On Meinardus’ examples for the conjugate gradient method
Ren‐Cang Li · Mathematics of Computation · 2007
The conjugate gradient (CG) method is widely used to solve a positive definite linear system A x = b Ax=b of order N N . It is well known that the relative residual of the k k th approximate solution by CG (with the initial approximation x 0 = 0 x_0=0 ) is bounded above by \[ 2 [ Δ κ k + Δ κ − k ] − 1 with Δ κ = κ + 1 κ − 1 , 2\left [\Delta _{\kappa }^k+\Delta _{\kappa }^{-k}\right ]^{-1} \quad \mbox {with}\quad \Delta _{\kappa }=\frac {\sqrt {\kappa }+1}{\sqrt {\kappa }-1}, \] where κ ≡ κ ( A ) = ‖ A ‖ 2 ‖ A − 1