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**1. Homework Statement**

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. Homework 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?