The Scaling and Squaring Method for the Matrix Exponential Revisited
The Scaling and Squaring Method for the Matrix Exponential Revisited
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DOI：
10.1137/090768539
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发表时间：
2009-12-01
期刊：
影响因子：
10.2
通讯作者：
Higham, Nicholas J.
中科院分区：
文献类型：
--
作者：
Higham, Nicholas J.
缩放和平方方法是计算矩阵指数最广泛使用的方法。尤其是因为它是在 MATLAB 函数 expm 中实现的方法。该方法按 2 的幂缩放矩阵，将范数降低到 1 阶，计算 a。填充近似于矩阵指数，然后重复平方以消除缩放的影响。我们对该方法（以精确算术）进行了新的后向误差分析，该方法对截断误差采用锐界，并实现了本质上最佳效率的实现。我们还给出了新的舍入误差分析，表明计算出的缩放矩阵的 Pade 近似值非常准确。对于 IEEE 双精度算术，Pade 近似次数的最佳选择是 13，而不是先前作者使用的 6 或 8。当矩阵范数超过 1 时，我们的缩放和平方方法的实现总是需要比 MATLAB 7.0 中的 expm 函数至少少两次的矩阵乘法，这相当于节省了 37% 的乘法次数，而且由于所需的平方更少，因此通常更准确。我们还研究了 Najfeld 和 Havel 提出的不同的缩放和平方算法，该算法采用函数 x coth(x) 的 Pade 近似。人们发现这种方法本质上是标准方法的一种变体，支持误差分析较弱。
The scaling and squaring method is the most widely used method for Computing the matrix exponential. not least because it is the method implemented in the MATLAB function expm. The method scales the matrix by a power of 2 to reduce the norm to order 1, computes a. Pade approximant to the matrix exponential, and then repeatedly squares to undo the effect of the scaling. We give a new backward error analysis of the method (in exact arithmetic) that employs sharp bounds for the truncation errors and leads to an implementation of essentially optimal efficiency. We also give a new rounding error analysis that shows the computed Pade approximant of the scaled matrix to be highly accurate. For IEEE double precision arithmetic the best choice of degree of Pade approximant turns out to be 13, rather than the 6 or 8 used by previous authors. Our implementation of the scaling and Squaring method always requires at least two fewer matrix multiplications than the expm function in MATLAB 7.0 when the matrix norm exceeds 1, which call amount to a 37% saving in the number of multiplications, and it is typically more accurate, owing to the fewer required squarings. We also investigate a different scaling and squaring algorithm proposed by Najfeld and Havel that employs a Pade approximation to the function x coth(x). This method is found to be essentially a variation of the standard one with weaker supporting error analysis.