On error estimation in the conjugate gradient method and why it works in finite precision computations.

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Recent computational developments in Krylov subspace methods for linear systems (2007, methodology): “Identities relating the error A-norm and 2-norm with quantities available in the algorithm, were already present in the original paper by Hestenes and Stiefel [180], but apparently they have been reconsidered only recently by Strakoš and Tichý [320].”
A preconditioned low‐rank CG method for parameter‐dependent Lyapunov matrix equations (2014, background): “In finite-precision arithmetic, ‖Rk‖F remains a faithful convergence indicator (up to machine precision) [41].”