## 2006.394: The Scaling and Squaring Method for the Matrix Exponential Revisited

2006.394:
Nicholas J. Higham
(2005)
*The Scaling and Squaring Method for the Matrix Exponential Revisited.*
SIAM Journal On Matrix Analysis and Applications, 26 (4).
pp. 1179-1193.
ISSN 1095-7162

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DOI: 10.1137/04061101X

## 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 {\tt 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 {\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 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.

Item Type: | Article |
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Uncontrolled Keywords: | matrix function; matrix exponential; Padé approximation; matrix polynomial evaluation; scaling and squaring method; MATLAB; \tt expm; backward error analysis; performance profile |

Subjects: | MSC 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis |

MIMS number: | 2006.394 |

Deposited By: | Nick Higham |

Deposited On: | 27 October 2006 |

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