扰动矩阵指数的缩放、分裂和平方方法
The Scaling, Splitting and Squaring Method for the Exponential of Perturbed Matrices
AI总结：
本文提出用于计算扰动矩阵指数的分裂方法，通过缩放和平方技术提高计算效率，实验证明在中等精度下性能优于现有方法。
摘要：
我们提出了一种用于计算扰动矩阵指数的分裂方法，该方法将矩阵表示为稀疏且可高效指数化的矩阵D与小范数的密集矩阵εB之和。主要算法通过将大矩阵A缩放为小数2^{-s}，然后通过高效的Padé或泰勒方法进行指数运算，最后平方以近似完整指数。计算成本主要来自密集矩阵乘法，我们提出一种改进的平方方法，利用扰动矩阵B的小性来减少所需的平方次数。理论结果和数值实验表明，在中等精度下，该方法优于现有方法。
英文摘要：
We propose splitting methods for the computation of the exponential of perturbed matrices which can be written as the sum $A=D+\varepsilon B$ of a sparse and efficiently exponentiable matrix $D$ with sparse exponential $e^D$ and a dense matrix $\varepsilon B$ which is of small norm in comparison with $D$. The predominant algorithm is based on scaling the large matrix $A$ by a small number $2^{-s}$, which is then exponentiated by efficient Padé or Taylor methods and finally squared in order to obtain an approximation for the full exponential. In this setting, the main portion of the computational cost arises from dense-matrix multiplications and we present a modified squaring which takes advantage of the smallness of the perturbed matrix $B$ in order to reduce the number of squarings necessary. Theoretical results on local error and error propagation for splitting methods are complemented with numerical experiments and show a clear improvement over existing methods when medium precision is sought.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
