We develop a mixed-precision iterative refinement framework for solving low-rank Lyapunov matrix equations $AX + XA^T + W =0$, where $W=LL^T$ or $W=LSL^T$. Via rounding error analysis of the algorithms we derive sufficient conditions for the attainable normwise residuals in different precision settings and show how the algorithmic parameters should be chosen. Using the sign function Newton iteration as the solver, we show that reduced precisions, such as the half precision, can be used as the solver precision (with unit roundoff $u_s$) to accelerate the solution of Lyapunov equations of condition number up to $1/u_s$ without compromising its quality.

翻译：我们开发了一种混合精度迭代精化框架，用于求解低秩李雅普诺夫矩阵方程 $AX + XA^T + W =0$，其中 $W=LL^T$ 或 $W=LSL^T$。通过对算法的舍入误差分析，我们推导了在不同精度设置下可达到的范数残差的充分条件，并说明了应如何选择算法参数。使用符号函数牛顿迭代作为求解器，我们证明了降低的精度（例如半精度）可用作求解器精度（单位舍入为 $u_s$），以加速求解条件数高达 $1/u_s$ 的李雅普诺夫方程，且不损害其求解质量。