Accurate approximation of the hyperbolic matrix cosine using Bernoulli matrix polynomials

Abstract

Hyperbolic Matrix functions cosh (A) and sinh (A) emerge in various areas of science and technology, and its computation has attracted significant attention due to their usefulness in the solution of systems of second-order linear differential equations. In this work, we introduce Bernoulli matrix polynomial series expansions for hyperbolic matrix cosine function cosh (A) in order to obtain accurate and powerful methods for their computation.

Full text

Juan Ramón Torregrosa Juan Carlos Cortés Antonio Hervás Antoni Vidal Elena López-Navarro

Modelling for Engineering
& Human Behaviour 2021
Val`encia, July 14th-16th, 2021
This book includes the extended abstracts of papers presented at XXIII Edition of the Mathemat-
ical Modelling Conference Series at the Institute for Multidisciplinary Mathematics Mathematical
Modelling in Engineering & Human Behaviour.

I.S.B.N.: 978-84-09-36287-5
November 30th, 2021
Report any problems with this document to imm@imm.upv.es.
Edited by: I.U. de Matem`atica Multidisciplinar, Universitat Polit`ecnica de Val`encia.
J.R. Torregrosa, J-C. Cort´es, J. A. Herv´as, A. Vidal-Ferr`andiz and E. L´opez-Navarro

Contents
Density-based uncertainty quantification in a generalized Logistic-type model . . . . . . . . . . . 1
Combined and updated H–matrices.....................................................7
Solving random fractional second-order linear equations via the mean square Laplace
transform................................................................................ 13
Conformable fractional iterative methods for solving nonlinear problems . . . . . . . . . . . . . . . 19
Construction of totally nonpositive matrices associated with a triple negatively realizable24
Modeling excess weight in Spain by using deterministic and random differential equations31
A new family for solving nonlinear systems based on weight functions Kalitkin-Ermankov
type......................................................................................36
Solving random free boundary problems of Stefan type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Modeling one species population growth with delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
On a Ermakov–Kalitkin scheme based family of fourth order . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A new mathematical structure with applications to computational linguistics and spe-
cialized text translation .................................................................. 60
Accurate approximation of the Hyperbolic matrix cosine using Bernoulli matrix polyno-
mials.....................................................................................67
Full probabilistic analysis of random first-order linear differential equations with Dirac
delta impulses appearing in control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Some advances in Relativistic Positioning Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
A Graph–Based Algorithm for the Inference of Boolean Networks . . . . . . . . . . . . . . . . . . . . . . 84
Stability comparison of self-accelerating parameter approximation on one-step iterative
methods..................................................................................90
Mathematical modelling of kidney disease stages in patients diagnosed with diabetes
mellitus II................................................................................96
The effect of the memory on the spread of a disease through the environtment . . . . . . . . 101
Improved pairwise comparison transitivity using strategically selected reduced informa-
tion.....................................................................................106
Contingency plan selection under interdependent risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Some techniques for solving the random Burgers’ equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Probabilistic analysis of a class of impulsive linear random differential equations via
density functions........................................................................ 122
v

Modelling for Engineering & Human Behaviour 2021
Probabilistic evolution of the bladder cancer growth considering transurethral resection127
Study of a symmetric family of anomalies to approach the elliptical two body problem
with special emphasis in the semifocal case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132
Advances in the physical approach to personality dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A Laplacian approach to the Greedy Rank-One Algorithm for a class of linear systems 143
Using STRESS to compute the agreement between computed image quality measures and
observer scores: advantanges and open issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Probabilistic analysis of the random logistic differential equation with stochastic jumps156
Introducing a new parametric family for solving nonlinear systems of equations . . . . . . . 162
Optimization of the cognitive processes involved in the learning of university students in
a virtual classroom......................................................................167
Parametric family of root-finding iterative methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Subdirect sums of matrices. Definitions, methodology and known results. . . . . . . . . . . . . . 180
On the dynamics of a predator-prey metapopulation on two patches . . . . . . . . . . . . . . . . . . .186
Prognostic Model of Cost / Effectiveness in the therapeutic Pharmacy Treatment of Lung
Cancer in a University Hospital of Spain: Discriminant Analysis and Logit . . . . . . . . . . . . . . . 192
Stability, bifurcations, and recovery from perturbations in a mean-field semiarid vegeta-
tion model with delay................................................................... 197
The random variable transformation method to solve some randomized first-order linear
control difference equations..............................................................202
Acoustic modelling of large aftertreatment devices with multimodal incident sound fields
208
Solving non homogeneous linear second order difference equations with random initial
values: Theory and simulations..........................................................216
A realistic proposal to considerably improve the energy footprint and energy efficiency of
a standard house of social interest in Chile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
Multiobjective Optimization of Impulsive Orbital Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . 230
Mathematical Modeling about Emigration/Immigration in Spain: Causes, magnitude,
consequences............................................................................236
New scheme with memory for solving nonlinear problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
SP
NNeutron Noise Calculations...................................................... 246
Analysis of a reinterpretation of grey models applied to measuring laboratory equipment
uncertainties............................................................................252
An Optimal Eighth Order Derivative-Free Scheme for Multiple Roots of Non-linear Equa-
tions....................................................................................257
A population-based study of COVID-19 patient’s survival prediction and the potential
biases in machine learning...............................................................262
A procedure for detection of border communities using convolution techniques. . . . . . . . . 267
vi

12
Accurate approximation of the Hyperbolic matrix
cosine using Bernoulli matrix polynomials
E. Defezı,1J.J. Ib´a˜nezıJ.M. Alonso¯J. Peinado˜and J. Sastreú
(ı) Instituto Universitario de Matem´atica Multidisciplinar,
(¯) Instituto de Instrumentaci´on para Imagen Molecular,
(˜) Departamento de Sistemas Inform´aticos y Computaci´on,
(ú) Instituto de Telecomunicaciones y Aplicaciones Multimedia,
Universitat Polit`ecnica de Val`encia
Cam´ı de Vera s/n, Valencia, Spain.
1 Introduction and motivation
The evaluation of matrix functions plays an important and relevant role in many scientific ap-
plications because matrix functions have proven to be an efficient tool in applications such as
reduced order models [1], [2, pp. 275–303], image denoising [3] and graph neural networks [4],
among others.
Among the different matrix functions, we must highlight hyperbolic matrix functions. The com-
putation of the hyperbolic matrix functions has received remarkable attention in the last decades
due to its usefulness in the solution of systems of partial differential problems, see references [5,6]
for example. For this reason, several algorithms have been provided recently for computing these
matrix functions, looking for high precision in the approximation and economy of computational
cost, see [7, pp.403–407], [8–11] and references therein.
Also, the generalizations of some known classical special functions into matrix framework are
important both from the theoretical and applied point of view. These new extensions (Laguerre,
Hermite, Chebyshev, Jacobi matrix polynomials, etc.) have proved to be very useful in vari-
ous fields such as physics, engineering, statistics and telecommunications. Recently, Bernoulli
polynomials Bn(x), who are defined in [12] as the coefficients of the generating function
g(x, t)= tetx
et≠1=ÿ
nØ0
Bn(x)
n!tn,|t|<2fi,(1)
and that have the explicit expression for Bn(x)
Bn(x)=
n
ÿ
k=0 An
kBBkxn≠k,(2)
where the Bernoulli numbers are defined by Bn=Bn(0), satisfying the explicit recurrence
B0=1,Bk=≠
k≠1
ÿ
i=0
Bi
k+1≠i,k Ø1.(3)
1edefez@imm.upv.es

Modelling for Engineering & Human Behaviour 2021
have been generalized to the matrix framework in [13]: For a matrix AœCr◊r,thenth Bernoulli
matrix polynomial it is defined by the expression
Bn(A)=
n
ÿ
k=0 An
kBBkAn≠k.(4)
This matrix polynomials have the series expansion
eAt =Aet≠1
tBÿ
nØ0
Bn(A)tn
n!,|t|<2fi.(5)
To obtain practical approximations of the exponential matrix using the expansion (5), let’s take
“s” as the scaling of the matrix Aand take the degree of the approximation “m”, and then
eA2≠s¥(e≠1)
m
ÿ
n=0
Bn(A2≠s)
n!.(6)
The use of expansion (5) to approximate matrix exponential with good results of precision and
computational cost can be found in [13]. For a matrix AœCr◊r, using expression (5) we obtain
cosh (A) = sinh (1) ÿ
nØ0
B2n(A)
(2n)! + (cosh (1) ≠1)ÿ
nØ0
B2n+1(A)
(2n+ 1)! .(7)
Note that unlike the Taylor (and Hermite) polynomials that are even or odd, depending on the
parity of the polynomial degree n, the Bernoulli matrix polynomials do not verify this property,
so in the development of cosh(A) all Bernoulli polynomials are needed (and not just the even-
numbered). We can also obtain, for CœCr◊r, the expression:
cosh (C) = sinh (1) ÿ
nØ0
22nB2n11
2(C+I)2
(2n)! .(8)
The objective of this work is to present algorithms based on the approximations (7) and (8) for
the matrix hyperbolic cosine, trying to choose the most precise and with the lowest computational
cost.
2 The proposed Algorithms
From (7) one gets the approximation
cosh (A)¥sinh (1)
m
ÿ
n=0
B2n(A)
(2n)! + (cosh (1)≠1)
m
ÿ
n=0
B2n+1(A)
(2n+ 1)! ,(9)
and from (8) one gets the alternative approximation
cosh (C)¥sinh (1)
m
ÿ
n=0
22nB2n11
2(C+I)2
(2n)! .(10)
We are going to try to compare algorithms based on the approximations in practice (9)-(10). As
different algorithms are going to be used, we will to establish the following identification code
denoted by coshmber≠x≠y, where the argument is chosen according to the following criteria:
•We denote x= 1 if we use directly formula (9).
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Modelling for Engineering & Human Behaviour 2021
Numerical test 1
E(coshmber≠1≠3) <E(coshmber≠1≠4) 1.23% E(coshmber≠1≠3) <E(coshmber≠1≠5) 0.61%
E(coshmber≠1≠3) >E(coshmber≠1≠4) 40.49% E(coshmber≠1≠3) >E(coshmber≠1≠5) 0.00%
E(coshmber≠1≠3) = E(coshmber≠1≠4) 58.28% E(coshmber≠1≠3) = E(coshmber≠1≠5) 99.39%
Table 1: Errors in test 1
•We denote x= 2 if we use directly formula (10).
•We use x= 3 if formula (10) is used, but terms with odd powers have been removed.
By other hand, we have the argument yœ{3,4,5}, it is chosen according to the following criteria:
•We denote y= 3 if the evaluation of mand suse a norm estimation, similar to the given in
reference [14].
•We denote y= 4 if the evaluation of mand suse other algorithm for the norm estimation, see
reference [14] for more details.
•We denote y= 5 if the evaluation of mand sis made without norm estimation (calculating
the norms), see [14].
Our algorithm has been compared with algorithm funmcosh. This functions is funm MATLAB
function to compute matrix functions, such as the matrix hyperbolic cosine. All computations
was implemented on MATLAB 2020b.
Matrices and numerical test
For the numerical experiments a set of 153 test matrices matrices has been selected: 60 diagonal-
izable (Hadamard matrices), 60 non-diagonalizable, 39 from toolbox [15] and 13 from Eigtool [16].
Size 128◊128. We have performed a series of experiments to determine the best algorithm choice.
First we carry out the following tests:
•test 1: we compare each coshmber ≠1≠3, coshmber ≠1≠4, coshmber ≠1≠5.
•test 2: we compare each coshmber ≠2≠3, coshmber ≠2≠4, coshmber ≠2≠5.
•test 3: we compare each coshmber ≠3≠3, coshmber ≠3≠4, coshmber ≠3≠5.
Analysis of results of test 1
We compare algorithms coshmber ≠1≠3, coshmber≠1≠4, coshmber≠1≠5, obtaining the following
table 1 of results. With respect the computational cost, the total number of matrix products of
each algorithm was: coshmber≠1≠3 (1940), coshmber≠1≠4 (1872) and coshmber≠1≠5 (1939).
Among the three proposed algorithms (coshmber≠1≠3, coshmber≠1≠4, coshmber≠1≠5) we choose
algorithm coshmber ≠1≠4 because E(coshmber≠1≠3) >E(coshmber≠1≠4) in the 40.49% and the
number of matrix products is 1872, therefore, this algorithm coshmber ≠1≠4 has the lowest com-
putational cost. Regarding errors, algorithms coshmber ≠1≠3 and coshmber ≠1≠5 are practically
the same.
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Modelling for Engineering & Human Behaviour 2021
12345
0.5
0.6
0.7
0.8
0.9
1
p
coshmber_1_3
coshmber_1_4
coshmber_1_5
(a) Profile test 1.
Lowest relative error rate
27%
45%
27%
Highest relative error rate
38%
23%
39%
coshmber_1_3
coshmber_1_4
coshmber_1_5
(b) Pie charts Test 1.
Numerical test 2
E(coshmber≠2≠3) <E(coshmber≠2≠4) 23.93% E(coshmber≠2≠3) <E(coshmber≠2≠5) 0.61%
E(coshmber≠2≠3) >E(coshmber≠2≠4) 17.18% E(coshmber≠2≠3) >E(coshmber≠2≠5) 0.00%
E(coshmber≠2≠3) = E(coshmber≠2≠4) 58.90% E(coshmber≠2≠3) = E(coshmber≠2≠5) 99.39%
Table 2: Errors in test 2
Analysis of results of test 2
We compare algorithms coshmber ≠2≠3, coshmber≠2≠4, coshmber≠2≠5, obtaining the table 2
of results. With respect the computational cost, the total number of matrix products of each
algorithm was: coshmber≠2≠3 (1940), coshmber≠2≠4 (1872) and coshmber≠2≠5 (1939).
12345
0.75
0.8
0.85
0.9
0.95
p
coshmber_2_3
coshmber_2_4
coshmber_2_5
(a) Profile test 2.
Lowest relative error rate
34%
32%
34%
Highest relative error rate
32%
35%
33%
coshmber_2_3
coshmber_2_4
coshmber_2_5
(b) Pie charts Test 2.
Among the three proposed algorithms (coshmber≠2≠3, coshmber≠2≠4, coshmber≠2≠5) we choose
algorithm coshmber ≠2≠3 because E(coshmber≠2≠3) <E(coshmber≠2≠4) in the 23.93% despite
the fact that it has a higher computational cost (the number of matrix products is 1940). Re-
garding errors, algorithms coshmber ≠2≠3 and coshmber ≠2≠5 are practically the same.
Analysis of results of test 3
We compare algorithms coshmber ≠2≠3, coshmber≠2≠4, coshmber≠2≠5, obtaining the table 3
of results. With respect the computational cost, the total number of matrix products of each
algorithm was: coshmber≠3≠3 (1435), coshmber≠3≠4 (1336) and coshmber≠3≠5 (1325).
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Modelling for Engineering & Human Behaviour 2021
Numerical test 3
E(coshmber≠3≠3) <E(coshmber≠3≠4) 0.61% E(coshmber≠3≠3) <E(coshmber≠3≠5) 0.61%
E(coshmber≠3≠3) >E(coshmber≠3≠4) 64.42% E(coshmber≠3≠3) >E(coshmber≠3≠5) 0.00%
E(coshmber≠3≠3) = E(coshmber≠3≠4) 34.97% E(coshmber≠3≠3) = E(coshmber≠3≠5) 99.39%
Table 3: Errors in test 3
12345
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
p
coshmber_3_3
coshmber_3_4
coshmber_3_5
(a) Profile test 3.
Lowest relative error rate
21%
58%
21%
Highest relative error rate
42%
15%
43%
coshmber_3_3
coshmber_3_4
coshmber_3_5
(b) Pie charts Test 3.
Among the three proposed algorithms (coshmber≠3≠3, coshmber≠3≠4, coshmber≠3≠5) we choose
algorithm coshmber ≠3≠4 because E(coshmber≠3≠3) >E(coshmber≠3≠4) in the 64.42% and has
a lower computational cost (the number of matrix products is 1336). Regarding errors, algorithms
coshmber ≠3≠3 and coshmber ≠2≠5 are practically the same.
Analysis of results with MATLAB function funmcosh (Numerical test 4)
Finally, we will compare the selected algorithms coshmber≠1≠4, coshmber≠2≠3, coshmber≠3≠4
and the MATLAB function funmcosh, see Table 4. With respect the computational cost, the
total number of matrix products of each algorithm was: funmcosh: (2282), coshmber≠1≠4 (1872),
coshmber ≠2≠3 (1940) and coshmber ≠3≠4 (1336).
12345
0
0.2
0.4
0.6
0.8
1
p
funmcosh
coshmber_1_4
coshmber_2_3
coshmber_3_4
(a) Profile test 4.
Lowest relative error rate
2%
16%
38%
43%
Highest relative error rate
90%
3%
5%
2%
funmcosh
coshmber_1_4
coshmber_2_3
coshmber_3_4
(b) Pie charts Test 4.
In general, the relative error improvements over the MATLAB function funmcosh exceed 94% in
all cases. Between algorithms coshmber≠1≠4, coshmber≠2≠3, coshmber≠3≠4, we choose algorithm
coshmber ≠3≠4 because it has a lower computational cost (the number of total matrix products is
1336).
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Modelling for Engineering & Human Behaviour 2021
Numerical test 4
E(funmcosh)<E(coshmber≠1≠4) 1.84%
E(funmcosh)>E(coshmber≠1≠4) 96.32%
E(funmcosh)=E(coshmber≠1≠4) 1.84%
E(funmcosh)<E(coshmber≠2≠3) 3.68%
E(funmcosh)>E(coshmber≠2≠3) 94.48%
E(funmcosh)=E(coshmber≠2≠3) 1.84%
E(funmcosh)<E(coshmber≠3≠4) 0.61%
E(funmcosh)>E(coshmber≠3≠4) 97.55%
E(funmcosh)=E(coshmber≠3≠4) 1.84%
Table 4: Errors in test 4
3 Conclusions
In this work, different variations of algorithms have been presented to calculate the matrix hy-
perbolic cosine based on new Bernoulli matrix polynomials series expansions (7) and (8). These
algorithms have been tested on a battery of test matrices in order to select the best variants,
both in terms of computational cost as in terms of error in the approximation. The best selec-
tion (algorithm coshmber ≠3≠4) is based in formula (10), but terms with odd powers have been
removed, and in the evaluation of mand swhich use the algorithm for the norm estimation given
in reference [14].
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