Papers › Lanczos with compression for symmetric matrix Lyapunov equations
Lanczos with compression for symmetric matrix Lyapunov equations
Angelo A. Casulli, Francesco Hrobat, Daniel Kressner
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This work considers large-scale Lyapunov matrix equations of the form AX + XA = ccᵀ, 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.
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