I have the following quadratic matrix equation:

$ XAX+X = B $

where $A$ and $B$ are all positive definite matrix.

The constraint here is that $X$ is actually a covariance matrix and hence should be positive definite.

All the things I have got is that when there is no constraint, the equation can be solved via Bernoulli iteration in the following form:

$X_{k+1} = -A^{-1}(I-BX_k^{-1})$

However, this seems cannot preserve the constraint.

Any guidances would be appreciated, thank you.