In this article, an efficient numerical method for computing finite-horizon controllability Gramians in Cholesky-factored form is proposed. The method is applicable to general dense matrices of moderate size and produces a Cholesky factor of the Gramian without computing the full product. In contrast to other methods applicable to this task, the proposed method is a generalization of the scaling-and-squaring approach for approximating the matrix exponential. It exploits a similar doubling formula for the Gramian, and thereby keeps the required computational effort modest. Most importantly, a rigorous backward error analysis is provided, which guarantees that the approximation is accurate to the round-off error level in double precision. This accuracy is illustrated in practice on a large number of standard test examples. The method has been implemented in the Julia package FiniteHorizonGramians.jl, which is available online under the MIT license. Code for reproducing the experimental results is included in this package, as well as code for determining the optimal method parameters. The analysis can thus easily be adapted to a different finite-precision arithmetic.

翻译：本文提出了一种高效数值方法，用于计算乔列斯基分解形式的有限时域可控性格兰姆。该方法适用于中等规模的一般稠密矩阵，可在不计算完整乘积的情况下获得格兰姆的乔列斯基因子。与同类方法相比，本方法是对矩阵指数近似计算中缩放-平方方法的推广，通过利用格兰姆的类似倍增公式，有效降低了计算成本。尤为重要的是，本文给出了严格的向后误差分析，保证双精度计算下近似值可达到舍入误差级别。通过大量标准测试案例的实践验证了该精度。该方法已在Julia软件包FiniteHorizonGramians.jl中实现，该包基于MIT许可证开源提供。软件包内含重复实验结果的代码，以及确定最优方法参数的代码，因此该分析可轻松适配不同有限精度算术环境。