Numerical Analysis: the power method with shifts

[sarahr]

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 ?

Homework Equations

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?

 

[sarahr]

nevermind! i figured it out! :)

 

[andrewbda]

Answer?

Does anyone know how to solve this question?

I think the answer is beta = -(1/2)(lambda_2 + lambda_n), however I have no idea how to reach this. Any help appreciated!