Consider the CG method for the iterative solution of the system of equations $A \mathrm{x}=\mathrm{b}$ with a positive definite and symmetric matrix $A \in \mathbb{R}^{n \times n}$ and $b \in \mathbb{R}^{n}$. Show that to reduce the initial error by the factor $\varepsilon$ at most $t(\varepsilon) \leq \frac{1}{2} \sqrt{\kappa} > \log (2 / \varepsilon)+1$ many iteration steps are required. Here $\kappa$ is the spectral condition number of A.
If I understand correctly, the CG method has the property that the error after $\mathrm{t}$ iterations is orthogonal to the initial residual in the A-norm $\left(g^{t}=A\left(x^{*}-x^{t}\right)\right.$, where $x^{*}$ is the exact solution, $x^{t}$ is the approximate solution after $t$ iterations and $g^{t}$ is the residual after $\mathrm{t}$ iterations.
The A-norm of a vector $v$ is defined as $\|v\|_{A}=\sqrt{v^{T} A v}$ --> error after t iterations with respect to the A-norm: $\left\|x^{*}-x^{t}\right\|_{A}=\left\|g^{t}\right\|_{A}$
After $t$ iterations, the error is reduced by a factor of at least $\left(\frac{2}{1+\sqrt{\kappa}}\right)^{t}$: $$\left\|x^{*}-x^{t}\right\|_{A} \leq \frac{2}{(1+\sqrt{\kappa})^{t}}\left\|x^{*}-x^{0}\right\|_{A}$$
Do I now have to set this inequality to $\varepsilon$ and solve for $\mathrm{t}$? $$ \begin{array}{l} \varepsilon \geq \frac{2}{(1+\sqrt{\kappa})^{t}} \\ t(\varepsilon) \leq \frac{1}{2} \sqrt{\kappa} \log \left(\frac{2}{\varepsilon}\right)+1 \end{array} $$
Can I proceed like this with my idea?