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I'm having a difficult time working out the details of the following problem. I'm hoping someone may be able to point me to a reference or suggest an approach.

I have three matrices $(A, B, C)$ and three sequences of matrices $(\bar{A}, \bar{B}, \bar{C})_n$ that converge to $(A, B, C)$ respectively. The convergence rate of $||\bar{A} - A||$ is $\mathcal{O}(a)$, $||\bar{B} - B||$ is $\mathcal{O}(b)$ and $||\bar{C} - C||$ is $\mathcal{O}(c)$. Can any explicit rate of convergence be set on:

$$||f(\bar{A}, \bar{B}, \bar{C})- f(A, B, C)|| $$ where $f(X, Y, Z) = (X - YZY)^{-1}$. Assume that $A,C$ and $\bar{A}, \bar{C}$ are always positive definite.

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You haven't said anything about how close $A$ and $BCB$ might be, despite your convergence bounds. If it turns out that $A - BCB$ is very close to not invertible, then as you approximate the $A - BCB$ you can get blow up in the approximation of the inverse. This isn't a complete answer, just an observation that I believe can be made rigorous. It seems like no matter how close you say you can approximate $A,B,C$, that so long as $A -BCB$ is arbitrarily close to non-invertible, you can't give any meaningful bound for the approximation of your inverse in terms of the approximation error bounds for the original matrices $A,B,C$. Basically, because it also should depend on the conditioning number of $A - BCB$.

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