I have the following quadratic matrix equation:
$$ XAX+X = B $$
where both $A$ and $B$ are given positive definite matrices, and $X$ is a covariance matrix and, hence, positive definite.
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 does not seems to preserve positive semidefinite.
Any guidance would be appreciated. Thank you.