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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.

MSC:

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

Citations:

Zbl 0048.09901

Software:

mctoolbox; BiCGstab
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