The saddle point problem and the Manteuffel algorithm

In the last two decades, the augmented linear systems and the saddle point problems have been solved by many researchers who have used the conjugate gradient method or the generalized SOR iterative method and variants of them. In the latter class of methods, when the block $$A \in {\mathrm{I\!R}}^{m\times m}$$A∈IRm×m of the matrix coefficient $$\mathcal {A} = \left[ \begin{array}{cc} A &{} B \\ -B^T &{} 0\end{array}\right] \in {\mathrm{I\!R}}^{(m+n) \times (m+n)}$$A=AB-BT0∈IR(m+n)×(m+n), $$m \ge n,$$m≥n, of the linear system to be solved, is symmetric positive definite and $${\mathrm{rank}}(B) =r \le n$$rank(B)=r≤n, convergence regions and optimal values of the parameters involved have been determined. In this work, we consider the block A to be nonsymmetric positive definite, $${\mathrm{rank}}(B) =r < n $$rank(B)=r<n, and use a two-level stationary iterative method whose main step is the linear second-order stationary iterative method for the solution of this class of problems. This method leads to the singular Manteuffel algorithm and the determination of its optimal parameters. As a byproduct, the optimal parameters of the Generalized Modified SSOR method in a particular case are also determined. Numerical examples verify our theoretical findings.