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的数值方法，用于计算Cholesky分解形式的有限时域能控性格拉姆矩阵。该方法适用于中等规模的一般稠密矩阵，可在不计算完整矩阵乘积的情况下直接获得格拉姆矩阵的Cholesky因子。与现有其他方法不同，本方法是对矩阵指数逼近中缩放及平方方法（scaling-and-squaring approach）的推广，通过利用格拉姆矩阵的类似倍乘公式，有效控制计算量。尤为重要的是，本文提供了严格的向后误差分析，保证近似结果在双精度浮点运算中达到舍入误差级别的精度。通过大量标准测试实例验证了实际计算精度。该方法已在Julia语言包FiniteHorizonGramians.jl中实现，并以MIT许可证在线开源。该软件包包含复现实验结果的代码及确定最优方法参数的代码，因此本文的分析可便捷地适配不同精度浮点运算环境。