QR The algorithm The most common method for solving small (dense) eigenvalue problems. The basic algorithm: QR without shifts 1. Until Convergence Do: 2. Compute the QR factorization A = QR 3. Set A := RQ EndDo 4. "Until Convergence" means "Until A becomes close enough to an upper triangular matrix"
Note: Anew = RQ = QH (QR)Q = QH AQ Anew is similar to A throughout the algorithm . Above basic algorithm is never used in practice. Two variations: (1) use shift of origin and (2) Transform A into Hessenberg form..
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Practical QR algorithms: Shifts of origin Observation: (from theory): Last row converges fastest. | | Convergence is dictated by | n |
n-1
We will now consider only the real symmetric case. Eigenvalues are real. A(k) remains symmetric throughout process. As k goes to infinity the last column and row (except (k) ann ) converge to zero quickly.,, and ann converges to lowest eigenvalue.
(k)
A(k) =
. . . . . . . . . . . . . . . . . . . . . . . . . a a a a a
a a a a a a
(k)
Idea: Apply QR algorithm to A(k) - I with = ann . Note: eigenvalues of A(k) - I are shifted by , and eigenvectors are the same.
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QR with shifts 1. Until row ain, 1 i < n converges to zero DO: 2. Obtain next shift (e.g. = ann) 3. A - I = QR 5. Set A := RQ + I EndDo 6. Convergence is cubic at the limit! [for symmetric case]
Result of algorithm: . . . . . 0 . . . . . 0 . . . . . 0 . . . . . 0 . 0 . 0 . 0 . 0 . 0 0 n
(k) = A
Next step: deflate, i.e., apply above algorithm to (n - 1) (n - 1) upper triangular matrix.
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Practical QR algorithms: Use of the Hessenberg Form Recall: Upper Hessenberg matrix is such that aij = 0 for j < i - 1 Observation: The QR algorithm preserves Hessenberg form (tridiagonal form in symmetric case). Results in substantial savings. Consider the first step only on a 6 6 T We want H1AH1 = 0 H1AH1 to have the 0 form: 0 0
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Choose a w in H1 = I -2ww T to make the first column have zeros from position 2 to So n. w1 = 0. Apply to left: B = H1A Apply to right: A1 = BH1. Main observation: the Householder matrix H1 which transforms the column A(2 : n, 1) into e1 works only on rows 2 to n. When applying the transpose H1 to the right of B = H1A, we observe that only columns 2 to n will be altered. So the first column will retain the desired pattern (zeros below row 2). Algorithm continues the same way for columns 2, ...,n- 2.
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QR for Hessenberg matrices Need the "implicit Q theorem" Suppose that QT AQ is an unreduced upper Hessenberg matrix. Then columns 2 to n of Q are determined uniquely (up to signs) by the first column of Q. Implication: to compute Ai+1 = QT AQi we can: i Example: With n = 6 :
A = 0 0 0
0 0
0
1. Choose G1 = G(1, 2, 1) so that Q(:, 1) = scalA(:, 1) T AG = + A1 = G1 1 0 0 0 0 0
Compute 1st column of Qi [== scalar A(:, 1)] Choose other columns so Qi = unitary, and Ai+1 = Hessenberg.
2. Choose G2 = G(2, 3, 2) so that (G2A1)31 = 0
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A2 = GT A1G2 = 0 2 0 0 + 0 0
A4 = GT A3G4 = 0 4 0 0 0 0 0
3. Choose G3 = G(3, 4, 3) so that (G3A2)42 = 0 A3 = GT A2G3 = ...