Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators
SIAM Journal on Scientific Computing
Journal of Computational and Applied Mathematics
Original article: Further properties of random orthogonal matrix simulation
Mathematics and Computers in Simulation
Static Network Reliability Estimation via Generalized Splitting
INFORMS Journal on Computing
Reducing the influence of tiny normwise relative errors on performance profiles
ACM Transactions on Mathematical Software (TOMS)
Hi-index | 0.00 |
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 Padé 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 Padé approximant of the scaled matrix to be highly accurate. For IEEE double precision arithmetic the best choice of degree of Padé 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 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 Padé 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.