  The delay Lyapunov equation is an important matrix boundary-value problem
which arises as an analogue of the Lyapunov equation in the study of time-delay
systems $\dot{x}(t) = A_0x(t)+A_1x(t-\tau)+B_0u(t)$. We propose a new algorithm
for the solution of the delay Lyapunov equation. Our method is based on the
fact that the delay Lyapunov equation can be expressed as a linear system of
equations, whose unknown is the value $U(\tau/2)\in\mathbb{R}^{n\times n}$,
i.e., the delay Lyapunov matrix at time $\tau/2$. This linear matrix equation
with $n^2$ unknowns is solved by adapting a preconditioned iterative method
such as GMRES. The action of the $n^2\times n^2$ matrix associated to this
linear system can be computed by solving a coupled matrix initial-value
problem. A preconditioner for the iterative method is proposed based on solving
a T-Sylvester equation $MX+X^TN=C$, for which there are methods available in
the literature. We prove that the preconditioner is effective under certain
assumptions. The efficiency of the approach is illustrated by applying it to a
time-delay system stemmingfrom the discretization of a partial differential
equation with delay. Approximate solutions to this problem can be obtained for
problems of size up to $n\approx 1000$, i.e., a linear system with $n^2\approx
10^6$ unknowns, a dimension which is outside of the capabilities of the other
existing methods for the delay Lyapunov equation.
