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Approximate eigenvalues

  1. Apr 25, 2008 #1
    [SOLVED] Approximate eigenvalues

    1. The problem statement, all variables and given/known data

    Use some QR method to approximate the eigenvalues of
    [4 3]
    [3 5]
    and compare with the actual values.

    The actual values are (9±√37)/2


    2. Relevant equations

    A(0)=Q(0)R(0)
    A(1)=R(0)Q(0)

    A-α(0)I=Q(0)R(0)
    A(1)=R(0)Q(0) + α(0)I


    3. The attempt at a solution

    A(0) = [4 3] = [(4/5) (-3/5)] [5 (27/5)]
    [3 5] [(3/5) (4/5)] [0 (11/5)]

    A(1)= [5 (27/5)] [(4/5) (-3/5)] = [(181/25) (33/25)]
    [0 (11/5)] [(3/5) (4/5)] [(33/25) (44/25)]

    When I get to A(2), its no longer symmetric

    Same if I try it the Shifted QR Method

    A(0) = [4 3] α(0) = 5
    [3 5]

    A(0)-α(0)I = [-1 3] = [(-1/sqrt 10) (3/sqrt 10)] [(sqrt 10) (-3/sqrt 10)]
    [3 0] [(3/sqrt 10) (1/sqrt 10)] [ 0 (9/sqrt 10)]

    A(1) = [(sqrt 10) (-3/sqrt 10)] [(-1/sqrt 10) (3/sqrt 10)] + [5 0]
    [ 0 (9/sqrt 10)] [(3/sqrt 10) (1/sqrt 10)] [0 5]

    = [(51/10) (27/10)] α(1) = (59/10)
    [(27/10) (59/10)]

    When I get to A(2), its no longer symmetric

    So I'm not sure what I'm doing wrong... or if there's another QR method to solve it.
     
  2. jcsd
  3. Apr 29, 2008 #2
    Well, since no one can help.... :(

    I don't need it anymore, thanks for looking!
     
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