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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…
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:
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 |
|||||
= [−1 |
2 |
−1] |
|||||
0 |
−1 |
3 |
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.)