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Given a real symmetric matrix $A$, say we already know its spectral decomposition $A = UDU^T$. We wish to compute the spectral decomposition of $A+E$, where $E$ is a small matrix (I am particularly interested in the case where E is nonzero in only one entry).

I know this places us squarely in the domain of perturbation theory, which is of course a richly studied topic. However, I am interested in an algorithm which can numerically solve this quickly, and which I could quickly implement or find prepackaged in Matlab or Python. I am sure such a thing exists but I am having trouble finding it. Can anyone provide me with a reference to such an algorithm?

For context, I need to compute the eigenvectors of $A$, then perturb it a tiny bit, and then recompute the eigenvectors, and then repeat many times. Recomputing the vectors from scratch each time is slow and seems wasteful.

Van Latimer
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    This may answer your question. https://math.stackexchange.com/questions/3052997/rank-one-update-of-eigenvalues

    If $E$ has one non-zero entry it's rank-one. The author lists the Netlib routine. I don't think it'd be right to copy it.

    http://www.netlib.org/lapack/explore-html/d2/d24/group__aux_o_t_h_e_rcomputational_ga3c4a943599132aea3ac964c08392853a.html

     – Ryan Howe Dec 19 '20 at 19:58
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    +1 for @RyanHowe: this is definitely the canonical way to solve this problem. Be aware of implementing this yourself: it is very tricky to solve this problem accurately. Demmel's book as referenced in the linked answer does a good job of explaining the difficulties. For particularly structured eigenvector computations, this method can further be accelerated by the fast multipole method. There's some discussion in this paper if you're interested. Unfortunately, I don't believe there's a publically available high-quality implementation of this accelerated variant. – eepperly16 Dec 21 '20 at 07:17
  • @eepperly16 thanks – Ryan Howe Dec 21 '20 at 16:39

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