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.

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