# Solve $(A+B)x=y$ given Cholesky decomposition of A and B

I wish to solve for $x$ in

$$(A+B)x=y$$

given square symmetric matrices $A$ and $B$. For certain reasons I have already computed the Cholesky decompositions for A and B:

$$A = L^T L$$ $$B = M^T M$$

Can I use these solve for $x$ more efficiently than by naively computing the Cholesky decomposition of $A+B$?

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For the Cholesky decomposition we should also assume that $A$ and $B$ are positive definite, I thought. –  Dietrich Burde May 22 '13 at 19:13
In general, no. One can make low-rank updates of factorization easily, but for anything different the usual answer is "you have to compute everything from scratch". Maybe you can use CG with $A$ (or $B$) as a preconditioner, but that will be numerically convenient only in some special cases. –  Federico Poloni May 22 '13 at 19:50