|,\nu\omega^2\}$ are explored. The optimal and approximate optimal parameters are iteration-independent and the bigger value of $\nu$ 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.Comment: 23 pages, 7 figures, 7 table

M-step preconditioned conjugate gradient methods

Preconditioned conjugate gradient methods for solving sparse symmetric and positive finite systems of linear equations are described. Necessary and sufficient conditions are given for when these preconditioners can be used and an analysis of their effectiveness is given. Efficient computer implementations of these methods are discussed and results on the CYBER 203 and the Finite Element Machine under construction at NASA Langley Research Center are included

An M-step preconditioned conjugate gradient method for parallel computation

This paper describes a preconditioned conjugate gradient method that can be effectively implemented on both vector machines and parallel arrays to solve sparse symmetric and positive definite systems of linear equations. The implementation on the CYBER 203/205 and on the Finite Element Machine is discussed and results obtained using the method on these machines are given

Newton-sor iterative method for solving the two-dimensional porous medium equation

In this paper, we consider the application of the Newton-SOR iterative method in obtainingthe approximate solution of the two-dimensional porous medium equation (2D PME). Thenonlinear finite difference approximation equation to the 2D PME is derived by using theimplicit finite difference scheme. The developed nonlinear system is linearized by using theNewton method. At each temporal step, the corresponding linear systems are solved by usingSOR iteration. We investigate the efficiency of the Newton-SOR iterative method by solvingthree examples of 2D PME and the performance is compared with the Newton-GS iterativemethod. Numerical results show that the Newton-SOR iterative method is better than theNewton-GS iterative method in terms of a number of iterations, computer time and maximum absolute errors.Keywords: porous medium equation; finite difference scheme; Newton; Successive OverRelaxation, Gauss-Seidel