1. The problem statement, all variables and given/known data Consider a symmetric matrix, A, n x n with distinct eigenvalues lambda_1 > lambda_2 > ... > lambda_n (note: i didnt miss anything here typing this, there are no absolute values here). What value of the shift beta will give fastest convergence to lamba_1 and its eigenvalue when the power method is applied to A + betaI ? 2. Relevant equations 3. The attempt at a solution First, I know that when beta is very large positive or negative: the power method works badly or not at all for computing lambda_1. Second, the rate of convergence is best when the ratio between the largest and second largest eigenvalues (in magnitude) is large. But.. these are not necessarily the shifted versons of lambda_1 and lambda_2 (lambda_1+ beta, lambda_2 + beta). For some shifts, the shifted version lambda_n at the other end might be the largest or second largest in magnitude. I've been trying to look at all the different cases (like the largest eig & second largest eig being both positive, both negative, both centered around zero, etc). I was thinking that the the ratio was largest when the largest eig and the second largest eig are equally centered around zero.. but not completely sure how i should state beta.. any further ideas?