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On Bernoulli series approximation for the matrix cosine

DOI: 10.1002/mma.7041
Authors:
Emilio Defez at Universitat Politècnica de València
Emilio Defez
Jacinto Javier Ibáñez at Universitat Politècnica de València
Jacinto Javier Ibáñez
José M. Alonso at Universitat Politècnica de València
José M. Alonso
Pedro Alonso at Universitat Politècnica de València
Pedro Alonso
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Abstract and Figures

This paper presents a new series expansion based on Bernoulli matrix polynomials to approximate the matrix cosine function. An approximation based on this series is not a straightforward exercise since there exist different options to implement such a solution. We dive into these options and include a thorough comparative of performance and accuracy in the experimental results section that shows benefits and downsides of each one. Also, a comparison with the Padé approximation is included. The algorithms have been implemented in MATLAB and in CUDA for NVIDIA GPUs.
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Received: 30 April 2020
DOI: 10.1002/mma.7041
SPECIAL ISSUE PAPER
On Bernoulli series approximation for the matrix cosine
Emilio Defez1Javier Ibáñez2José M. Alonso2Pedro Alonso-Jordá3
1Instituto de Matemática Multidisciplinar,
Universitat Politècnica de València,
Valencia, Spain
2Instituto de Instrumentación para
Imagen Molecular, Universitat Politècnica
de València, Valencia, Spain
3Department of Information Systems and
Computation, Universitat Politècnica de
València, Valencia, Spain
Correspondence
Pedro Alonso-Jordá, Department of
Information Systems and Computation,
Universitat Politècnica de València,
Camino de Vera s/n, 46022 Valencia,
Spain.
Email: palonso@upv.es
Communicated by: M. Tosun
Funding information
Spanish Ministerio de Econom𝚤ay
Competitividad and European Regional
Development Fund, Grant/Award
Number: TIN2017-89314-P; Universitat
Politecnica de Valencia, Grant/Award
Number: SP20180016
This paper presents a new series expansion based on Bernoulli matrix polyno-
mials to approximate the matrix cosine function. An approximation based on
this series is not a straightforward exercise since there exist different options to
implement such a solution. We dive into these options and include a thorough
comparative of performance and accuracy in the experimental results section
that shows benefits and downsides of each one. Also, a comparison with the Padé
approximation is included. The algorithms have been implemented in MATLAB
and in CUDA for NVIDIA GPUs.
KEYWORDS
matrix exponential and similar functions of matrices, polynomials and matrices
MSC CLASSIFICATION
65F60; 68W10
1INTRODUCTION AND NOTATION
In recent years, the study of matrix functions has been the subject of increasing focus due to its usefulness in various areas
of science and engineering, providing new and interesting problems to those already existing and already well known. Of
all matrix functions, it is certainly the matrix exponential which attracts much of the attention because of its connection
with systems of first order linear differential equations
Y′(t)=AY (t)
Y(0)=Y0,A∈Cr×r,
whose solution is given by Y(t)=eAtY0and where Cr×rrepresents the set of all complex square matrices of size r.The
hyperbolic matrix functions are applied in the study of the communicability analysis in complex networks1-3 and also in
the solution of coupled hyperbolic systems of partial differential equations.4In particular, the sine and cosine trigono-
metric matrix functions have been proven to be especially useful for solving systems of second-order linear differential
equations of the form:
d2
dt2Y(t)+A2Y(t)=0
Y(0)=Y0
Y′(0)=Y′
0
,A∈Cr×r,
Math Meth Appl Sci. 2022;45:3239–3253. wileyonlinelibrary.com/journal/mma © 2020 John Wiley & Sons, Ltd. 3239
... Taking into account the precomputed values of Θ m k andΘ m k of Tables 1-3, Algorithm 4 computes the most appropriate values of polynomial order m and scaling parameter s. In fact, Algorithm 4 is an improvement on [38]'s Algorithm 4, where it is explained in further detail. The main difference with respect to [38] is that the new code can be used to determine the values of m and s independently of the nature of the error, covering relative, absolute, forward and backward errors. ...
... In fact, Algorithm 4 is an improvement on [38]'s Algorithm 4, where it is explained in further detail. The main difference with respect to [38] is that the new code can be used to determine the values of m and s independently of the nature of the error, covering relative, absolute, forward and backward errors. This is accommodated by means of Line 8 in Algorithm 4, which takes into account that the first non-zero term occupies position m i , for the relative backward error series, or m i + 1, for the absolute/relative forward or absolute backward error ones. ...
... Additionally, α m is approximated as α m ≈ A m 1/m , as justified in [38]. In line 18, p m i is the highest-order coefficient of the approximating polynomial. ...
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