Let $A$ be a large sparse symmetric matrix (say of dimension $N\geq 10^6$). I compute its largest eigenvalues using the power method. Suppose I got approximate eigenvalue $\lambda'$ with eigenvector $v'$ at some step of the computation.
Question: If $|Av'-\lambda'v'|\leq 10^{-k}$, can one estimate how close is $\lambda'$ to the actual eigenvalue?