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Algorithm for efficiently calculating $(A+\sum_{i=1}^n B_i)^{-1}$ where $A^{-1}\in\mathbb S^n_+$ is known and $B_i$ are sparse matrices

Let $A\in\mathbb R^{n\times n}$ be a symmetric positive-definite matrix and $A^{-1}$ is already known. Now I want to compute the matrix $(A+\sum_{i=1}^n B_i)^{-1}$ where each $B_i$ is a sparse ...
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