# Furthermore suppose qj v0 0 then after k iterations vk

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Unformatted text preview: envalue of µ and λK is the second closest, that is |µ − λJ | < |µ − λK | < |µ − λj | for each j = J . Furthermore, suppose qJ v(0) = 0, then after k iterations v(k ) − (±qJ ) = O µ−λJ µ−λK k , |λ(k ) − λJ | = O µ−λJ µ−λK 2k as k → ∞. The ± sign means that at each step k , one or the other choice of sign is to be taken, and then the indicated bound holds 5 / 21 Inverse iteration and Rayleigh quotient iteration Inverse iteration: one of the most valuable tools of numerical linear algebra a standard method of calculating one or more eigenvectors of a matrix if the eigenvalues are already known Obtaining an eigenvalue estimate from an eigenvector estimate (the Rayleigh quotient) Obtaining an eigenvector estimate from an eigenvalue estimate (inverse iteration) Rayleigh quotient algorithm: Initialize v(0) randomly with v(0) = 1 λ(0) = (v(0) ) Av(0) = corresponding Rayleigh quotient for k = 1, 2, . . . do Solve w = (A − λ(k −1) I )−1 v(k −1) // apply (A − µ(k −1) I )−1 v(k ) = w // normalize w λ(k ) = (v(k ) ) Av(k ) end for // Rayleigh quotient 6 / 21 Rayleigh quotient iteration Theorem Rayleigh quotient iteration converges to an eigenvalue/eigenvector pair for all except a set of measure zero of starting vectors v(0) . When it converges, the convergence is ultimately c...
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## This document was uploaded on 02/10/2014.

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