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The question "given the inverse/a certain factorization of $A$, can I compute cheaply one of $A+B$?" is one that gets asked a lot here, in slightly different forms: for instance when $A$ is symmetric, or $B$ diagonal (and I have seen it pop up several times on MO as well). The answer is always "no, unless $B$ has small rank (and then you can use SMW or QR/Cholesky updates) or small norm (and then you can use a perturbation approach and/or use $A$ as a preconditioner in an iterative algorithm).

I think it would be useful to have a canonical question in which we explain this in full generality, so that we can refer people to it instead of re-explaining the same thing continuously. What is your opinion? I volunteer to write one in the next days, if you wish.

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    Over at Math.SE we settled on a notion of "abstract duplicates". See the List of Generalizations of Common Questions at Meta Math.SE for an illustration of how existing Questions and Answers, often with some editing, can be harnessed. – hardmath Nov 4 '18 at 21:30
  • I like the approach that @hardmath proposed. – Anton Menshov Nov 4 '18 at 23:48
  • @hardmath I read your link but I don't really understand --- what is the difference between the concept of canonical question (as used across the whole SE network, see e.g. 1 2 3) and this "abstract duplicate" thing that they have at math.se? Is it just the same thing with a different name? – Federico Poloni Nov 5 '18 at 7:37
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    I have decided to ask this also on meta.math.se, since it seems a natural question to me: math.meta.stackexchange.com/questions/29339 – Federico Poloni Nov 5 '18 at 7:42
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    @FedericoPoloni and that thread convinced me now that there is no clear answer to the difference between them... – Anton Menshov Dec 19 '18 at 21:20
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List of possible candidates for closing as abstract duplicates/canonical questions (for the matrix-inverse/factorization update):

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