AI 中文总结
针对大规模稀疏连续时间代数黎卡提方程，提出基于牛顿法的混合精度迭代细化框架，通过特定因子分解程序和误差分析，结合ADI实现，减少计算量，数值实验显示其在精度相同情况下比全双精度更快。
AI 中文摘要
我们提出了一种基于牛顿法的混合精度迭代细化框架来求解大规模稀疏连续时间代数黎卡提方程（CAREs）。该框架以较低精度计算初始近似和内部李雅普诺夫校正方程，同时以较高精度评估残差和更新解。为处理低秩形式的不定残差和牛顿校正项，引入了具有截断策略的因子分解程序。通过一阶舍入误差分析得出细化过程的残差递推关系，并将稳定的混合精度细化与由较低精度内部求解的单位舍入控制的李雅普诺夫算子条件阈值相关联。还给出了基于ADI的具体实现，与密集李雅普诺夫校正实现相比，减少了主要计算量。数值实验表明该混合精度框架在保持精度的同时比全双精度实现更快。
英文摘要
We propose a Newton-based mixed precision iterative refinement framework for solving large-scale sparse continuous-time algebraic Riccati equations (CAREs). The framework computes the initial approximation and the inner Lyapunov correction equations in lower precision, while evaluating residuals and updating the solution in higher precision. To handle indefinite residuals and Newton correction terms in low-rank form, we introduce factor decomposition procedures with truncation strategies that preserve positive semidefiniteness and control rank growth. A first-order rounding error analysis derives a residual recurrence for the refinement process and relates stable mixed precision refinement to a Lyapunov operator conditioning threshold governed by the unit roundoff of the lower precision inner solves. We then present a concrete ADI-based realization, using NLR-ADI for the initial CARE approximation and LR-ADI for the inner Lyapunov correction equations. Compared with dense Lyapunov correction implementations, this realization reduces the main computations to shifted linear solves and low-rank factor operations, and we provide a solver-dependent complexity analysis. Numerical experiments on dense CARE over a range of condition numbers illustrate the conditioning effect described by the error analysis, and experiments on large-scale sparse CAREs show that the mixed precision framework is faster than the full double precision implementation while maintaining the same level of accuracy.
Comments18 pages, 5 tables
