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The algorithm for the inverse power iteration works as following : \begin{align} &v^{(0)} =\text{ some vector with }\|v^{(0)}\|=1\\ &\text{for }k = 1, 2, \ldots\\ &\qquad\text{Solve } (A - \mu I)w = v^{(k-1)}\text{ for }w\\ &\qquad v^{(k)}= w/\|w\|\\ &\qquad \lambda^{(k)} = (v^{(k)})^TAv^{(k)}\\ \end{align}

However, If we use the exact (or a very close) eigenvalue $\mu$ of the matrix $A$, wouldn't solving $(A - \mu I)w = v^{(k-1)}$ be impossible/ill-conditioned? How is this handled?

edit : for example $\begin{bmatrix}2 & 1\\1 & 2\end{bmatrix}$ has eigenvalues $1$ and $3$ but if we have as input $\mu=1$ or $\mu=3$ to the algorithm it would not work (julia script).

using LinearAlgebra

function invIter(A, n, mu)
    v = [1;0]
    lambda = 0
    for k = 1:n
        w = (A-mu*I)\v
        v = w/norm(w)
        lambda = v'*A*v
    end
    return lambda, v
end
    
A = [2 1;1 2]
lambda, v = invIter(A, 100, 1)
println("lambda, v are : ", lambda, v)
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1 Answer 1

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I don't have much detail here but I'm posting a quick answer because it seems one else steps in.

In practice the ill-conditioning does not matter in this application, because the perturbations to the computed solution $w$ caused by the ill-conditioning are in the direction of the eigenvector that you wish to find.

To better understand this, take an SVD $A-\mu I = U\Sigma V^T$, and assume $\sigma_n \approx 0$ and $\sigma_n \ll \sigma_{n-1}$. Then, the change in the solution of a linear system $(A-\mu I)w = b$ due to a perturbation $b \to b+f$ of small norm is $$ (A-\mu I)^{-1}f = V\Sigma^{-1}U^Tf $$ and has a large component in the direction of $v_n$ (the last column of $V$ and smaller components in the orthogonal directions. However, $(A-\mu I)v_n = U\Sigma e_n = u_n \sigma_n \approx 0$, so $v_n$ is close to the eigenvector of $A$ with eigenvalue $\mu$.

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  • $\begingroup$ Thank you for the answer. I confess it is still not 100% clear but I am going te reread and reread the answer. Quick question : how can I assume $\sigma_n \ll\sigma_{n-1}$? $\endgroup$
    – edamondo
    Dec 14, 2022 at 15:17
  • $\begingroup$ Good question -- it's not something that holds in general; I was analyzing that case in particular. If there is more than one singular value that is very small, then it means that $A$ is close to a matrix that has a double eigenvalue in $\mu$, and you have uncertainty in a subspace of larger dimension, but they are all close to eigenvectors. $\endgroup$ Dec 14, 2022 at 15:31
  • $\begingroup$ But just to be sure. Using the reverse power iteration with the exact eigenvalue as input would not work right (which is a bit frustrating)? (see the edit to my post) $\endgroup$
    – edamondo
    Dec 15, 2022 at 10:03
  • $\begingroup$ Also how do you deal in the case of the double eigenvalue in order to find both eigenvectors? $\endgroup$
    – edamondo
    Dec 15, 2022 at 10:06
  • $\begingroup$ No, it would not work, you are right. And in the case of a double eigenvalue v does not converge in the usual sense. $\endgroup$ Dec 15, 2022 at 10:30

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