- #1
angelz429
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[SOLVED] Approximate eigenvalues
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
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
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.
Homework Statement
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
Homework 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
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.