On Meinardus' examples for the conjugate gradient method

@article{Li2008OnME,
  title={On Meinardus' examples for the conjugate gradient method},
  author={Ren-Cang Li},
  journal={Math. Comput.},
  year={2008},
  volume={77},
  pages={335-352}
}
The Conjugate Gradient (CG) method is widely used to solve a positive definite linear system Ax = b of order N . In 1963, Meinardus (Numer. Math., 5 (1963), pp. 14–23.) proved that the relative residual of the kth approximate solution by CG (with the initial approximation x0 = 0) is bounded above by 2 [ ∆κ + ∆ −k κ ]−1 with ∆κ = √ κ + 1 √ κ− 1 , where κ ≡ κ(A) = ‖A‖2‖A−1‖2 is A’s spectral condition number. In the same paper he also gave an example to achieve this bound for k = N − 1 but without… CONTINUE READING
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Über eine Verallgemeinerung einer Ungleichung von L

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4 Excerpts

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