The Scaling and Squaring Method for the Matrix Exponential Revisited

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 MATLAB's {\tt expm} function. 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 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 {\tt expm} when the matrix norm exceeds 1, which can 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.