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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