I have a set of $c$ matrices $A_1 ... A_c$ which are all symmetric and positive definite. I would like to calculate the inverses of all the possible sums, i.e.

$(A_1+A_2)^{-1},(A_1+A_3)^{-1},(A_2+A_3)^{-1},(A_1+A_2+A_3)^{-1}$

My main limitation is memory, i.e. I don't want to store the $2^c$ inverses, but find a way given a limited number of inverses to be able to compute the inverse I want from those I stored.

To go a bit farther, the matrices will be continuously updated using rank-one updates. Thus I would like to compute the rank-one updates of a limited subset of these $2^c$ inverses and be able to compute the inverse I want in $O(n^2)$.