IEEE - Aerospace and Electronic Systems - May 2020 - 49

DOI. No. 10.1109/MAES.2020.2982757

Industry Insights

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Industry Tip: Quick and Easy Matrix Exponentials
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James Corwell, William Dale Blair, Georgia Tech Research Institute

INTRODUCTION
Matrix exponential computation is famously detailed by
Moler and Van Loan in [1]. The most popular method for
computing the matrix exponential involves the scaled and
squared Pade approximations as described in [2]. Despite
the popularity of this approach, it is often computationally
slower than necessary. A simple method of computing the
matrix exponential is the reverse summation truncated
Taylor series utilizing Horner's Rule-a computational
trick dating back to the 11th century in China [3].
The definition of matrix exponentiation is given by
eA ¼

1
X
An
n¼0

n!

Forward summation of Taylor series approximation for matrix
exponentials as detailed in [1], utilizing a relative stopping parameter.

:

In practice, this infinite sum must be approximated with a
finite sum of N terms. The summation is made forward
from n ¼ 0 to N À 1; where N can be determined by
monitoring the incremental change to the sum and stopping the summation when the increment falls below a
meaningful value relative to the desired precision. A Matlab function that efficiently performs the forward summation is given in Figure 1.
A Matlab function for the reverse finite summation is
shown in Figure 2, where N is specified by the user and
determined prior to the function call. However, N that is
required for the summation to achieve a desired precision
depends on the matrix. If one has some domain knowledge
of the types of matrices that one will be processing, the
number of terms needed for a successful approximation
may be found intuitively. For a more robust calculation
to find the proper amount of terms, according to the literature [1], [4], a reasonable value of N is given by the
Authors' current addresses: James Corwell, William D.
Blair, Georgia Tech Research Institute, Georgia Institute of Technology, Atlanta, GA, 30332 USA (e-mails:
james.corwell@gtri.gatech.edu; dale.blair@gtri.gatech.
edu).
Manuscript received August 15, 2019, revised
November 27, 2019; accepted January 26, 2020, and
ready for publication March 20, 2020.
Review handled by Haiying Liu.
0885-8985/20/$26.00 ß 2020 UGGoc
MAY 2020

Figure 1.

Figure 2.
Reverse summation of Taylor series approximation for matrix
exponentials. Here, the number of iterations is passed in, but it
may be a fixed value within the function or dynamically allocated.

minimum N that satisfies
jjTN ðAÞ À eA jj

jjAjjNþ1
1
ðN þ 1Þ! 1 À jjAjj=ðN þ 2Þ

d

(1)

where TN ðAÞ is the finite sum of N terms, d is the prescribed error tolerance, and jj Á jj is a matrix norm.
The accuracy of this method is demonstrated via an
example of a linear system with zero input and specified
initial conditions. Given the linear system specified by
2

3
6
3 À2
A ¼ 4 À4 À1 2 5
13
9 À3
with initial conditions
3
À2
~
xð0Þ ¼ 4 1 5:
4

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