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Estimating the error in CG-like algorithms for least-squares and least-norm problems. (arXiv:2305.02044v1 [math.NA])

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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.

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