New matrix series expansions for the matrix cosine approximation

Abstract

The computation of matrix trigonometric functions has received remarkable attention in the last decades due to its usefulness in the solution of systems of second order linear differential equations. Recently, several state-of-the-art algorithms have been provided for computing these matrix functions, in particular for the matrix cosine function.

Full text

Modelling for Engineering
& Human Behaviour 2019
Val`encia, 10 −12 July 2019
This book includes the extended abstracts of papers presented at XXIst Edition of the Mathe-
matical Modelling Conference Series at the Institute for Multidisciplinary Mathematics “Math-
ematical Modelling in Engineering & Human Behaviour”.

I.S.B.N.: 978-84-09-16428-8
Version: 18/11/19
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ropean Social Fund (ESF)] of the Valencian Com-
munity 2014-2020. [Record: GJIDI/2018/A/010].

Contents
A personality mathematical model of placebo with or without deception: an application
of the Self-Regulation Therapy..........................................................1
The role of police deterrence in urban burglary prevention: a new mathematical ap-
proach..................................................................................9
A Heuristic optimization approach to solve berth allocation problem . . . . . . . . . . . . . . . . .14
Improving the efficiency of orbit determination processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
A new three-steps iterative method for solving nonlinear systems . . . . . . . . . . . . . . . . . . . . 22
Adaptive modal methods to integrate the neutron diffusion equation . . . . . . . . . . .. . . . . . 26
Numerical integral transform methods for random hyperbolic models . . . . . . . . . . . . . . . . 32
Nonstandard finite difference schemes for coupled delay differential models . . . . . . . . . . .37
Semilocal convergence for new Chebyshev-type iterative methods . . . . . . . . . . . . . . . . . . . 42
Mathematical modeling of Myocardial Infarction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Symmetry relations between dynamical planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Econometric methodology applied to financial systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56
New matrix series expansions for the matrix cosine approximation . . . . . . . . . . . . . . . . . . . 64
Modeling the political corruption in Spain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Exponential time differencing schemes for pricing American option under the Heston
model................................................................................. 75
Chromium layer thickness forecast in hard chromium plating process using gradient
boosted regression trees: a case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Design and convergence of new iterative methods with memory for solving nonlinear
problems ..............................................................................83
Study of the influence falling friction on the wheel/rail contact in railway dynamics . . 88
Extension of the modal superposition method for general damping applied in railway
dynamics..............................................................................94
Predicting healthcare cost of diabetes using machine learning models . . . . . . . . . . . . . . . . 99
iv

Modelling for Engineering & Human Behaviour 2019
Sampling of pairwise comparisons in decision-making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A multi-objective and multi-criteria approach for district metered area design: water
operation and quality analysis ........................................................110
Updating the OSPF routing protocol for communication networks by optimal decision-
making over the k-shortest path algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Optimal placement of quality sensors in water distribution systems . . . . . . . . . . . . . . . . . 124
Mapping musical notes to socio-political events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Comparison between DKGA optimization algorithm and Grammar Swarm surrogated
model applied to CEC2005 optimization benchmark. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
The quantum brain model ......................................................... 142
Probabilistic solution of a randomized first order differential equation with discrete
delay.................................................................................151
A predictive method for bridge health monitoring under operational conditions . . . . . 155
Comparison of a new maximum power point tracking based on neural network with
conventional methodologies ...........................................................160
Influence of different pathologies on the dynamic behaviour and against fatigue of
railway steel bridges..................................................................166
Statistical-vibratory analysis of wind turbine multipliers under different working con-
ditions................................................................................171
Analysis of finite dimensional linear control systems subject to uncertainties via prob-
abilistic densities ..................................................................... 176
Topographic representation of cancer data using Boolean Networks . . . . . . . . . . . . . . . . . 180
Trying to stabilize the population and mean temperature of the World . . . . . . . . . . . . . 185
Optimizing the demographic rates to control the dependency ratio in Spain . . . . . . . . .193
An integer linear programming approach to check the embodied CO2emissions of the
opaque part of a fa¸cade...............................................................199
Acoustics on the Poincar´e Disk.....................................................206
Network computational model to estimate the effectiveness of the influenza vaccine a
posteriori .............................................................................211
The key role of Liouville-Gibbs equation for solving random differential equations:
Some insights and applications........................................................217
v

13
New matrix series expansions for the matrix cosine
approximation
Emilio Defez [1, Javier Ib´a˜nez\, Jos´e M. Alonso\, Jes´us Peinado]and Pedro Alonso-Jord´a∗
([) Institut Universitari de Matem`atica Multidisciplinar,
Universitat Polit`ecnica de Val`encia,
(\) Instituto de Instrumentaci´on para Imagen Molecular,
Universitat Polit`ecnica de Val`encia,
(]) Departamento de Sistemas Inform´aticos y Computaci´on,
Universitat Polit`ecnica de Val`encia,
(∗) Grupo Interdisciplinar de Computaci´on y Comunicaciones,
Universitat Polit`ecnica de Val`encia.
1 Introduction and notation
The computation of matrix trigonometric functions has received remarkable attention in the
last decades due to its usefulness in the solution of systems of second order linear differential
equations. Recently, several state-of-the-art algorithms have been provided for computing these
matrix functions, see [1–4], in particular for the matrix cosine function.
Among the proposed methods for the approximate computation of the matrix cosine, two fun-
damental ones stand out: those based on rational approximations [1,5–7], and those related to
polynomial approximations, using either Taylor series developments [8,9] or serial developments
of Hermite matrix polynomials [10]. In general, polynomial approximations showed to be more
efficient than the rational algorithms in tests because they are more accurate despite a slightly
higher cost.
Bernoulli polynomials and Bernoulli numbers have been extensively used in several areas of
mathematics (an excelent survey about Bernoulli polynomials and its applicacions can be found
in [11]).
In this paper, we will present a new series development of the matrix cosine in terms of the
Bernoulli matrix polynomials. We are going to verify that its use allows obtaining a new and
competitive method for the approximation of the matrix cosine.
The organization of the paper is as follows: In Section 2, we will obtain two serial developments
of the matrix cosine in terms of the Bernoulli matrix polynomials. In Section 3, we will present
1e-mail: edefez@imm.upv.es
64

Modelling for Engineering & Human Behaviour 2019
the different numerical tests performed. Conclusions are given in Section 4.
Throughout this paper, we denote by Cr×rthe set of all the complex square matrices of size
r. Besides, we denote Ias the identity matrix in Cr×r. A polynomial of degree mis given by
an expression of the form Pm(t) = amtm+am−1tm−1+··· +a1t+a0, where tis a real variable
and aj, for 0 ≤j≤m, are complex numbers. Moreover, we can define the matrix polynomial
Pm(B) for B∈Cr×ras Pm(B) = amBm+am−1Bm−1+· ·· +a1B+a0I. As usual, the matrix
norm k···k denotes any subordinate matrix norm; in particular k···k1is the usual 1−norm.
2 On Bernoulli matrix polynomials
The Bernoulli polynomials Bn(x) are defined in [12, p.588] as the coefficients of the generating
function
g(x, t) = tetx
et−1=X
n≥0
Bn(x)
n!tn,|t|<2π, (1)
where g(x, t) is an holomorphic function in Cfor the variable t(it has an avoidable singularity
in t= 0). Bernoulli polynomials Bn(x) has the explicit expression
Bn(x) =
n
X
k=0 n
k!Bkxn−k,(2)
where the Bernoulli numbers are defined by Bn=Bn(0). Therefore, it follows that the Bernoulli
numbers satisfy
z
ez−1=X
n≥0
Bn
n!zn,|z|<2π, (3)
where
B0= 1, Bk=−
k−1
X
i=0 k
i!Bi
k+ 1 −i, k ≥1.(4)
Note that B3=B5=··· =B2k+1 = 0, for k≥1. For a matrix A∈Cr×r, we define the m−th
Bernoulli matrix polynomial by the expression
Bm(A) =
m
X
k=0 m
k!BkAm−k.(5)
We can use the series expansion
eAt = et−1
t!X
n≥0
Bn(A)tn
n!,|t|<2π, (6)
to obtain approximations of the matrix exponential. A method based in (6) to approximate
the exponential matrix has been presented in [13].
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Modelling for Engineering & Human Behaviour 2019
From (6), we obtain the following expression for the matrix cosine and sine:
cos (A) = (cos (1) −1) X
n≥0
(−1)nB2n+1(A)
(2n+ 1)! + sin (1) X
n≥0
(−1)nB2n(A)
(2n)! ,
sin (A) = sin (1) X
n≥0
(−1)nB2n+1(A)
(2n+ 1)! −(cos (1) −1) X
n≥0
(−1)nB2n(A)
(2n)! .

















(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 polynomials do not verify this property. Thus,
in the development of cos(A) and sin (A), all Bernoulli polynomials are needed (and not just
the even-numbered ones).
Replacing in (6) the value tfor it and −it respectively and taking the arithmetic mean, we
obtain the expression
X
n≥0
(−1)nB2n(A)
(2n)! t2n=t
2 sin t
2cos tA −t
2I ,|t|<2π. (8)
Taking t= 2 in (8) it follows that
cos (A) = sin (1) X
n≥0
(−1)n22nB2nA+I
2
(2n)! ,(9)
Note that in formula (9) only even grade Bernoulli’s polynomials appear.
3 Numerical Experiments
Having in mind expressions (7) and (9), two different approximations are given to compute
cosine matrix function.
To test the proposed method and the two distinct approximations, and to compare them with
other approaches, the following algorithms have been implemented on MATLAB R2018b:
-cosmber. New code based on the new developments of Bernoulli matrix polynomials (formulae
(7) and (9)). The maximum value of mto be used is m= 36, with even and odd terms.
-cosmtay. Code based on the Taylor series for the cosine [8]. It will provide a maximum value
of m= 16, considering only the even terms, which would be equivalent to m= 32 using
the even and odd terms.
-cosmtayher. Code based on the Hermite series for the cosine [10]. As mentioned before, it
will provide a maximum value of m= 16.
-cosm. Code based on the Pad´e rational approximation for the cosine [7].
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Modelling for Engineering & Human Behaviour 2019
The following sets of matrices have been used:
a) Diagonalizable matrices. The matrices have been obtained as A=V·D·VT, where D
is a diagonal matrix (with complex or real values) and matrix Vis an orthogonal matrix,
V=H/16, where His a Hadamard matrix. We have 2.18 ≤ kAk1≤207.52. The matrix
cosine is exactly calculated as cos (A) = V·cos (D)·VT.
b) Non-diagonalizables matrices. The matrices have been computed as A=V·J·V−1,
where Jis a Jordan matrix with complex eigenvalues with module less than 10 and random
algebraic multiplicity between 1 and 5. Matrix Vis a random matrix with elements in
the interval [−0.5,0.5]. We have 1279.16 ≤ kAk1≤87886.4. The matrix cosine is exactly
calculated as cos (A) = V·cos (J)·V−1.
c) Matrices from the Matrix Computation Toolbox [14] and from the Eigtool Matlab
package [15]. These matrices have been chosen because they have more varied and
significant characteristics.
In the numerical test, we used 259 matrices of size 128 ×128: 100 from the diagonalizable
set, 100 from the non-diagonalizable set, 42 from Matrix Computation Toolbox and 17 from
Eigtool Matlab package. Results are given in Tables 1 and 2. The rows of each table show the
percentage of cases in which the relative errors of cosmber (Bernoulli) is lower, greater or equal
than the relative errors of cosmtay (Taylor), cosmtayher (Hermite) and cosm (Pad´e). Graphics
of the Normwise relative errors and the Performance Profile are given in Figures 1 and 2. The
total number of matrix products was: 3202 (cosmber ), 2391 (cosmtay), 1782 (cosmtayher) and
3016 (cosm). Recall that in the Bernoulli implementation, the maximum value of mto be used
was m= 36 considering all the terms and, in the rest of algorithms, was m= 32 but just
having into account the even terms.
E(cosmber)< E(cosmtay) 55.60%
E(cosmber)> E(cosmtay) 44.40%
E(cosmber) = E(cosmtay) 0%
E(cosmber)< E(cosmtayher) 50.97%
E(cosmber)> E(cosmtayher) 49.03%
E(cosmber) = E(cosmtayher) 0%
E(cosmber)< E(cosm) 76.83%
E(cosmber)> E(cosm) 23.17%
E(cosmber) = E(cosm) 0%
Table 1: Using approximation (7)
E(cosmber)< E(cosmtay) 65.64%
E(cosmber)> E(cosmtay) 34.36%
E(cosmber) = E(cosmtay) 0%
E(cosmber)< E(cosmtayher) 60.62%
E(cosmber)> E(cosmtayher) 39.38%
E(cosmber) = E(cosmtayher) 0%
E(cosmber)< E(cosm) 73.75%
E(cosmber)> E(cosm) 26.25%
E(cosmber) = E(cosm) 0%
Table 2: Using approximation (9)
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Modelling for Engineering & Human Behaviour 2019
0 50 100 150 200 250 300
Matrix
10-150
10-100
10-50
100
Er
cond*u
cosmber
cosmtay
cosmtayher
cosm
(a) Using formula (7)
0 50 100 150 200 250 300
Matrix
10-150
10-100
10-50
100
Er
cond*u
cosmber
cosmtay
cosmtayher
cosm
(b) Using formula (9)
Figure 1: Normwise relative errors.
12345
0
0.2
0.4
0.6
0.8
1
p
cosmber
cosmtay
cosmtayher
cosm
(a) Using formula (7)
12345
0
0.2
0.4
0.6
0.8
1
p
cosmber
cosmtay
cosmtayher
cosm
(b) Using formula (9)
Figure 2: Performance Profile.
4 Conclusions
In general, the implementation based on the new Bernoulli series (9) is more accurate than (7),
comparing it with the one based on the Taylor series, algorithm (cosmtay) and Hermite series,
algorithm (cosmtayher), and the one based in Pad´e rational approximation, algorithm (cosm).
Acknowledgement
This work has been partially supported by Spanish Ministerio de Econom´ıa y Competitividad
and European Regional Development Fund (ERDF) grants TIN2017-89314-P and by the Pro-
grama de Apoyo a la Investigaci´on y Desarrollo 2018 of the Universitat Polit`ecnica de Val`encia
(PAID-06-18) grants SP20180016.
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Modelling for Engineering & Human Behaviour 2019
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