Description
Problem 1:
Determine one eigenvalue of the following matrix using Rayleigh Quotient iteration, starting with initial guess (0) = [0 1]T and initial eigenvalue estimate (0) = ( (0)) (0). Terminate iteration after 3 steps, i.e., after you obtain ( 3 ) . What is the approximate eigenvector (3)? What is the error of each ( )?
-
= [−6
2
]
2
−3
Problem 2:
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Perform the first two iterations of the QR algorithm (i.e., compute |
(2) |
̃ |
(2) |
) for the following matrix. How |
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|
and |
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close are the diagonal elements of (2) to the eigenvalues of ? |
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|
3 |
−1 |
0 |
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|
= [−1 |
2 |
−1] |
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|
0 |
−1 |
3 |
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Problem 3:
Reduce the following matrix to Hessenberg form using Householder reflector.
-
3
−2
4
4
= [−2
1
9
−4]
4
9
2
−4
4
−4
−4
2
Problem 4:
Let Q and R be the QR factors of a symmetric tridiagonal matrix H. Show that the product = is again a symmetric tridiagonal matrix.
(Hint: Prove the symmetry of K. Show that Q has Hessenberg form and that the product of an upper triangular matrix and a Hessenberg matrix is again a Hessenberg matrix. Then use the symmetry of K.)
