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Numerical Analysis: the power method with shifts

  1. Apr 25, 2007 #1
    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

    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?
  2. jcsd
  3. Apr 26, 2007 #2
    nevermind! i figured it out!! :)
  4. Apr 30, 2007 #3

    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!
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