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Unformatted text preview: online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 538 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 is revealed by looking at a generic 4 × 4 pattern for Hk+1 = RQ + αI ⎛ ∗ ∗ 0 0 ∗ ∗ ∗ 0 ∗ ⎝∗ 0 0 ≈ ∗ ⎝0 0 0 ∗ ∗ ∗ 0 ∗ ∗ ∗ ⎛ It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu = ⎞⎛ ∗ ∗ ∗⎠⎝∗ ∗ 0 0 ∗ ∗ ∗ α± ⎞ ∗ ∗ ∗ 0 ⎠≈ ∗ ∗ ∗ ⎛ ∗ ⎝∗ 0 0 ⎞ ⎛ 0 α 0⎠ +⎝ 0 ±1 ∗ ∗ ∗ 0 ∗ ∗ ∗ 0 ∗ ∗ ∗ α± ⎞ ⎠ α α ⎞ α ⎠. D E The strength of the last approximation rests not only on the size of , but it is also reinforced by the fact that ≈ 0 because the 2-norm of the last row of Q must be 1. This indicates why this technique (called the single shifted QR iteration) can provide rapid convergence to a real eigenvalue. To extract complex eigenvalues, a double shift strategy is employed in which the eigenvalues αk and βk of the lower 2 × 2 block of Hk are used as shifts as indicated below: Factor: Hk − αk I = Qk Rk , Set: T H IG R (so Hk+1 = QT Hk Qk ), k Hk+1 = Rk Qk + αk I Factor: Hk+1 − βk I = Qk+1 Rk+1 , Set: Hk+2 = Rk+1 Qk+1 + βk I . . . (so Hk+2 = QT+1 QT Hk Qk Qk+1 ), k k Y P The nice thing about the double shift strategy is that even when αk is complex (so that βk = αk ) the matrix Qk Qk+1 (and hence Hk+2 ) is real, and there are efficient ways to form Qk Qk+1 by computing only the first column of the product. The double shift method typically requires very few iterations (using only real arithmetic) to produce a small entry in the (n − 2, 2)-position as depicted in (7.3.19) for a generic 4 × 4 pattern. O C Exercises for section 7.3 7.3.1. Determine cos A for A = −π/2 π/2 π/2 −π/2 . 7.3.2. For the matrix A in Example 7.3.3, verify with direct computation that λ1 t 0 eλ1 t G1 + eλ2 t G2 = P e 0 eλ2 t P−1 = eAt . 7.3.3. Explain why si...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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