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MODELLING FOR ENGINEERING AND

HUMAN BEHAVIOUR

2018

Instituto Universitario de Matemática Multidisciplinar

L. Jódar, J. C. Cortés and L. Acedo( Editors )

Instituto Universitario de

Matemática Multidisciplinar

MODELLING FOR ENGINEERING,

& HUMAN BEHAVIOUR 2018

Instituto Universitario de Matem´atica Multidisciplinar

Universitat Polit`ecnica de Val`encia

Valencia 46022, SPAIN

Edited by

Lucas J´odar, Juan Carlos Cort´es and Luis Acedo

Instituto Universitario de Matem´atica Multidisciplinar

Universitat Polit`ecnica de Val`encia

I.S.B.N.: 978-84-09-07541-6

CONTENTS

1. A model for making choices with fuzzy soft sets in an intertemporal framework,

by J. C. R. Alcantud, and M. J. Mu˜noz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 1-5

2. Methylphenidate and the Self-Regulation Therapy increase happiness and reduce

depression: a dynamical mathematical model, by S. Amig´o, J. C. Mic´o, and A. Caselles

Pag: 6-9

3. A procedure to predict the short-term glucose level in a diabetic patient which

captures the uncertainty of the data, by C. Burgos, J. C. Cort´es, D. Mart´ınez-Rodr´ıguez,

J. I. Hidalgo, and R.J. Villanueva. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 10-15

4. Assessing organizational risk in industry by evaluating interdependencies among

human factors through the DEMATEL methodology, by S. Carpitella, F. Carpitella,

A. Certa, J. Ben´ıtez, and J. Izquierdo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 16-24

5. Selection of an anti-torpedo decoy for the new frigate F-110 by using the GMUBO

method, by R. M. Carre˜no, J. Mart´ınez, and J. Benito . . . . . . . . . . . . . . . . . . . . . . . . Pag: 25-30

6. A modelling methodology based on General Systems Theory, by A. Caselles .Pag:

31-34

7. Dynamics of the general factor of personality as a consequence of alcohol con-

sumption, by S. Amig´o, A. Caselles, J. C. Mic´o, M. T. Sanz, and D. Soler . . . . . . . . . . .Pag:

35-38

8. An optimal eighth-order scheme for multiple roots applied to some real life prob-

lems, by R. Behl, E. Mart´ınez, F. Cevallos, and A. S. Alshomrani . . . . . . . . . . . . . . . . . . . . Pag:

39-43

9. Optimal Control of Plant Virus Propagation, by B. Chen and M. Jackson Pag: 44-49

10. On the inclusion of memory in Traub-type iterative methods for solving nonlinear

equations, by F. I. Chicharro, A. Cordero, N. Garrido, and J. R. Torregrosa . . Pag: 50-55

11. Mean square analysis of non-autonomous second-order linear diﬀerential equa-

tions with randomness, by J. Calatayud, J. C. Cort´es, M. Jornet and L. Villafuerte Pag:

56-62

12. A Statistical Model with a Lotka-Volterra Structure for Microbiota Data, by I.

Creus, A. Moya, and F. J. Santonja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 63-68

13. Some new Hermite matrix polynomials series expansions and their applications in

hyperbolic matrix sine and cosine approximation, by E. Defez, J. Ib´a˜nez, J. Peinado,

P. Alonso, J. M. Alonso, and J. Sastre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 69-78

14. A novel optimization technique for railway wheel rolling noise reduction, by J.

Guti´errez, X. Garc´ıa, J. Mart´ınez, E. Nadal, and F. D. Denia . . . . . . . . . . . . . . . . . . Pag: 79-84

15. Improving the order of convergence of Traub-type derivative-free methods, by F.

I. Chicharro, A. Cordero, N. Garrido, and J. R. Torregrosa . . . . . . . . . . . . . . . . . . . . . Pag: 85-90

16. Eﬃcient decoupling technique applied to the numerical time integration of ad-

vanced interaction models for railway dynamics, by J. Giner, J. Mar´ınez, F. D. Denia,

and L. Baeza .................................................................. Pag: 91-96

17. Matrix-free block Newton method to compute the dominant λ-modes of a nuclear

power reactor, by A. Carre˜no, L. Bergamaschi, A. Mart´ınez, A. Vidal, D. Ginestar, and

G. Verd´u ..................................................................... Pag: 97-103

18. A new automatic gonad diﬀerentiation for salmon gender identiﬁcation based on

Echography image treatment, by A. Sancho, L. Andr´es, B. Baydal, and J. Real . . Pag:

104-109

19. A New Earthwork Measurement System based on Stereoscopic Vision by Un-

manned Aerial System ﬂights, by V. Espert, P. Moscoso, T. Real, M. Mart´ınez, and J.

Real .........................................................................Pag: 110-115

20. A New Forest Measurement and Monitoring System Based on Unmanned Aerial

Vehicles Imaging, by F. Ribes, V. Ramos, V. Espert, and J. Real . . . . . . . . . . Pag: 116-121

21. A new non-intrusive and real time monitoring technique for pavement execution

based on Unmanned Aerial Vehicles ﬂights, by T. Real, P. Moscoso, V. Espert, A.

Sancho, and J. Real ......................................................... Pag: 122-127

22. A New Road Type Response Roughness Measurement System for existent de-

fects localization and quantiﬁcation, by F. J. Vea, C. Masanet, M. Ballester, R. Red´on,

and J. Real ..................................................................Pag: 128-133

23. Application of an analytical solution based on beams on elastic foundation model

for precast railway transition wedge design automatization, by J. L. P´erez, M.

Labrado, T. Real, A. Zorzona, and J. Real . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 134-139

24. Development of an innovative wheel damage detection system based on track

vibration response on frequency domain, by R. Au˜non, B. Baydal, S. Nu˜nez, and J.

Real .........................................................................Pag: 140-145

25. Mathematical characterization of liquefaction phaenomena for structure founda-

tion monitoring, by P. Moscoso, R. Sancho, E. Colomer, and J. Real . . . . . . . . . . . . . . . . Pag:

146-151

26. Neural Network application for concrete compression strength evolution predic-

tion, by T. Real, M. Labrado, B. Baydal, and J . Real . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag:

152-157

27. Numerical simulation of lateral railway dynamic eﬀects for a new stabilizer

sleeper design, by F. J. Fern´andez, T. Real, A. Zorzona, and J. Real . . . . . . . Pag: 158-162

28. Operational costs optimization method in transport systems for open-pit mines,

by F. Halles, F. Ribes, C. Masanet, R. Red´on, and J. Real . . . . . . . . . . . . . . . . . . . Pag: 163-168

29. Structural Railway Bridge Health monitoring by means of data analysis, by F.

Ribes, C. Zamorano, P. Moscoso, and J. Real . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 169-174

30. A spatial model for mean house mortgage appraisal value in boroughs of the city

of Valencia, by M. A. L´op ez, N. Guadalajara, A Iftimi, and A. Usai . . . . . . . . . . . . . 175-180

31. Third order root-ﬁnding methods based on a generalization of Gander’s result,

by S. Busquier, J. M. Guti´errez, and H. Ramos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 181-184

32. ASSESSMENT OF A GRAPHIC MODEL FOR SOLVING DELAY TIME MODEL

INSPECTION CASES OF REPAIRABLE MACHINERY. PREDICTION OF

RISK WHEN SELECTING INSPECTION PERIODS, by F. Pascual, E. Larrod´e,

and V. Muerza .............................................................. Pag: 185-189

33. A high order iterative scheme of ﬁxed point for solving nonlinear Fredholm in-

tegral equations, by M. A. Hern´andez, M. Ib´a˜nez, E. Mart´ınez, and S. Singh . . . . . . . Pag:

190-194

34. Some parametric families improving Newton’s method, by A. Cordero, S. Masall´en,

and J. R. Torregrosa ........................................................ Pag: 195-200

35. Modeling consumer behavior in Spain, by P. Merello, L. J´odar, G. Douklia, and E. de

la Poza ......................................................................Pag: 201-208

36. Hamiltonian approach to human personality dynamics: an experiment with

methylphenidate, by J. C. Mic´o, S. Amig´o, and A. Caselles . . . . . . . . . . . . . . . . Pag: 209-212

37. A Pattern Recognition Bayesian Model for the appearance of Pathologies in

Automated Systems, by M. Alacreu, N. Montes, E. Garc´ıa, and A. Falco . . Pag: 213-218

38. A study of the seasonal forcing in SIRS models for Respiratory Syncytial Virus

(RSV) using a constant period of temporary immunity, by L. Acedo, J. A. Mora˜no,

and R. J. Villanueva .........................................................Pag: 219-226

39. Improving urban freight distribution through techniques of multicriteria decision

making. An AHP-GIS approach, by V. Muerza, C. Thaller, and E. Larrod´e . . . . . .Pag:

227-232

40. Nonlinear transport through thin heterogeneous membranes, by A. Muntean Pag:

233-236

41. Application of the transfer matrix method for modelling Cardan mechanism of

a real vehicle, by P. Hubr´y and T. Nhl´ık . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 237-242

42. The RVT method to solve random non-autonomous second-order linear diﬀer-

ential equations about singular-regular points, by J. C. Cort´es, A. Navarro, J. V.

Romero, and M. D. Rosell´o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 243-248

43. On some properties of the PageRank versatility, by F. Pedroche, R. Criado, E. Garc´ıa,

and M. Romance ............................................................Pag: 249-254

44. Network clustering strategies for setting degree predictors based on deep learning

architectures, by F. J. P´erez, E. Navarro, J. M. Garc´ıa, and J. Alberto Conejero . . . . Pag:

255-261

45. Qualitative preserving stable diﬀerence methods for solving nonlocal biological

dynamic problems, by M. A. Piqueras, R. Company, and L. J´odar . . . . . . . . . Pag: 262-267

46. Probabilistic solution of a random model to study the eﬀectiveness of anti-

epileptic drugs, by E. M. S´anchez-Orgaz, I. Barrachina, A. Navarro, and M. Ramos Pag:

268-273

47. Weighted graphs to redeﬁne the centrality measures, by M. D. L´opez, J. Rodrigo, C.

Puente, and J. A. Olivas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 274-279

48. Numerical solution to the random heat equation with zero Cauchy-type boundary

conditions, by J. C. Cort´es, A. Navarro, J. V. Romero, and M. D. Rosell´o . . Pag: 280-285

49. A Multistate Model for Non Muscle Invasive Bladder Carcinoma, by C. Santamar´ıa,

B. Garc´ıa, and G. Rubio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 286-291

50. Birth rate and population pyramid: A stochastic dynamical model, by J. C. Mic´o,

D. Soler, M. T. Sanz, A. Caselles, and S. Amig´o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 292-297

51. Application of the ﬁnite element method in the analysis of oscillations of rotating

parts of machine mechanisms, by P. Hubr´y, and D. Smetanov´a . . . . . . . . . . Pag: 298-302

52. Using Integer Linear Programming to minimize the cost of the thermal refur-

bishment of a faade: An application to building 1B of the Universitat Polit`ecnica

de Val`encia, Spain, by D. Soler, A. Salandin, and M. Bevivino . . . . . . . . . . . . Pag: 303-308

53. Modeling the Eﬀects of the Immune System on Bone Fracture Healing, by I. Trejo,

H. Kojouharov, and B. Chen-Charpentier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 309-314

54. Metamaterial Acoustics on the Einstein Cylinder, by M. M. Tung . . . . Pag: 315-324

55. Extrapolated Stabilized Explicit Runge-Kutta methods, by J. Mart´ın and A. Kleefeld

Pag: 325-331

56. Modelling and simulation of biological pest control in broccoli production, by L.

V. Vela-Ar´evalo, R. A. Ku-Carrilo, and S. E. Delgadillo-Alem´an . . . . . . . . . . . . . Pag: 332-337

57. Preliminary study of fuel assembly vibrations in a nuclear reactor, by A. Vidal, D.

Ginestar, A. Carre˜no and G. Verd´u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 338-343

58. Evolution and prediction with uncertainty of the bladder cancer of a patient

using a dynamic model, by C. Burgos, N. Garc´ıa, D. Mart´ınez, and R. J. Villanueva Pag:

344-348

59. Dynamics of a family of Ermakov-Kalitlin type methods, by A. Cordero, J. R.

Torregrosa, and P. Vindel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 349-353

60. A Family of Optimal Fourth Order Methods for Multiple Roots of Non-linear

Equations, by F. Zafar, A. Cordero, and J. R. Torregrosa . . . . . . . . . . . . . . . . . . . Pag: 354-359

61. Randomizing the von Bertalanﬀy growth model: Theoretical analysis and com-

puting, by J. Calatayud, J.-C. Cort´es, and M. Jornet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag:

360-365

62. A Gauss-Legendre Product Quadrature for the Neutron Transport Equation, by

A. Bernal, S. Morat´o, R. Mir´o, and G. Verd´u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pag: 366-371

63. PGD path planning for dynamic obstacle robotic problems, by L. Hilario, N. Mont´es,

M. C. Mora, E. Nadal, A. Falc´o, F. Chinesta and J. L. Duval . . . . . . . . . . . . . . . . Pag: 372-376

Some new Hermite matrix polynomials series

expansions and their applications in hyperbolic

matrix sine and cosine approximation ∗

E. Defez⋆, J. Ib´a˜nez†, J. Peinado§, P. Alonso♮, J. M. Alonso†, J. Sastre‡

⋆Instituto de Matem´atica Multidisciplinar.

†Instituto de Instrumentaci´on para Imagen Molecular.

§Departamento de Sistemas Inform´aticos y Computaci´on.

♮Grupo Interdisciplinar de Computaci´on y Comunicaciones.

‡Instituto de Telecomunicaciones y Aplicaciones Multimedia.

Universitat Polit`ecnica de Val`encia, Camino de Vera s/n, 46022, Valencia, Espa˜na.

edefez@imm.upv.es, {jjibanez, jpeinado, palonso, jmalonso }@dsic.upv.es, jorsasma@iteam.upv.es

1 Introduction and notation

Hermite matrix polynomial Hn(x, A) has the generating function, see [1]:

ext√2A=et2∑

n≥0

Hn(x, A)

n!tn,(1)

from following expressions for the matrix hyperbolic sine and cosine are derived:

cosh (xt√2A)=et2∑

n≥0

H2n(x, A)

(2n)! t2n

sinh (xt√2A)=et2∑

n≥0

H2n+1(x, A)

(2n+ 1)! t2n+1

, x ∈R,|t|<∞.(2)

∗Acknowledgements: 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 Programa de Apoyo a la Investigaci´on y Desarrollo 2018 of the

Universitat Polit`ecnica de Val`encia (PAID-06-18) grants SP20180016.

69

Recently we have shown the following formulas which are a generalization of

formulas (2):

n≥0

H2n+1(x, A)

(2n)! t2n=e−t2H1(x, A) cosh xt√2A−2tsinh xt√2A,

n≥0

H2n+2(x, A)

(2n+ 1)! t2n+1 =e−t2H1(x, A) sinh xt√2A−2tcosh xt√2A,

n≥0

H2n+3(x, A)

(2n+ 1)! t2n+1 =e−t2H2(x, A) +4t2Isinh xt√2A−4tH1(x, A) cosh xt√2A.

(3)

We will use formulas (3) to obtain a new expansion of the hyperbolic matrix

sine and cosine in Hermite matrix polynomials series.

Throughout this paper, we denote by Cr×rthe set of all the complex square

matrices of size r. We denote by Θ and I, respectively, the zero and the identity

matrix in Cr×r. If A∈Cr×r, we denote by σ(A) the set of all the eigenvalues

of A. For a real number x,⌊x⌋denotes the lowest integer not less than xand

⌈x⌉denotes the highest integer not exceeding x.

We recall that for a positive stable matrix A∈Cr×rthe n−th Hermite

matrix polynomial is deﬁned in [1] by:

Hn(x, A) = n!⌊n

2⌋

∑

k=0

(−1)k(√2A)n−2k

k!(n−2k)! xn−2k,(4)

which satisﬁes the three-term matrix recurrence:

Hm(x, A) = x√2AHm−1(x, A)−2(m−1)Hm−2(x, A), m ≥1,

H−1(x, A) = Θ , H0(x, A) = I .

(5)

2 Some new Hermite matrix series expansions

for the hyperbolic matrix cosine and sine

Let A∈Cr×rbe a positive stable matrix, then the matrix polynomial H1(x, A) =

√2Ax is invertible if x̸= 0. Substituting sinh (xt√2A)given in (2) into the

ﬁrst expression of (3) we obtain the following new rational expression for the

hyperbolic matrix cosine in terms of Hermite matrix polynomials:

70

cosh xt√2A=et2

n≥0

H2n+1(x, A)

(2n)! 1 + 2t2

2n+ 1t2n

[H1(x, A)]−1,

x∈R∼ {0},|t|<+∞.

(6)

Substituting sinh xt√2Agiven in (2) into the second expression of (3)

and using the three-term matrix recurrence (5) we obtain the expression of

cosh xt√2Agiven in (2).

On the other hand, replacing the expression of sin xt√2Agiven in (2) into

the third expression of (3), we obtain another new rational expression for the

hyperbolic matrix cosine in terms of Hermite matrix polynomials:

cosh xt√2A=

=−et2

4

n≥0

H2n+3(x, A)

(2n+ 1)! t2n−H2(x, A) + 4t2I⋆

n≥0

H2n+1(x, A)

(2n+ 1)! t2n+1

[H1(x, A)]−1,

x∈R∼ {0},|t|<+∞.(7)

Comparing (7) with (6), we observe that it always has a matrix product

more when evaluating (7), the matrix product remarked by symbol “⋆” in (7).

Due to the importance of reducing the number of matrix products, see [2–4] for

more details, we will focus mainly on the expansion (6).

From (4), it follows that, for x̸= 0:

H2n+1 (x, A) [H1(x, A)]−1=(2n+ 1)!

x

n

k=0

(−1)kx2(n−k)+1(2A)n−k

k!(2(n−k) + 1)!

=

H2n+1 (x, A),(8)

where

Hn(x, A) = n!⌊n

2⌋

k=0

(−1)k√2An−2k−1

k!(n−2k)! xn−2k,(9)

so the right side of (8) is still deﬁned in the case where the matrix Ais

singular. In this way, we can re-write the relation (6) in terms of the matrix

polynomial

H2n+1 (x, A):

71

cosh xt√2A=et2

n≥0

H2n+1 (x, A)

(2n)! 1 + 2t2

2n+ 1t2n

,

x∈R,|t|<+∞.

(10)

Replacing the matrix Aby matrix A2/2 in (10) we can avoid the square

roots of matrices, and taking x=λ, λ ̸= 0, t = 1/λ, we ﬁnally obtain

cosh (A) = e1

λ2

n≥0

H2n+1 λ, 1

2A2

(2n)!λ2n+1 1 + 2

(2n+ 1)λ2

,0< λ < +∞.(11)

3 Numerical approximations

Truncating the given series (11) until order m, we obtain the approximation

CHm(λ, A)≈cosh (A) deﬁned by

CHm(λ, A) = e1

λ2m

n=0

H2n+1 λ, 1

2A2

(2n)!λ2n+1 1 + 2

(2n+ 1)λ2,0< λ < +∞.

(12)

Working analogously to the proof of the formula (3.6) of [5] one gets, for

x̸= 0 the following bound:

H2n+1 x, 1

2A2

2≤(2n+ 1)!

esinh |x|

A2

1/2

2

|x|∥A2∥1/2

2

.(13)

Then we can obtain the following expression for the approximation error:

∥cosh (A)−CHm(λ, A)∥2≤e1

λ2

n≥m+1

H2n+1 λ, 1

2A2

2

(2n)!λ2n+1 1 + 2

(2n+ 1)λ2(14)

≤

e1+ 1

λ2sinh λ

A2

1/2

2

λ2∥A2∥1/2

2

n≥m+1

2n+ 1

λ2n1 + 2

(2n+ 1)λ2.

Taking λ > 1 it follows that 2

(2n+ 1)λ2<1, and one gets

n≥m+1

2n+ 1

λ2n1 + 2

(2n+ 1)λ2≤2

n≥m+1

2n+ 1

λ2n

=4 + (4m+ 6)(λ2−1)

λ2m(λ2−1)2,

72

m zmλm

2 0.0020000000061361199 909.39256098888882

4 0.079956209874370632 99.997970988888895

6 0.34561400005673254 39.999499988888893

9 1.1120032200657 17.997896988889799

12 2.2373014291079998 11.882978988901458

16 4.1086396680000004 7.9999999964157498

Table 1: Values of zmand λmfor cosh (A).

m1= 2 m2= 4 m3= 6 m4= 9 m5= 12 m6= 16

¯mk1 2 3 5 7 11

˜mk1 2 4 10 13 17

fmk(max) 0 0 1.9·10−17 6.0·10−19 1.4·10−26 1.3·10−35

Table 2: Values ¯mk, ˜mk, and fmax .

thus from (14) we ﬁnally obtain:

∥cosh (A)−CHm(λ, A)∥2≤

e1+ 1

λ2sinh λ

A2

1/2

24 + (4m+ 6)(λ2−1)

∥A2∥1/2

2λ2m+2 (λ2−1)2.

(15)

From this expression (15) we derived the optimal values (λm;zm) such that

zm= max

z=

A2

2;

e1+ 1

λ2sinh λz1/24 + (4m+ 6)(λ2−1)

z1/2λ2m+2 (λ2−1)2< u

where uis the unit roundoﬀ in IEEE double precision arithmetic, u= 2−53. The

optimal values of m,zand λhave been obtained with MATLAB. The results

are given in the Table 1.

If cosh(A) is calculated from the Taylor series, then the absolute forward

error of the Hermite approximation of cosh(A), denoted by Ef, can be computed

as

Ef=∥cosh (A)−Pmk(B)∥=

i>¯mk

fmk,iBi

∼

=

i>˜mk

fmk,iBi

,

where the values of ¯mkand ˜mkfor each mk∈ {2,4,6,9,12,16}appear in the

Table 2.

Scaling factor sand the order of Hermite approximation mkare obtained

by the following:

73

Theorem 3.1 ( [6]) Let hl(x) = ∑

i≥l

pixibe a power series with radius of con-

vergence w,˜

hl(x) = ∑

i≥l|pi|xi,B∈Cn×nwith ρ(B)< w,l∈Nand t∈Nwith

16t6l. If t0is the multiple of tsuch that l6t06l+t−1and

βt= max{d1/j

j:j=t, l, l + 1, . . . , t0−1, t0+ 1, t0+ 2, . . . , l +t−1},

where djis an upper bound for ||Bj||,dj>||Bj||, then

||hl(B)|| 6˜

hl(βt).

We have empirically veriﬁed that by neglecting the coeﬃcients whose abso-

lute value is lower than u, the eﬃciency results are far superior to the state-of-

the-art algorithms, with also excellent accuracy.

4 Numerical experiments

The MATLAB’s implementation coshmtayher is a modiﬁcation of the MAT-

LAB’s code coshher given in [5], replacing the original Hermite approxima-

tion coshher by the new Hermite matrix polynomial obtained from (11). In

this section, we compare the new MATLAB function developed in this paper,

coshmtayher, with the functions coshher and funmcosh:

•coshmtayher. New code based on the new developments of Hermites matrix

polynomials (11).

•coshher. Code based on the Hermite series for the hyperbolic matrix cosine

[5].

•funmcosh. MATLAB function funm for compute matrix functions, i. e. the

hyperbolic matrix cosine.

The tests have been develop using MATLAB (R2017b), runing on an Apple

Macintosh iMac 27” (iMac retina 5K 27” late 2015) with a quadcore INTEL

i7-6700K 4 Ghz processor and 16 Gb of RAM.

The following sets of matrices have been used:

a) One hundred diagonalizable matrices of size 128 ×128. Table 3 show the

percentage of cases in which the relative errors of coshmtayher (new

Hermite code) are lower, greater or equal than the relative errors of

coshher(Hermite code) and funmcosh (funm code). Table 4 shows the

matrix products of each method. Graphics with the Normwise relative

errors, see [7, p. 253] and Performance Proﬁle, see [7, p. 254], are given

in Figure 1.

b) One hundred non diagonalizables matrices of size 128 ×128 with multiple

eigenvalues randomly generated. Table 5 shows the percentage of cases in

74

which the relative errors of coshmtayher are lower, greater or equal than

the relative errors of coshher and funmcosh. Table 6 shows the matrix

products of each method. Graphics of the Normwise relative errors and

the Performance Proﬁle are given in Figure 2.

c) Ten matrices from the Eigtool MATLAB [8] package with size 128 ×128,

and thirty matrices from the function matrix of the Matrix Computa-

tion Toolbox [9] with dimensions lower or equal than 128. These matrices

have been chosen because they have more varied and signiﬁcant char-

acteristics. Table 7 shows the percentage of cases in which the relative

errors of coshmtayher are lower, greater or equal than the relative errors

of coshher and funmcosh. Table 8 shows the matrix products of each

method. Graphics of the Normwise relative errors and the Performance

Proﬁle are given Figure 3.

E(coshmtayher)< E(coshher) 47.50%

E(coshmtayher)> E(coshher) 50.00%

E(coshmtayher) = E(coshher) 3.00%

E(coshmtayher)< E(f unmcosh) 100.00%

E(coshmtayher)> E(f unmcosh) 0.00%

E(coshmtayher) = E(f unmcosh) 0.00%

Table 3: Comparative between the methods

cosmtayher coshher funmcosh

671 973 1500

Table 4: Matrix products

E(coshmtayher)< E(coshher) 52.50%

E(coshmtayher)> E(coshher) 47.00%

E(coshmtayher) = E(coshher) 1.00%

E(coshmtayher)< E(f unmcosh) 100.00%

E(coshmtayher)> E(f unmcosh) 0.00%

E(coshmtayher) = E(f unmcosh) 0.00%

Table 5: Comparative between the methods

75

0 20 40 60 80 100

Matrix

10-17

10-16

10-15

10-14

10-13

Er

cond*u

coshmtayher

coshher

funmcosh

(a) Normwise relative errors

1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

p

coshmtayher

coshher

funmcosh

(b) Performance Proﬁle

Figure 1: Diagonalizable matrices

cosmtayher coshher funmcosh

685 989 1500

Table 6: Matrix products

0 20 40 60 80 100

Matrix

10-16

10-15

10-14

10-13

Er

cond*u

coshmtayher

coshher

funmcosh

(a) Normwise relative errors

1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

p

coshmtayher

coshher

funmcosh

(b) Performance Proﬁle

Figure 2: Non diagonalizable matrices

E(coshmtayher)< E(coshher) 57.50%

E(coshmtayher)> E(coshher) 30.00%

E(coshmtayher) = E(coshher) 12.50%

E(coshmtayher)< E(f unmcosh) 97.50%

E(coshmtayher)> E(f unmcosh) 2.50%

E(coshmtayher) = E(f unmcosh) 0.00%

Table 7: Comparative between the methods

76

cosmtayher coshher funmcosh

191 315 600

Table 8: Matrix products

0 5 10 15 20 25 30 35 40

Matrix

10-15

10-10

10-5

100

Er

cond*u

coshmtayher

coshher

funmcosh

(a) Normwise relative errors

1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

p

coshmtayher

coshher

funmcosh

(b) Performance Proﬁle

Figure 3: Matrices from the Eigtool and the Matrix Computation Toolbox

packages

5 Conclusions

The more accurate are the implementations based on the Hermite series: the

initial MATLAB implementation (coshher) and the proposed MATLAB im-

plementation based on (11) (coshmtayher). Also, the new implementation

(coshmtayher) have considerably lower computational costs than the other func-

tions.

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