IEEE - Aerospace and Electronic Systems - May 2020 - 51

Corwell and Blair

Figure 4.
Computational time of the Matlab function, expm.m, and forward
summation of the Taylor series with relativistic truncation for
matrix norm < 40.

Figure 6.

of sparse positive-definite 14 Â 14 matrices with average
jjAjj % 5800, the superiority of the Taylor exponential is
further illustrated in Figure 5. Here, the computation time
for the Taylor exponential per the computation time for
the Matlab function, expm.m, for over 1000 different
matrices is shown. Remarkably, the Taylor approximation
achieved near perfect accuracy in as few as N ¼ 3 terms,
while regularly requiring less than a tenth of the time
required by the Matlab function.
Consider one more example where complete failure
results from significant digit error as detailed in [1]. Given
the matrix

the computation of eA fails using single-precision arithmetic with the reverse summation of the truncated Taylor
series. This is caused by catastrophic cancellation of significant digits. Note this example is an extreme case of
failure. For a more robust, conceptually simple calculation, scaling and squaring must be introduced.

À49
A¼
À64

24
31

Expm utilizing scaled and squared Taylor series approximations.

EXTENSIONS AND TIPS
Scaling and squaring is a method that aims to decrease the
scale of the matrix initially, run the exponential calculation,
and eventually reintroduce the scale of the matrix. This process manages the magnitude of the matrix to prevent errors
of extreme scale. This is described in [7] and [8] as
s

expðAÞ ¼ ðexpð2Às AÞÞ2

(2)

where s is an integer scaling parameter and A is the matrix
of interest. Figure 6 gives a sample implementation and
provides a means of finding s. It is desirable to keep s as
low as required to minimize rounding errors [7].
Another beneficial precomputation is shifting of the
matrix in order to cluster the eigenvalues as well as decrease
the scale of the matrix. Defining of the relationship
~

eAt ¼ eAt ebt

(3)

one may wish to find b such that the eigenvalues are
shifted around zero at the beginning of the computation.
~ and b may be selected such that
A

Figure 5.
Ratio of computational time of the reverse Taylor series approximation to the Matlab function, expm.m, on large (14 Â 14) positive-definite matrices. = cputime(Taylor expm)/cputime(Matlab
expm). Taylor approximations are roughly ten times faster and
require only three terms for an accurate approximation.

MAY 2020

~ ¼ A À bI
A
b¼

trðAÞ
n

IEEE A&E SYSTEMS MAGAZINE

(4)

(5)

51

IEEE - Aerospace and Electronic Systems - May 2020

Table of Contents for the Digital Edition of IEEE - Aerospace and Electronic Systems - May 2020

Contents
IEEE - Aerospace and Electronic Systems - May 2020 - Cover1
IEEE - Aerospace and Electronic Systems - May 2020 - Cover2
IEEE - Aerospace and Electronic Systems - May 2020 - Contents
IEEE - Aerospace and Electronic Systems - May 2020 - 2
IEEE - Aerospace and Electronic Systems - May 2020 - 3
IEEE - Aerospace and Electronic Systems - May 2020 - 4
IEEE - Aerospace and Electronic Systems - May 2020 - 5
IEEE - Aerospace and Electronic Systems - May 2020 - 6
IEEE - Aerospace and Electronic Systems - May 2020 - 7
IEEE - Aerospace and Electronic Systems - May 2020 - 8
IEEE - Aerospace and Electronic Systems - May 2020 - 9
IEEE - Aerospace and Electronic Systems - May 2020 - 10
IEEE - Aerospace and Electronic Systems - May 2020 - 11
IEEE - Aerospace and Electronic Systems - May 2020 - 12
IEEE - Aerospace and Electronic Systems - May 2020 - 13
IEEE - Aerospace and Electronic Systems - May 2020 - 14
IEEE - Aerospace and Electronic Systems - May 2020 - 15
IEEE - Aerospace and Electronic Systems - May 2020 - 16
IEEE - Aerospace and Electronic Systems - May 2020 - 17
IEEE - Aerospace and Electronic Systems - May 2020 - 18
IEEE - Aerospace and Electronic Systems - May 2020 - 19
IEEE - Aerospace and Electronic Systems - May 2020 - 20
IEEE - Aerospace and Electronic Systems - May 2020 - 21
IEEE - Aerospace and Electronic Systems - May 2020 - 22
IEEE - Aerospace and Electronic Systems - May 2020 - 23
IEEE - Aerospace and Electronic Systems - May 2020 - 24
IEEE - Aerospace and Electronic Systems - May 2020 - 25
IEEE - Aerospace and Electronic Systems - May 2020 - 26
IEEE - Aerospace and Electronic Systems - May 2020 - 27
IEEE - Aerospace and Electronic Systems - May 2020 - 28
IEEE - Aerospace and Electronic Systems - May 2020 - 29
IEEE - Aerospace and Electronic Systems - May 2020 - 30
IEEE - Aerospace and Electronic Systems - May 2020 - 31
IEEE - Aerospace and Electronic Systems - May 2020 - 32
IEEE - Aerospace and Electronic Systems - May 2020 - 33
IEEE - Aerospace and Electronic Systems - May 2020 - 34
IEEE - Aerospace and Electronic Systems - May 2020 - 35
IEEE - Aerospace and Electronic Systems - May 2020 - 36
IEEE - Aerospace and Electronic Systems - May 2020 - 37
IEEE - Aerospace and Electronic Systems - May 2020 - 38
IEEE - Aerospace and Electronic Systems - May 2020 - 39
IEEE - Aerospace and Electronic Systems - May 2020 - 40
IEEE - Aerospace and Electronic Systems - May 2020 - 41
IEEE - Aerospace and Electronic Systems - May 2020 - 42
IEEE - Aerospace and Electronic Systems - May 2020 - 43
IEEE - Aerospace and Electronic Systems - May 2020 - 44
IEEE - Aerospace and Electronic Systems - May 2020 - 45
IEEE - Aerospace and Electronic Systems - May 2020 - 46
IEEE - Aerospace and Electronic Systems - May 2020 - 47
IEEE - Aerospace and Electronic Systems - May 2020 - 48
IEEE - Aerospace and Electronic Systems - May 2020 - 49
IEEE - Aerospace and Electronic Systems - May 2020 - 50
IEEE - Aerospace and Electronic Systems - May 2020 - 51
IEEE - Aerospace and Electronic Systems - May 2020 - 52
IEEE - Aerospace and Electronic Systems - May 2020 - Cover3
IEEE - Aerospace and Electronic Systems - May 2020 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/aerospace_december2023
https://www.nxtbook.com/nxtbooks/ieee/aerospace_november2023
https://www.nxtbook.com/nxtbooks/ieee/aerospace_october2023
https://www.nxtbook.com/nxtbooks/ieee/aerospace_september2023
https://www.nxtbook.com/nxtbooks/ieee/aerospace_august2023
https://www.nxtbook.com/nxtbooks/ieee/aerospace_july2023
https://www.nxtbook.com/nxtbooks/ieee/aerospace_june2023
https://www.nxtbook.com/nxtbooks/ieee/aerospace_may2023
https://www.nxtbook.com/nxtbooks/ieee/aerospace_april2023
https://www.nxtbook.com/nxtbooks/ieee/aerospace_march2023
https://www.nxtbook.com/nxtbooks/ieee/aerospace_february2023
https://www.nxtbook.com/nxtbooks/ieee/aerospace_january2023
https://www.nxtbook.com/nxtbooks/ieee/aerospace_december2022
https://www.nxtbook.com/nxtbooks/ieee/aerospace_november2022
https://www.nxtbook.com/nxtbooks/ieee/aerospace_october2022
https://www.nxtbook.com/nxtbooks/ieee/aerospace_september2022
https://www.nxtbook.com/nxtbooks/ieee/aerospace_august2022
https://www.nxtbook.com/nxtbooks/ieee/aerospace_july2022
https://www.nxtbook.com/nxtbooks/ieee/aerospace_june2022
https://www.nxtbook.com/nxtbooks/ieee/aerospace_may2022_tutorial
https://www.nxtbook.com/nxtbooks/ieee/aerospace_may2022
https://www.nxtbook.com/nxtbooks/ieee/aerospace_april2022
https://www.nxtbook.com/nxtbooks/ieee/aerospace_march2022
https://www.nxtbook.com/nxtbooks/ieee/aerospace_february2022
https://www.nxtbook.com/nxtbooks/ieee/aerospace_january2022
https://www.nxtbook.com/nxtbooks/ieee/aerospace_december2021
https://www.nxtbook.com/nxtbooks/ieee/aerospace_november2021
https://www.nxtbook.com/nxtbooks/ieee/aerospace_october2021
https://www.nxtbook.com/nxtbooks/ieee/aerospace_september2021
https://www.nxtbook.com/nxtbooks/ieee/aerospace_august2021
https://www.nxtbook.com/nxtbooks/ieee/aerospace_july2021
https://www.nxtbook.com/nxtbooks/ieee/aerospace_june2021
https://www.nxtbook.com/nxtbooks/ieee/aerospace_may2021
https://www.nxtbook.com/nxtbooks/ieee/aerospace_april2021
https://www.nxtbook.com/nxtbooks/ieee/aerospace_march2021
https://www.nxtbook.com/nxtbooks/ieee/aerospace_march2021_tutorials
https://www.nxtbook.com/nxtbooks/ieee/aerospace_february2021
https://www.nxtbook.com/nxtbooks/ieee/aerospace_january2021
https://www.nxtbook.com/nxtbooks/ieee/aerospace_november2020
https://www.nxtbook.com/nxtbooks/ieee/aerospace_december2020
https://www.nxtbook.com/nxtbooks/ieee/aerospace_october2020
https://www.nxtbook.com/nxtbooks/ieee/aerospace_september2020
https://www.nxtbook.com/nxtbooks/ieee/aerospace_august2020
https://www.nxtbook.com/nxtbooks/ieee/aerospace_july2020
https://www.nxtbook.com/nxtbooks/ieee/aerospace_june2020
https://www.nxtbook.com/nxtbooks/ieee/aerospace_may2020
https://www.nxtbook.com/nxtbooks/ieee/aerospace_april2020
https://www.nxtbook.com/nxtbooks/ieee/aerospace_march2020
https://www.nxtbook.com/nxtbooks/ieee/aerospace_february2020
https://www.nxtbook.com/nxtbooks/ieee/aerospace_january2020
https://www.nxtbook.com/nxtbooks/ieee/aerospace_december2019
https://www.nxtbook.com/nxtbooks/ieee/aerospace_october2019partII
https://www.nxtbook.com/nxtbooks/ieee/aerospace_november2019
https://www.nxtbook.com/nxtbooks/ieee/aerospace_october2019
https://www.nxtbook.com/nxtbooks/ieee/aerospace_july2019
https://www.nxtbook.com/nxtbooks/ieee/aerospace_september2019
https://www.nxtbook.com/nxtbooks/ieee/aerospace_august2019
https://www.nxtbook.com/nxtbooks/ieee/aerospace_june2019
https://www.nxtbook.com/nxtbooks/ieee/aerospace_april2019
https://www.nxtbook.com/nxtbooks/ieee/aerospace_may2019
https://www.nxtbook.com/nxtbooks/ieee/aerospace_march2019
https://www.nxtbook.com/nxtbooks/ieee/aerospace_december2018
https://www.nxtbook.com/nxtbooks/ieee/aerospace_august2018
https://www.nxtbook.com/nxtbooks/ieee/aerospace_october2018
https://www.nxtbook.com/nxtbooks/ieee/aerospace_september2018
https://www.nxtbook.com/nxtbooks/ieee/aerospace_november2018
https://www.nxtbookmedia.com