As discussed in example 574 p 350 householder

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Unformatted text preview: vector associated with λ1 . But rather than successively powering A, the sequence An x0 /m(An x0 ) is more efficiently generated by starting with / x0 ∈ R (A − λ1 I) and setting yn = Axn , νn = m(yn ), xn+1 = yn , νn D E for n = 0, 1, 2, . . . . (7.3.17) Not only does xn → x, but as a bonus we get νn → λ1 because for all n, Axn+1 = A2 xn /νn , so if νn → ν as n → ∞, the limit on the left-hand side is Ax = λ1 x, while the limit on the right-hand side is A2 x/ν = λ2 x/ν. Since 1 these two limits must agree, λ1 x = (λ2 /ν )x, and this implies ν = λ1 . 1 T H Summary. The sequence (νn , xn ) defined by (7.3.17) converges to an eigenpair / (λ1 , x) for A provided that G1 x0 = 0 or, equivalently, x0 ∈ R (A − λ1 I). Advantages. Each iteration requires only one matrix–vector product, and this can be exploited to reduce the computational effort when A is large and sparse—assuming that a dominant eigenpair is the only one of interest. Disadvantages. Only a dominant eigenpair is determined—something else must be done if others are desired. Furthermore, it’s clear from (7.3.16) that the rate at which (7.3.17) converges depends on how fast (λ2 /λ1 )n → 0, so convergence is slow when |λ1 | is close to |λ2 |. IG R Example 7.3.8 Y P O C Inverse Power Method. Given a real approximation α ∈ σ (A) to any real / λ ∈ σ (A), this algorithm (also called the inverse iteration ) determines an eigenpair (λ, x) for a diagonalizable matrix A ∈ m×m by applying the power 75 method to B = (A − αI)−1 . Recall from Exercise 7.1.9 that x is an eigenvector for A ⇐⇒ x is an eigenvector for B, λ ∈ σ (A) ⇐⇒ (λ − α)−1 ∈ σ (B). (7.3.18) If |λ − α| < |λi − α| for all other λi ∈ σ (A), then (λ − α)−1 is the dominant eigenvalue of B because |λ − α|−1 > |λi − α|−1 . Therefore, applying the power 75 The relation between the power method and inverse iteration is clear to us now, but it originally took 15 y...
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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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