In [Meurant, Pape\v{z}, Tich\'y; Numerical Algorithms 88, 2021], we presented an adaptive estimate for the energy norm of the error in the conjugate gradient (CG) method. In this paper, we extend the estimate to algorithms for solving linear approximation problems with a general, possibly rectangular matrix that are based on applying CG to a system with a positive (semi-)definite matrix build from the original matrix. We show that the resulting estimate preserves its key properties: it can be very cheaply evaluated, and it is numerically reliable in finite-precision arithmetic under some mild assumptions. We discuss algorithms based on Hestenes-Stiefel-like implementation (often called CGLS and CGNE in the literature) as well as on bidiagonalization (LSQR and CRAIG), and both unpreconditioned and preconditioned variants. The numerical experiments confirm the robustness and very satisfactory behaviour of the estimate.

翻译：在[Meurant, Pape\v{z}, Tich\'y; Numerical Algorithms 88, 2021]中，我们提出了共轭梯度(CG)方法中误差能量范数的自适应估计。本文将这一估计推广至求解一般（可能为矩形）矩阵线性逼近问题的算法，此类算法通过将CG应用于由原始矩阵构造的正（半）定矩阵系统来实现。我们证明该估计保持了其关键特性：计算成本极低，且在适度假设下，在有限精度计算中具有数值可靠性。我们讨论了基于Hestenes-Stiefel式实现（文献中常称为CGLS和CGNE）以及基于双对角化（LSQR和CRAIG）的算法，涵盖了非预处理和预处理两种变体。数值实验证实了该估计的稳健性与非常令人满意的性能表现。