Power iterations for computing v 1 To start lets consider the simpler problem

Power iterations for computing v 1 to start lets

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Power iterations for computing v 1 To start, let’s consider the simpler problem of computing the largest eigenvalue λ 1 and corresponding eigenvector v 1 of A . We do this with the following iteration. Let q 0 be an arbitrary vector in R N with unit norm, k q 0 k 2 = 1. Then for k = 1 , 2 , . . . compute z k = Aq k - 1 q k = z k k z k k 2 γ k = q T k Aq k Then, as long as q 0 is not orthogonal to v 1 , h q 0 , v 1 i 6 = 0, and λ 1 > λ 2 , as k gets large q k v 1 and γ k λ 1 . A detailed proof of this, including rates of convergence, can be found in Section 8.2 of the Golub/van Loan book. But we can see roughly why it works with a simple calculation. 9 Georgia Tech ECE 6250 Fall 2019; Notes by J. Romberg and M. Davenport. Last updated 13:22, November 13, 2019
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The vector q k above can be written as q k = A k q 0 k A k q 0 k 2 . Since the eigenvectors of A , v 1 , v 2 , . . . , v N form an orthobasis for R N , we can write q 0 = α 1 v 1 + α 2 v 2 + · · · + α N v N , for α n = h q 0 , v n i . Then the expression for q k above becomes q k = α 1 λ k 1 v 1 + α 2 λ k 2 v 2 + · · · + α N λ k N v N p α 2 1 λ 2 k 1 + α 2 2 λ 2 k 2 + · · · + α 2 N λ 2 k N . If α 1 6 = 0 and λ 1 > λ 2 ≥ · · · ≥ λ N , then as k gets large, the first term in each of the sums above will dominate. Thus for large k , q k α 1 λ k 1 v 1 p α 2 1 λ 2 k 1 = v 1 . Since v T 1 Av 1 = λ 1 and q k v 1 , we also have that q T k Aq k λ 1 . QR iterations Now consider the problem of computing all the eigenvectors and eigenvalues of A . We might be tempted to extend the the power method by starting with an entire orthobasis 1 Q 0 , then take Z k = AQ k - 1 , 1 So Q 0 is N × N and satisfies Q T 0 Q 0 = I . 10 Georgia Tech ECE 6250 Fall 2019; Notes by J. Romberg and M. Davenport. Last updated 13:22, November 13, 2019
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and then renormalize the columns of Z k to get Q k . The problem with this is that all of the columns of Z k will converge to v 1 — this is just running the power method with N different starting points. What we do instead is “orthonormalize” the columns of Z k , we make them orthogonal to each other at every iteration as well as unit norm. This gives us the following iteration: Let Q 0 be any orthonormal matrix. For k = 1 , 2 , . . . , take Z k = AQ k - 1 (1) [ Q k , R k ] = qr( Z k ) (so Z k = Q k R k ) . (2) Using arguments not too different than for the power method, you can show (again, see Golub/van Loan Section 8.2 for details) that Q k V , and Γ k = Q T k AQ k Λ . A more popular way to state the iteration ( 1 )–( 2 ) above, and the one you will see in almost every textbook on numerical linear algebra, is the following. Set Γ 0 = A . Then for k = 1 , 2 , . . . , take [ U k , R k ] = qr( Γ k - 1 ) (so Γ k - 1 = U k R k ) (3) Γ k = R k U k (4) So at each iteration, we are computing a QR factorization, then reversing it (cute!). This gives us the relation R k - 1 U k - 1 = U k R k . To see the relationship between version 1 in ( 1 )–( 2 ) and version 2 in 11 Georgia Tech ECE 6250 Fall 2019; Notes by J. Romberg and M. Davenport. Last updated 13:22, November 13, 2019
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( 3 )–( 4 ), notice that if we initialize version 1 with Q 0 = I , then A = Q 1 R 1 Q 1 = AR - 1 1 AQ 1 = Q 2 R 2 Q 2 = A 2 R - 1 1 R - 1 2 .
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