We consider the iterative solution of generalized saddle point systems. When the right bottom block is zero, Arioli [SIAM J. Matrix Anal. Appl., 34 (2013), pp. 571--592] proposed a CRAIG algorithm based on generalized Golub-Kahan Bidiagonalization (GKB) for the augmented systems with the leading block being symmetric and positive definite (SPD), and then Dumitrasc et al. [SIAM J. Matrix Anal. Appl., 46 (2025), pp. 370--392] extended the GKB for the case where the symmetry condition of the leading block no longer holds and then proposed nonsymmetric version of the CRAIG (nsCRAIG) algorithm. The CRAIG and nsCRAIG algorithms are theoretically equivalent to the Schur complement reduction (SCR) methods where the Conjugate Gradient (CG) method and the Full Orthogonalization Method (FOM) are applied to the associated Schur-complement equation, respectively. We extend the GKB and its nonsymmetric counterpart used separately in CRAIG and nsCRAIG algorithms for the case where the right bottom block of saddle point system is nonzero. On this basis, we propose CRAIG and nsCRAIG algorithms for the solution of the generalized saddle point problems with the leading block being SPD and nonsymmetric positive definite (NSPD), respectively. They are also theoretically equivalent to the SCR methods with inner CG and FOM iterations for the associated Schur-complement equation, respectively. Moreover, we give algorithm steps of the two new solvers and propose appropriate stopping criteria based on an estimate of the energy norm for the error and the residual norm. Numerical comparison with MINRES or GMRES highlights the advantages of our proposed strategies regarding its high computational efficiency and/or low memory requirements and the associated implications.

翻译：我们考虑广义鞍点系统的迭代求解。当右下角块为零时，Arioli [SIAM J. Matrix Anal. Appl., 34 (2013), pp. 571--592] 针对主块为对称正定（SPD）的增广系统，提出了一种基于广义Golub-Kahan双对角化（GKB）的CRAIG算法。随后，Dumitrasc等人 [SIAM J. Matrix Anal. Appl., 46 (2025), pp. 370--392] 将GKB推广至主块不再满足对称性的情形，并提出了非对称版本的CRAIG（nsCRAIG）算法。CRAIG和nsCRAIG算法在理论上分别等价于将共轭梯度（CG）法和完全正交化方法（FOM）应用于相关Schur补方程的Schur补约化（SCR）方法。我们将分别用于CRAIG和nsCRAIG算法的GKB及其非对称对应形式，推广至鞍点系统右下角块非零的情形。在此基础上，我们分别针对主块为SPD和非对称正定（NSPD）的广义鞍点问题，提出了相应的CRAIG和nsCRAIG算法。它们在理论上也分别等价于对相关Schur补方程进行内层CG迭代和FOM迭代的SCR方法。此外，我们给出了这两种新求解器的算法步骤，并基于误差的能量范数估计和残差范数提出了合适的停止准则。与MINRES或GMRES的数值比较凸显了我们所提出策略在高计算效率和/或低内存需求方面的优势及其相关意义。