The generalized Golub-Kahan bidiagonalization has been used to solve saddle-point systems where the leading block is symmetric and positive definite. We extend this iterative method for the case where the symmetry condition no longer holds. We do so by relying on the known connection the algorithm has with the Conjugate Gradient method and following the line of reasoning that adapts the latter into the Full Orthogonalization Method. We propose appropriate stopping criteria based on the residual and an estimate of the energy norm for the error associated with the primal variable. Numerical comparison with GMRES highlights the advantages of our proposed strategy regarding its low memory requirements and the associated implications.

翻译：广义Golub-Kahan双对角化方法此前被用于求解主导块对称正定的鞍点系统。我们将该迭代方法拓展至对称性条件不再成立的情形。具体实现基于该算法与共轭梯度法之间的已知关联，并遵循将后者适配为完全正交化方法的推理思路。我们提出了基于残差和与原变量相关误差的能量范数估计的适当停止准则。与GMRES方法的数值对比凸显了本策略在低内存需求及其相关影响方面的优势。