On error estimation in the conjugate gradient method and why it works in finite precision computations (Q1866487)

The authors show that the lower bound for the \(A\)-norm of the error based on Gauss quadrature is mathematically equivalent to the original formula of \textit{M. R. Hestenes} and \textit{E. Stiefel} [J. Res. Natl. Bur. Stand. 49, 409-435 (1952; Zbl 0048.09901)]. Existing bounds are compared and the authors demonstrate the necessity of a proper roundoff error analysis with an example of the well-known bound that fails in finite precision arithmetic. The numerical stability of the simplest bound is proved. In addition a lower bound for the Euclidean norm is described. The results are illustrated by numerical examples.