Solving large-scale symmetric Lyapunov equations via a Lanczos method with compression

This work considers large-scale Lyapunov matrix equations of the form AX + XA = cc^T, where A is a symmetric positive definite matrix and c is a vector. Motivated by the need to solve such equations in a wide range of applications, various numerical methods have been developed to compute low-rank approximations of the solution matrix X. In this work, we focus on the Lanczos method, which has the distinct advantage of requiring only matrix-vector products with A, making it broadly applicable. However, the Lanczos method may suffer from slow convergence when A is ill-conditioned, leading to excessive memory requirements for storing the Krylov subspace basis generated by the algorithm. To address this issue, we propose a novel compression strategy for the Krylov subspace basis that significantly reduces memory usage without hindering convergence. This is supported by both numerical experiments and a convergence analysis. Our analysis also accounts for the loss of orthogonality due to round-off errors in the Lanczos process.