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Unformatted text preview: FF state change to ON, while 90% of those in the ON
state revert to the OFF state.
(a) Show that the device approaches an equilibrium in the sense
that the proportion of switches in each state eventually becomes
constant, and determine these equilibrium proportions.
(b) Independent of the initial proportions, about how many clock
cycles does it take for the device to become essentially stable? Copyright c 2000 SIAM Buy online from SIAM
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Eigenvalues and Eigenvectors
7.3.12. The spectral radius of A is ρ(A) = maxλi ∈σ(A) |λi | (p. 497). Prove
that if A is diagonalizable, then
An ρ(A) = lim n→∞ 1/n for every matrix norm. It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] This result is true for nondiagonalizable matrices as well, but the proof
at this point in the game is more involved. The full development is given
in Example 7.10.1 (p. 619). 7.3.13. Find a dominant eigenpair for A = 7
E by the power method. 7.3.14. Apply the inverse power method (Example 7.3.8, p. 534) to ﬁnd an eigenvector for each of the eigenvalues of the matrix A in Exercise 7.3.13. T
H 7.3.15. Explain why the function m(v) used in the development of the power
method in Example 7.3.7 is not a continuous function, so statements
like m(xn ) → m(x) when xn → x are not valid. Nevertheless, if
limn→∞ xn = 0, then limn→∞ m(xn ) = 0. 7.3.16. Let H = 1
1 . IG
P (a) Apply the “vanilla” QR iteration to H.
(b) Apply the the single shift QR iteration on H. 7.3.17. Show that the QR iteration can fail to converge using H = O
C Copyright c 2000 SIAM 0
0 . (a) First use the “vanilla” QR iteration on H to see what happens.
(b) Now try the single shift QR iteration on H.
(c) Finally, execute the double shift QR iteration on H. Buy online fr...
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