Conjugate gradient convergence bound ($\Vert e_k{\Vert }_H\le 2{\left(\frac{\sqrt{{\kappa }_2(H)}-1}{\sqrt{{\kappa }_2(H)}+1}\right)}^k\Vert e_0{\Vert }_H$)

For a symmetric positive-definite matrix $H$, the Conjugate gradient method minimizes the error in the induced weighted norm over its affine Krylov subspace. Writing the error as $p(H)e_0$ givesFor ${\lambda }_{\min }<{\lambda }_{\max }$, choose the scaled Chebyshev polynomial $p(\lambda )=T_k(({\lambda }_{\max }+{\lambda }_{\min }-2\lambda )/({\lambda }_{\max }-{\lambda }_{\min }))/T_k(({\lambda }_{\max }+{\lambda }_{\min })/({\lambda }_{\max }-{\lambda }_{\min }))$. On the spectral interval its numerator has absolute value at most one. With $\rho =(\sqrt{\kappa }+1)/(\sqrt{\kappa }-1)$, the denominator equals $({\rho }^k+{\rho }^{-k})/2\ge {\rho }^k/2$. This proves the displayed bound. If the two endpoints coincide, $H$ is a positive scalar multiple of the identity and one iteration solves the system exactly.