Content uploaded by Jesús Peinado

Author content

All content in this area was uploaded by Jesús Peinado on Nov 25, 2019

Content may be subject to copyright.

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

Report any problems with this document to ellona1@upvnet.upv.es.

Edited by: R. Company, J. C. Cort´es, L. J´odar and E. L´opez-Navarro.

Credits: The cover has been designed using images from kjpargeter/freepik.

Instituto Universitario de Matem´atica

Multidisciplinar

This book has been supported by the European

Union through the Operational Program of the [Eu-

ropean Regional Development Fund (ERDF) / Eu-

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].

65

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

2cos tA −t

2I ,|t|<2π. (8)

Taking t= 2 in (8) it follows that

cos (A) = sin (1) X

n≥0

(−1)n22nB2nA+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].

66

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)

67

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.

References

[1] Serbin, S.M. and Blalock, S.A., An algorithm for computing the matrix cosine. SIAM

Journal on Scientiﬁc Computing, 1(2): 198–204, 1980.

68

Modelling for Engineering & Human Behaviour 2019

[2] Dehghan, M. and Hajarian, M., Computing matrix functions using mixed interpolation

methods. Mathematical and Computer Mod- el ling, 52(5-6): 826–836, 2010.

[3] Higham, N.J., Functions of Matrices: Theory and Computation. SIAM, Philadelphia, PA,

USA, 2008.

[4] Alonso-Jord´a, P., Peinado, J., Ib´a˜nez, J., Sastre, J. and Defez, E., Computing matrix

trigonometric functions with GPUs through Matlab. The Journal of Supercomputing, pages

1–14, 2018.

[5] Tsitouras, Ch. and Katsikis, V., Bounds for variable degree ratio- nal L∞approximations

to the matrix cosine. Computer Physics Communications, 185(11): 2834–2840, 2014.

[6] Serbin, S.M., Rational approximations of trigonometric matrices with application to

second-order systems of diﬀerential equations. Applied Mathematics and Computation,

5(1): 75–92, 1979.

[7] Al-Mohy, A.H., Higham, N.J. and Relton, S.D., New algorithms for computing the matrix

sine and cosine separately or simultaneously. SIAM Journal on Scientiﬁc Computing, 37(1):

A456–A487, 2015.

[8] Sastre, J., Ib´a˜nez, J., Alonso-Jord´a, P., Peinado, J. and Defez, E., Two algorithms for com-

puting the matrix cosine function. Applied Mathe- matics and Computation, 312: 66–77,

2017.

[9] Sastre, J., Ib´a˜nez, J., Alonso-Jord´a, P., Peinado, J. and Defez, E., Fast Taylor polynomial

evaluation for the computation of the matrix cosine. Journal of Computational and Applied

Mathematics, 354: 641–650, 2019.

[10] Defez, E., Ib´a˜nez, J., Peinado, J., Sastre, J. and Alonso-Jord´a, P., An eﬃcient and accurate

algorithm for computing the matrix cosine based on new Hermite approximations. Journal

of Computational and Applied Mathematics, 348: 1–13, 2019.

[11] Kouba, O., Lecture Notes, Bernoulli Polynomials and Applications. arXiv preprint

arXiv:1309.7560, 2013.

[12] WJ Olver, F., W Lozier, D., F Boisvert, R. and W Clark, C., NIST handbook of mathe-

matical functions hardback and CD-ROM. Cambridge University Press, 2010.

[13] Defez, E., Ib´a˜nez, J., Peinado, J., Alonso-Jord´a, P. and Alonso, J.M., Computing matrix

functions by matrix Bernoulli series. In 19th International Conference on Computational

and Mathematical Methods in Science and Engineering (CMMSE-2019), From 30th of Juny

to 6th of July 2019. Poster presented at some conference, Rota (C´adiz), Spain.

[14] Higham, N.J., The test matrix toolbox for MATLAB (Version 3.0). University of Manch-

ester Manchester, 1995.

[15] Wright, TG., Eigtool, version 2.1. URL: web.comlab.ox.ac.uk/pseudospectra/eigtool,

2009.

69