The Gauss-Seidel iterative method with the preconditioning matrix \((I+S+S_m)\)
An improved preconditioner is discussed for the Gauss-Seidel method applied to a linear system \(Ax=b\) where \(A\) is a matrix with ones on the diagonal and all other elements non-positive. \textit{A. D.~Gunawardena, S. K.~Jain}, and \textit{L.~Snyder} [Linear Algebra Appl. 154--156, 123--143 (1991; Zbl 0731.65016)] have proposed a preconditioner of the form \(I+S\) with \(S\) the zero matrix, except for the elements \((i,i+1)\) which are replaced by \(a_{ij}\), the corresponding element from \(A\). Here it is proposed to add a correction term and use \(I+S+S_m\) where \(S_m\) is the zero matrix, except for the elements \((i,j)\) which is \(-a_{ij}\) where \(a_{ij}\) is the first element in row \(i\) of \(A\) satisfying \(| a_{ij}| =\max_{i+1<k}| a_{ik}| \). The different precontitioners \(I\) (no preconditioning), \(I+S\) (Gunawardena et al. [loc. cit.]) and \(I+S+S_m\) are compared and it is shown, in theory and by numerical examples, that the the latter gives indeed a slight improvement over \(I+S\) (the spectral radius is slightly smaller and execution time is shorter).
scientific article; zbMATH DE number 2103572