How to solve this special case of Algebraic Matrix Riccati equations?

Question

I have a particular "Ricatti equation", for $X \in \mathbb{R}^{2 \times 2}$, where $A \in \mathbb{R}^{2 \times 2}$ is symmetric positive definite:

$$ X^T + X + X^T \cdot A \cdot X = \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} $$

$X$ need neither be unique nor symmetric. Any solution will do.

Since it is only in two dimensions and I want to implement it for an embedded target, can someone help me with how to solve it? I know very little about nonlinear matrix equations.

I assume there is a concise iteration formula for $X$ that involves factorizations and can be interpreted as a Newton-type (or other rapidly convergent) iteration and is asserted to converge when started from a particular trivial initial guess. ..or at least that's what I am hoping for :)

Answer 1

Rewrite the equation as $$ (X+A^{-1})^TA\ (X+A^{-1}) = B:= \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} + A^{-1}. $$ This is in the form of $Y^TAY=B$, where $Y=X+A^{-1}$. Since $A$ is positive definite, the equation is solvable if and only if $B$ is positive semidefinite. In case $B$ is really positive semidefinite, one obvious solution is given by $Y=R_A^{-1}R_B$, where $R_A^TR_A=A$ and $R_B^TR_B=B$ are Cholesky decompositions of $A$ and $B$ respectively. In terms of $X$, this means $$ X=Y-A^{-1}=R_A^{-1}R_B-A^{-1}=R_A^{-1}\left[R_B-(R_A^{-1})^T\right] $$ is a solution.