The monotone convergence rate of the parallel nonlinear AOR method (Q1912855)

Systems of nonlinear equations (1) \(Ax+ \varphi(x)= b\) are solved iteratively using a combination of an accelerated overrelaxation (AOR) and a multisplitting method. The resulting algorithm is called ``Parallel nonlinear AOR method'' by the author. In (1) \(A\) is a real nonsingular \(n\times n\) matrix, \(b\), \(x= (x_i)\in \mathbb{R}^n\) are vectors and \(\varphi= (\varphi_i): \mathbb{R}^n\to \mathbb{R}^n\) is a continuous, diagonal and isotone function where diagonal means that \(\varphi_i\) depends on \(x_i\) only. For \(M\)-matrices \(A\) conditions on the underlying splittings and on the AOR-parameters are listed for the sequence \(\{x^m\}\) of iterates to converge to the unique solution of (1). Moreover, a comparison result is proved and a numerical example is given.