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I wish to solve numerically for $x$,

$$ y = (A+B^{-1})x $$

with $A, B$ positive definite. So,

$$ x = (A+B^{-1})^{-1}y $$

I would like to avoid calculating $B^{-1}$ since that's generally bad.

This question seems unusually short. I can provide extra info if needed.

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You can just do an intermediate solve: Solve $Bw=x$, then $y=Ax+w$ – KevinG Jan 26 '13 at 19:57
That sounds like just the kind of thing I need, but I don't know what that is. Can you elaborate? Maybe I'm just dense. – John Salvatier Jan 26 '13 at 19:58
Sorry about that... I always forget get that <return> submits a comment... editted above. – KevinG Jan 26 '13 at 19:59
@JohnSalvatier Could you clarify whether you are solving $x$ for a known $y$, or solving $y$ for a known $x$? – Erick Wong Jan 26 '13 at 20:00
You want to solve for $Ax + B^{-1}x = y$ i.e. you want to solve for $$(BA+I)x = By$$ If $\Vert BA \Vert < 1$, then we have $$x = \sum_{k=0}^{\infty} (-1)^k (BA)^k By$$ This can be truncated to get arbitrarily accurate results. IF you truncate after $r$ terms the cost to evaluate this goes as $\mathcal{O}(rn^2)$ where $n$ is the size of the matrices $A$ and $B$. – user17762 Jan 26 '13 at 20:08
up vote 2 down vote accepted

I would take Marvis' answer one step ahead. You want to solve $(BA+I)x=By$. Because $BA+I$ is positive semidefinite, you can use conjugate gradient. The only operation you need is matrix multiplication and convergence is guarantee to an accurate solution within a finite number of steps.

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