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There are two invertible symmetric matrices A and B, of which B is a block diagonal. A and B have the same dimensions. I need to iteratively calculate the inverse of M = s * A + B, where s is a positive scalar and will be changed during iterations.

Is there a simple approach to update the inverse of M with respect to different values of s?

Much appreciated for your help.

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    $\begingroup$ As far as I know, no, unless you can afford to compute a generalized Schur decomposition (i.e., essentially compute all eigenvalues). In that case, you can solve systems with $O(n^2)$ each afterwards. $\endgroup$
    – Federico Poloni
    Commented Oct 23, 2013 at 11:53
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    $\begingroup$ I suppose in some restricted circumstances you'd have a power series expansion for the inverse of $M$. It might not be useful, depending on your precise needs. $\endgroup$
    – Ryan Budney
    Commented Oct 23, 2013 at 12:00

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