invpower

# invpower - The Inverse Power Method Assume that A Cnn has a...

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The Inverse Power Method Assume that A C n × n has a n linearly independent eigenvectors v 1 , v 2 , . . . , v n , and associated eigenvalues λ 1 , λ 2 , . . . , λ n , with | λ 1 | > | λ 2 | ≥ | λ i | , i = 3 , 4 , . . . , n . Then ( λ 1 , v 1 ) is a dominant eigenpair of A , and for almost all x C n , x k A k x v 1 as k → ∞ . How fast does such an iteration converge? Write x = n X i =1 c i v i . Then A k x λ k 1 = c 1 v 1 + c 2 ( λ 2 λ 1 ) k v 2 + n X i =3 c i ( λ i λ 1 ) k v i (1) and it is clear (yes?) that the error gets multiplied by about | λ 2 1 | at each step (we say that the convergence is linear with asymptotic error constant | λ 2 1 | ). So the smaller the ratio | λ 2 1 | , the better, and if | λ 1 | ≈ | λ 2 | then we expect very slow convergence. Let B C n × n have eigenvalues μ i labeled so that | μ i | ≥ | μ i +1 | . The power method applied to B converges to the dominant eigenvector of B (if one exists) with a speed that depends
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## This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.

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