I have a problem of computing the dominant eigenvector. When I want to approximate the dominant eigenvector of a large sparse matrix via the famous Arnoldi method, I am wondering how to choose the reduced order $k$ (i.e., the number of Arnoldi iterations). I know that the approximation accuracy is relevant to the choice of $k$. As $k$ approaches to $n$ (the dimension of the full space) , the accuracy is very high. Now the problem is that given an accuracy $e$, can we find an approperate $k$ (depending on $e$) such that the difference between my approximate dominant eigenvector via Krylov subspace and the exact one is less than $e$? That's to say, I need to find an a-prior error bound, rather than an a-posterior error bound (however, I only found some a-posterior error bound results in some papers). Could you give me some suggestions? Thanks!