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In [Meurant, Pape\v{z}, Tich\'y; Numerical Algorithms 88, 2021], we presentedan 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 solvinglinear approximation problems with a general, possibly rectangular matrix thatare based on applying CG to a system with a positive (semi-)definite matrixbuild from the original matrix. We show that the resulting estimate preservesits key properties: it can be very cheaply evaluated, and it is numericallyreliable in finite-precision arithmetic under some mild assumptions. We discussalgorithms based on Hestenes-Stiefel-like implementation (often called CGLS andCGNE in the literature) as well as on bidiagonalization (LSQR and CRAIG), andboth unpreconditioned and preconditioned variants. The numerical experimentsconfirm the robustness and very satisfactory behaviour of the estimate.

Estimating the error in CG-like algorithms for least-squares and least-norm problems