Speaker
Description
Let $A,S,X_0\in \mathbb R^{n\times n}$ and $ B \in \mathbb R^{n\times b}$ be. Moreover assume that $S$, $X_0$ are symmetric positive semidefinite.
We consider
\begin{align}
\dot{X}(t) &= A^T X (t) + X(t) A - X(t) S X(t) + C^T C, \\
X(0)&=X_0.
\end{align}
It is well known that the solution $X$ can be obtained from the Hamiltonian system
\begin{align}
\begin{bmatrix} \dot{U}(t) \\ \dot{V}(t) \end{bmatrix}
&=
-H \begin{bmatrix} U(t) \\ V(t) \end{bmatrix}
=
-\begin{bmatrix} A & -S \\ -C^T C & -A^T \end{bmatrix}
\begin{bmatrix} U(t) \\ V(t) \end{bmatrix}, \\
\begin{bmatrix} U(0) \\ V(0) \end{bmatrix}
&=
\begin{bmatrix} I_n \\ X_0 \end{bmatrix},
\end{align}
by the formula $X(t) = V(t) U(t)^{-1}$.
The solution of the Hamiltonian system can be expressed in terms of the matrix exponential of the Hamiltonian $-H$, which is symplectic.
This motivates to approximate the flow by numerical schemes, which uses symplectic transformations.
