Edit: If any information is missing, please tell me and I'll edit the question. Thanks again!
The conjugate gradient (cg) method was applied to a positive definite Matrix $A$. It is only known that $||e||_A=1$ and $||e^{10}||_A=2^{-9}$ (where $e$ is the error $||e^k||_A= ||x-x^k||_A$). Calculate with this information a lower bound for $κ(A)$ (where $κ$ is the condition number) and compare it with the equation $k \geq \frac{1}{2}(\sqrt{κ(A)}\ln(2/ε))$ where ε is the factor by which the error is reduced, defined as $||e^k||_A= ||x-x^k||_A \leq ε||e^0||_A$
Here's what I have so far. If I have understood it correctly $ε=2^{-9}/1=2^{-9}$. I however don't understand how only from that can I calculate the condition number. Doesn't the condition number require knowing the matrix and it's inverse? $κ=||A||||A^{-1}||$
I have calculated what $κ$ should be using the equation $k \geq \frac{1}{2}(\sqrt{κ(A)}\ln(2/ε))$ and I got $10 \geq \frac{6.93}{2}(\sqrt{κ(A)})$ $2.89 \geq \sqrt{κ(A)}$ $8.33 \geq κ(A)$
How can I move forward? Thanks in advance!