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

Question

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.

Answer 1

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.