On error estimation in the conjugate gradient method and why it works in finite precision computations. (English) Zbl 1026.65027

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 M. R. Hestenes and 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.


65F10 Iterative numerical methods for linear systems
65F25 Orthogonalization in numerical linear algebra
65G50 Roundoff error


Zbl 0048.09901


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