Some new Hermite matrix polynomials series expansions and their applications in hyperbolic matrix sine and cosine approximation

Abstract

In this paper we show new algorithms for computing hyperbolic matrix sine and cosine approximations using Hermite matrix Series Expansions

Full text

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−t2H1(x, A) cosh xt√2A−2tsinh xt√2A,

n≥0
H2n+2(x, A)
(2n+ 1)! t2n+1 =e−t2H1(x, A) sinh xt√2A−2tcosh xt√2A,

n≥0
H2n+3(x, A)
(2n+ 1)! t2n+1 =e−t2H2(x, A) +4t2Isinh 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+ 1t2n
[H1(x, A)]−1,
x∈R∼ {0},|t|<+∞.
(6)
Substituting sinh xt√2Agiven in (2) into the second expression of (3)
and using the three-term matrix recurrence (5) we obtain the expression of
cosh xt√2Agiven in (2).
On the other hand, replacing the expression of sin xt√2Agiven 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√2An−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+ 1t2n
,
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
λ2m

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
λ2n1 + 2
(2n+ 1)λ2.
Taking λ > 1 it follows that 2
(2n+ 1)λ2<1, and one gets

n≥m+1
2n+ 1
λ2n1 + 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
24 + (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 λz1/24 + (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.
References
[1] J. J´odar, R. Company, Hermite matrix polynomials and second order matrix
differential equations, Approximation Theory and its Applications 12 (2)
(1996) 20–30.
[2] J. Sastre, J. Ib´a˜nez, E. Defez, P. Ruiz, New scaling-squaring Taylor algo-
rithms for computing the matrix exponential, SIAM Journal on Scientific
Computing 37 (1) (2015) A439–A455.
[3] P. Alonso, J. Peinado, J. Ib´a˜nez, J. Sastre, E. Defez, Computing matrix
trigonometric functions with gpus through matlab, The Journal of Super-
computing (2018) 1–14.
77

[4] J. Sastre, Efficient evaluation of matrix polynomials, Linear Algebra and its
Applications 539 (2018) 229–250.
[5] E. Defez, J. Sastre, J. Ib´a˜nez, J. Peinado, Solving engineering models using
hyperbolic matrix functions, Applied Mathematical Modelling 40 (4) (2016)
2837–2844.
[6] J. Sastre, J. Ib´a˜nez, P. Ruiz, E. Defez, Efficient computation of the matrix
cosine, Applied Mathematics and Computation 219 (14) (2013) 7575–7585.
[7] N. J. Higham, Functions of Matrices: Theory and Computation, SIAM,
Philadelphia, PA, USA, 2008.
[8] T. Wright, Eigtool, version 2.1, URL: web. comlab. ox. ac.
uk/pseudospectra/eigtool.
[9] N. J. Higham, The test matrix toolbox for MATLAB (Version 3.0), Univer-
sity of Manchester Manchester, 1995.
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