Abstract
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 expm function. 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 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 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. © 2005 Society for Industrial and Applied Mathematics.
Original language  English 

Pages (fromto)  11791193 
Number of pages  14 
Journal  SIAM Journal on Matrix Analysis and Applications 
Volume  26 
Issue number  4 
DOIs  
Publication status  Published  2005 
Keywords
 Backward error analysis
 Expm
 MATLAB
 Matrix exponential
 Matrix function
 Matrix polynomial evaluation
 Padé approximation
 Performance profile
 Scaling and squaring method
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Mathematical Software for Computing Matrix Functions
Nicholas Higham (Participant), Francoise Tisseur (Participant) & Philip Davies (Participant)
Impact: Economic, Technological