  In this paper, we consider finding a low-rank approximation to the solution
of a large-scale generalized Lyapunov matrix equation in the form of $A X M + M
X A = C$, where $A$ and $M$ are symmetric positive definite matrices. An
algorithm called an Increasing Rank Riemannian Method for Generalized Lyapunov
Equation (IRRLyap) is proposed by merging the increasing rank technique and
Riemannian optimization techniques on the quotient manifold $\mathbb{R}_*^{n
\times p} / \mathcal{O}_p$. To efficiently solve the optimization problem on
$\mathbb{R}_*^{n \times p} / \mathcal{O}_p$, a line-search-based Riemannian
inexact Newton's method is developed with its global convergence and local
superlinear convergence rate guaranteed. Moreover, we derive a preconditioner
which takes $M \neq I$ into consideration. Numerical experiments show that the
proposed Riemannian inexact Newton's method and preconditioner has superior
performance and IRRLyap is preferable compared to the tested state-of-the-art
methods when the lowest rank solution is desired.
