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Description
This work considers large-scale Lyapunov matrix equations of the form $AX+XA = cc^T$
with a symmetric positive definite matrix $A$ and a vector ${c}$. Motivated by the need for solving such equations in a broad range of applications, various numerical methods have been developed to compute a low-rank approximation to the solution
matrix $X$. In this work, we consider the Lanczos method, which has the distinct advantage that it only requires matrix-vector products with $A$ and makes it broadly applicable. At the same time, the Lanczos method may suffer from slow convergence when $A$ is ill-conditioned, leading to exessive memory requirements for storing the Krylov subspace basis generated by Lanczos. In this work, we alleviate this problem by developing a novel compression strategy for the Krylov subspace basis, which drastically reduces the memory requirement without impeding convergence. This is confirmed by both numerical experiments and convergence analysis.
