Advances in the Approximation of the Matrix Hyperbolic Tangent

Abstract

In this paper, we introduce two approaches to compute the matrix hyperbolic tangent. While one of them is based on its own definition and uses the matrix exponential, the other one is focused on the expansion of its Taylor series. For this second approximation, we analyse two different alternatives to evaluate the corresponding matrix polynomials. This resulted in three stable and accurate codes, which we implemented in MATLAB and numerically and computationally compared by means of a battery of tests composed of distinct state-of-the-art matrices. Our results show that the Taylor series-based methods were more accurate, although somewhat more computationally expensive, compared with the approach based on the exponential matrix. To avoid this drawback, we propose the use of a set of formulas that allows us to evaluate polynomials in a more efficient way compared with that of the traditional Paterson–Stockmeyer method, thus, substantially reducing the number of matrix products (practically equal in number to the approach based on the matrix exponential), without penalising the accuracy of the result.

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mathematics
Article
Advances in the Approximation of the Matrix
Hyperbolic Tangent
Javier Ibáñez 1, José M. Alonso 1, Jorge Sastre 2, Emilio Defez 3and Pedro Alonso-Jordá 4,*


Citation: Ibáñez, J.; Alonso, J.M.;
Sastre, J.; Defez, E.; Alonso-Jordá, P.
Advances in the Approximation of
the Matrix Hyperbolic Tangent.
Mathematics 2021,9, 1219. https://
doi.org/10.3390/math9111219
Academic Editor: Mariano Torrisi
Received: 23 March 2021
Accepted: 20 May 2021
Published: 27 May 2021
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Copyright: © 2021 by the authors.
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Attribution (CC BY) license (https://
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4.0/).
1
Instituto de Instrumentación para Imagen Molecular, Universitat Politècnica de València, Av. dels Tarongers,
14, 46011 Valencia, Spain; jjibanez@dsic.upv.es (J.I.);
jmalonso@dsic.upv.es (J.M.A.)
2Instituto de Telecomunicación y Aplicaciones Multimedia, Universitat Politècnica de València, Ed. 8G,
Camino de Vera s/n, 46022 Valencia, Spain; jsastrem@upv.es
3Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Ed. 8G, Camino de Vera s/n,
46022 Valencia, Spain; edefez@imm.upv.es
4Department of Computer Systems and Computation, Universitat Politècnica de València, Ed. 1F,
Camino de Vera s/n, 46022 Valencia, Spain
*Correspondence: palonso@upv.es
Abstract:
In this paper, we introduce two approaches to compute the matrix hyperbolic tangent.
While one of them is based on its own definition and uses the matrix exponential, the other one is
focused on the expansion of its Taylor series. For this second approximation, we analyse two different
alternatives to evaluate the corresponding matrix polynomials. This resulted in three stable and
accurate codes, which we implemented in MATLAB and numerically and computationally compared
by means of a battery of tests composed of distinct state-of-the-art matrices. Our results show that
the Taylor series-based methods were more accurate, although somewhat more computationally
expensive, compared with the approach based on the exponential matrix. To avoid this drawback,
we propose the use of a set of formulas that allows us to evaluate polynomials in a more efficient way
compared with that of the traditional Paterson–Stockmeyer method, thus, substantially reducing
the number of matrix products (practically equal in number to the approach based on the matrix
exponential), without penalising the accuracy of the result.
Keywords:
matrix functions; matrix hyperbolic tangent; matrix exponential; Taylor series; matrix
polynomial evaluation
1. Introduction and Notation
Matrix functions have been an increasing focus of attention due to their applications
to new and interesting problems related, e.g., to statistics [
1
], Lie theory [
2
], differential
equations (the matrix exponential function
eAt
can be considered as a classical example
for its application in the solution of first order differential systems
Y0(t) = AY(t)
with
A∈Cn×n
and for its use in the development of exponential integrators for nonlinear ODEs
and PDEs, see [
3
] for example), approximation theory, and many other areas of science and
engineering [4].
There are different ways to define the notion of the function
f(A)
of a square matrix
A
.
The most common are via the Jordan canonical form, via the Hermite interpolation, and via
the Cauchy integral formula. The equivalence among the different definitions of a matrix
function can be found in [
5
]. Several general methods have been proposed for evaluating
matrix functions, among which, we can highlight the Taylor or Padé approximations and
methods based on the Schur form of a matrix [4].
Among the most well-known matrix functions, we have the matrix hyperbolic cosine
and the matrix hyperbolic sine functions, respectively defined in terms of the matrix
exponential function eAby means of the following expressions:
Mathematics 2021,9, 1219. https://doi.org/10.3390/math9111219 https://www.mdpi.com/journal/mathematics

Mathematics 2021,9, 1219 2 of 20
cosh (A) = 1
2eA+e−A, sinh (A) = 1
2eA−e−A. (1)
These matrix functions are applied, e.g., in the study of the communicability analysis
in complex networks [
6
], or to construct the exact series solution of coupled hyperbolic
systems [
7
]. Precisely due to their applicability, the numerical computation of these func-
tions has received remarkable and growing attention in recent years. A set of state-of-the-art
algorithms to calculate these functions developed by the authors can be found in [8–11].
On the other hand, we have the matrix hyperbolic tangent function, defined as
tanh (A) = sinh (A)(cosh (A))−1=(cosh (A))−1sinh (A), (2)
and used, for instance, to give an analytical solution of the radiative transfer equation [
12
],
in the heat transference field [
13
,
14
], in the study of symplectic systems [
15
,
16
], in graph
theory [17], and in the development of special types of exponential integrators [18,19].
In this work, we propose and study two different implementations that compute the
matrix hyperbolic tangent function: the first uses the matrix exponential function whereas
the second is based on its Taylor series expansion. In addition, for the second approach, we
use and compare two different alternatives to evaluate the matrix polynomials involved in
the series expansion.
1.1. The Matrix Exponential Function-Based Approach
This first option is derived from the matrix hyperbolic tangent function definition as
expressed in Equations
(1)
and
(2)
, from which the following matrix rational expression is
immediately deduced:
tanh (A) = e2A−Ie2A+I−1=e2A+I−1e2A−I, (3)
where
I
denotes the identity matrix with the same order as
A
. Equation
(3)
reduces the
approximation of the matrix hyperbolic tangent function to the computation of the ma-
trix exponential e2A.
There exists profuse literature (see e.g., [
4
,
20
]) about the approximation of the matrix
exponential function and the inconveniences that its calculation leads to [
21
]. The most
competitive methods used in practice are either those based on polynomial approximations
or those based on Padé rational approaches, with the former, in general, being more
accurate and with lower computational costs [22].
In recent years, different polynomial approaches to the matrix exponential function
have been proposed, depending on the type of matrix polynomial used. For example,
some approximations use the Hermite matrix polynomials [
23
], while others derived from
on Taylor polynomials [
22
,
24
]. More recently, a new method based on Bernoulli matrix
polynomials was also proposed in [25].
All these methods use the scaling and squaring method based on the identity
eA=e2−sA2s
,
which satisfies the matrix exponential. In the scaling phase, an integer scaling factor
s
is
taken, and the approximation of
e2−sA
is computed using any of the proposed methods so
that the required precision is obtained with the lowest possible computational cost. In the
squaring phase, we obtain eAby srepeated squaring operations.

Mathematics 2021,9, 1219 3 of 20
1.2. The Taylor Series-Based Approach
The other possibility for computing the matrix hyperbolic tangent function is to use
its Taylor series expansion
tanh (z) = ∑
k≥1
22k(22k−1)B2k
(2k)!z2k−1,|z|<π
2,
where B2kare the Bernoulli’s numbers.
As in the case of the matrix exponential, it is highly recommended to use the scaling
and squaring technique to reduce the norm of the matrix to be computed and, thus, to obtain
a good approximation of the matrix hyperbolic tangent with an acceptable computational
cost. Due to the double angle formula for the matrix hyperbolic tangent function
tanh(2A) = 2I+tanh2(A)−1tanh(A), (4)
which is derived from the scalar one
tanh(2z) = 2 tanh(z)
1+tanh2(z),
it is possible to compute Ts=tanh(A)by using the following recurrence:
T0=tanh(2−sA),
Ti=2I+T2
i−1(A)−1Ti−1(A),i=1, . . . , s.(5)
Throughout this paper we will denote by
σ(A)
the set of eigenvalues of matrix
A∈Cn×n
and by
In
(or
I
) the matrix identity of order
n
. In addition,
ρ(A)
refers to the
spectral radius of A, defined as
ρ(A) = max{|λ|;λ∈σ(A)}.
With
dxe
, we denote the value obtained by rounding
x
to the nearest integer greater
than or equal to
x
, and
bxc
is the value obtained rounding
x
to the nearest integer less than
or equal to
x
. The matrix norm
|| ·||
will stand for any subordinate matrix norm and, in
particular, ||·||1denotes the 1-norm.
This work is organised as follows. First, Section 2incorporates the algorithms corre-
sponding to the different approaches previously described for approximating the matrix
hyperbolic tangent and for computing the scaling parameter and the order of the Taylor
polynomials. Next, Section 3details the experiments carried out to compare the numerical
properties of the codes to be evaluated. Finally, in Section 4, we present our conclusions.
2. Algorithms for Computing the Matrix Hyperbolic Tangent Function
2.1. The Matrix Exponential Function-Based Algorithm
The first algorithm designed, called Algorithm 1, computes the matrix hyperbolic
tangent by means of the matrix exponential according to Formula
(3)
. In Steps 1 and 2,
Algorithm 2from [
26
] is responsible for computing
e2−sB
by means of the Taylor approxima-
tion of order
mk
, being
B=
2
A
. In Step 3,
T'tanh(
2
−sB)
is worked out using Formula
(3)
.
In this phase,
T
is computed by solving a system of linear equations, with
e2−sB+I
being
the coefficient matrix and
e2−sB−I
being the right hand side term. Finally, through
Steps 4–8,
tanh(A)
is recovered from
T
by using the squaring technique and the double
angle Formula (5).

Mathematics 2021,9, 1219 4 of 20
Algorithm 1:
Given a matrix
A∈Cn×n
, this algorithm computes
T=tanh(A)
by means of the matrix exponential function.
1B=2A
2Calculate the scaling factor s∈N∪{0}, the order of Taylor polynomial
mk∈ {2, 4, 6, 9, 12, 16, 20, 25, 30}and compute e2−sBby using the Taylor
approximation /* Phase I (see Algorithm 2from [26]) */
3T=e2−sB+I−1e2−sB−I/* Phase II: Work out tanh(2−sB)by (3)*/
4for i=1to sdo /* Phase III: Recover tanh(A)by (5)*/
5B=I+T2
6Solve for Xthe system of linear equations BX =2T
7T=X
8end
Algorithm 2:
Given a matrix
A∈Cn×n
, this algorithm computes
T=
tanh(A)
by means of the Taylor approximation Equation
(8)
and the Paterson–
Stockmeyer method.
1Calculate the scaling factor s∈N∪{0}, the order of Taylor approximation
mk∈{2, 4, 6, 9, 12, 16, 20, 25, 30}, 2−sAand the required matrix powers of 4−sB
/* Phase I (Algorithm 4) */
2T=2−sAPmk(4−sB)/* Phase II: Compute Equation (8)*/
3for i=1to sdo /* Phase III: Recover tanh(A)by Equation (5)*/
4B=I+T2
5Solve for Xthe system of linear equations BX =2T
6T=X
7end
2.2. Taylor Approximation-Based Algorithms
Let
f(z) = ∑
n≥1
22n(22n−1)B2n
(2n)!z2n−1(6)
be the Taylor series expansion of the hyperbolic tangent function, with the radius of
convergence
r=π/
2, where
B2n
are the Bernoulli’s numbers, defined by the recursive
expression
B0=1 , Bk=−
k−1
∑
i=0k
iBi
k+1−i,k≥1.
The following proposition is easily obtained:
Proposition 1.
Let
A∈Cn×n
be a matrix satisfying
ρ(A)<π/
2. Then, the matrix hyperbolic
tangent tanh(A)can be defined for A as the Taylor series
tanh(A) = f(A) = ∑
n≥1
22n(22n−1)B2n
(2n)!A2n−1. (7)
From [
4
] Theorem 4.7, this series in Equation (7) converges if the distinct eigenvalues
λ1,λ2,··· ,λtof Asatisfy one of these conditions:
1. |λi|<π/2.
2. |λi|=π/
2 and the series
f(ni−1)(λ)
, where
f(z)
is given by Equation
(6)
and
ni
is the
index of λi, is convergent at the point λ=λi,i=1, . . . , t.

Mathematics 2021,9, 1219 5 of 20
To simplify the notation, we denote with
tanh(A) = ∑
k≥0
q2k+1A2k+1,
the Taylor series (7), and with
T2m+1(A) =
m
∑
k=0
q2k+1A2k+1=A
m
∑
k=0
pkBk=APm(B), (8)
the Taylor approximation of order 2m+1 of tanh(A), where B=A2.
There exist several alternatives that can be applied to obtain
Pm(B)
, such as the
Paterson–Stockmeyer method [
27
] or the Sastre formulas [
28
], with the latter being more
efficient, in terms of matrix products, compared with the former.
Algorithm 2works out
tanh(A)
by means of the Taylor approximation of the scaled
matrix 4
−sB
Equation
(8)
. In addition, it uses the Paterson–Stockmeyer method for the ma-
trix polynomial evaluation, and finally it applies the recurrence Equation
(5)
for recovering tanh(A).
Phase I of Algorithm 2is in charge of estimating the integers
m
and
s
so that the Taylor
approximation of the scaled matrix
B
is computed accurately and efficiently. Then, in
Phase II, once the integer mkhas been chosen from the set
M={2, 4, 6, 9, 12, 16, 20, 25, 30, . . . },
and powers
Bi
, 2
≤i≤q
are calculated, with
q=√mk
or
q=b√mkc
as an integer
divisor of
mk
, the Paterson–Stockmeyer method computes
Pmk(B)
with the necessary
accuracy and with a minimal computational cost as
Pmk(B) = ((( pmkBq+pmk−1Bq−1+pmk−2Bq−2+· · · +pmk−q+1B+pmk−qI)Bq
+pmk−q−1Bq−1+pmk−q−2Bq−2+··· +pmk−2q+1B+pmk−2qI)Bq
+pmk−2q−1Bq−1+pmk−2q−2Bq−2+··· +pmk−3q+1B+pmk−3qI)Bq
. . .
+pq−1Bq−1+pq−2Bq−2+··· +p1B+p0I.
(9)
Taking into account Equation
(9)
, the computational cost of Algorithm 2is
O2k+4+8s
3n3flops.
Finally, in Phase III, the matrix hyperbolic tangent of matrix
A
is recovered by squaring
and repeatedly solving a system of linear equations equivalent to Equation (4).
With the purpose of evaluating
Pm(B)
in Equation
(8)
in a more efficient way com-
pared with that offered by the Paterson–Stockmeyer method, as stated in the Phase II of
Algorithm 2, the formulas provided in [
28
] were taken into consideration into the design
of
Algorithm 3
. Concretely, we use the evaluation formulas for Taylor-based matrix poly-
nomial approximations of orders
m=
8, 14
+
, and 21
+
in a similar way to the evaluation
described in [
22
] (Sections 3.1–3.3) for the matrix exponential function. Nevertheless, the
Paterson–Stockmeyer method is still being used for orders equal to m=2 and m=4.
Following the notation given in [
22
] (Section 4) for an order
m
, the suffix “
+
” in
m=
14
+
and
m=
21
+
means that these Taylor approximations are more accurate than
those approximations of order
m=
14 and
m=
21, respectively, since the former will
be composed of a few more polynomial terms. The coefficients of these additional terms
will be similar but not identical to the corresponding traditional Taylor approximation
ones. It is convenient to clarify that we have used the order
m=
14
+
, instead of the
order
m=
15
+
, because we have not found a real solution for the coefficients of the
corresponding evaluation formula with order m=15+.

Mathematics 2021,9, 1219 6 of 20
The evaluation formulas for the order
m=
8 that comprise the system of non-linear
equations to be solved for determining the unknown coefficients ci,i=1, . . . , 6, are:
B=−A2,
B2=B2,
y0(B) = B2(c1B2+c2B), (10)
y1(B) = A((y0(B) + c3B2+c4B)(y0(B) + c5B2) + c6y0(B)
+2B2/15 +B/3 +I),
where
y1(B) = T17(A) = AP8(B),
and
T17(A)
, or
AP8(B)
, refers to the Taylor polynomial of order 17 of function
tanh(A)
given by Equation (8).
Algorithm 3:
Given a matrix
A∈Cn×n
, this algorithm computes
T=tanh(A)
by means of the Taylor approximation Equation (8) and the Sastre formulas.
1Calculate the scaling factor s∈N∪{0}, the order of Taylor approximation
mk∈{2, 4, 8, 14, 21}, 2−sAand the required matrix powers of 4−sB/* Phase I
(Algorithm 5) */
2T=2−sAPmk(4−sB)/* Phase II: Compute Equation (8)*/
3for i=1to sdo /* Phase III: Recover tanh(A)by Equation (5)*/
4B=I+T2
5Solve for Xthe system of linear equations BX =2T
6T=X
7end
Regarding the non-linear equations for order
m=
14
+
and its unknown coefficients
ci,i=1, .. . , 13, we have
y0(B) = B2(c1B2+c2B),
y1(B) = (y0(B) + c3B2+c4B)(y0(B) + c5B2) + c6y0(B), (11)
y2(B) = A((y1(B) + c7y0(B) + c8B2+c9B)(y1(B) + c10 B2+c11B)
+c12y1+c13 B2+B/3 +I),
where
y2(B) = A(P14 +b15B15 +b16 B16),
and
AP14
represents the Taylor polynomial of order 29 of function
tanh(A)
given by
Equation
(8)
. If we denote as
p15
and
p16
the Taylor polynomial coefficients corresponding
to the powers
B15
and
B16
, respectively, the relative error of coefficients
b15
and
b16
with
respect to them, with two decimal digits, are:
|(b15 −p15)/p15 |=0.38,
|(b16 −p16)/p16 |=0.85.
Taking
B3=B2B,

Mathematics 2021,9, 1219 7 of 20
the evaluation formulas related to the system of non-linear equations for order
m=
21
+
with the coefficients ci,i=1, . . . , 21 to be determined are
y0(B) = B3(c1B3+c2B2+c3B),
y1(B) = (y0(B) + c4B3+c5B2+c6B)(y0(B) + c7B3+c8B2)(12)
+c9y0(B) + c10B3,
y2(B) = A((y1(B) + c11B3+c12B2+c13 B)(y1(B) + c14 y0(B)
+c15B3+c16B2+c17 B) + c18y1+c19y0(B) + c20 B3
+c21B2+B/3 +I),
where
y2(B) = A(P21 +b22B22 +b23 B23 +b24 B24),
and
AP21
stands for the Taylor polynomial of order 43 of the function
tanh(A)
given by
Equation
(8)
. With two decimal digits of accuracy, the relative error made by the coefficients
b22
,
b23
, and
b24
with respect to their corresponding Taylor polynomial coefficients
p22
,
p23
,
and p24 that accompany their respective powers B22,B23 , and B24 , are the following:
|(b22 −p22)/p22 |=0.69,
|(b23 −p23)/p23 |=0.69,
|(b24 −p24)/p24 |=0.70.
Similarly to [
22
] (Sections 3.1–3.3), we obtained different sets of solutions for the coef-
ficients in Equations
(10)
–
(12)
using the
vpasolve
function (https://es.mathworks.com/
help/symbolic/vpasolve.html, accessed on 7 March 2020) from the MATLAB Symbolic
Computation Toolbox with variable precision arithmetic. For the case of
m=
21
+
, the
random option of
vpasolve
has been used, which allowed us to obtain different solutions
for the coefficients, after running it 1000 times. From all the sets of real solutions provided,
we selected the most stable ones according to the stability check proposed in [
28
] (Ex. 3.2).
2.3. Polynomial Order m and Scaling Value s Calculation
The computation of
m
and
s
from Phase I in Algorithms 2and 3is based on the
relative forward error of approximating
tanh(A)
by means of the Taylor approximation
Equation (8). This error, defined as Ef=

tanh (A)−1(I−T2m+1)

, can be expressed as
Ef=∑
k≥2m+2
ckAk,
and it can be bounded as (see Theorem 1.1 from [29]):
Ef=




∑
k≥2m+2
ckAk




=




∑
k≥m+1
¯
ckBk



≤∑
k≥m+1|¯
ck|βk
m≡hm(βm),
where βm=maxn||Bk||1/k:k>m+1, ¯
cm+16=0o.
Let Θmbe
Θm=max(θ≥0 : ∑
k≥m+1|¯
ck|θk≤u),
where
u=
2
−53
is the unit roundoff in IEEE double precision arithmetic. The values of
Θm
can be computed with any given precision by using symbolic computations as is shown in
Tables 1and 2, depending on the polynomial evaluation alternative selected.
Algorithm 4provides the Taylor approximation order
mk∈M
,
lower ≤k≤upper
,
where
mlower
and
mup per
are, respectively, the minimum and maximum order used, the
scaling factor
s
, together with 2
−sA
, and the necessary powers of 4
−sB
for computing
T
in

Mathematics 2021,9, 1219 8 of 20
Phase II of Algorithm 2. To simplify reading this algorithm, we use the following aliases:
βk≡βmkand Θk≡Θmk.
Algorithm 4:
Given a matrix
A∈Cn×n
, the values
Θ
from Table 1, a minimum order
mlower ∈M
, a maximum order
mup per ∈M
, with
M={2, 4, 6, 9, 12, 16, 20, 25, 30}
, and
a tolerance
tol
, this algorithm computes the order of Taylor approximation
m∈M
,
mlower ≤mk≤mup per
, and the scaling factor
s
, together with 2
−sA
and the necessary
powers of 4−sBfor computing Pmk(4−sB)from (9).
1B1=A2;k=lower;q=d√mke;f=0
2for j=2to qdo
3Bj=Bj−1B1
4end
5Compute βk≈
Bmk+1

1/(mk+1)from B1and Bq;/* see [30] */
6while f=0and k<up per do
7k=k+1
8if mod (k, 2) = 1then
9q=√mk
10 Bq=Bq−1B1
11 end
12 Compute βk≈kBmk+1k1/(mk+1)from B1and Bq;/* see [30] */
13 if |βk−βk−1|<tol and βk<Θkthen
14 f=1
15 end
16 end
17 s=max0, l1
2log2(βk/Θk)m
18 if s>0then
19 s0=max0, l1
2log2(βk−1/Θk−1)m
20 if s0=sthen
21 k=k−1
22 q=√mk
23 end
24 A=2−sA
25 for j=1to qdo
26 Bj=4−sj Bj
27 end
28 end
29 m=mk
In Steps 1–4 of Algorithm 4, the required powers of
B
for working out
Pmk(B)
are
computed. Then, in Step 5, βkis obtained by using Algorithm 1from [30].
As
lim
t→∞||Bt||1/t=ρ(B)
, where
ρ
is the spectral radius of matrix
B
, then
lim
m→∞|βm−
βm−1|=
0. Hence, given a small tolerance value
tol
, Steps 6–16 test if there is a value
βk
such that
|βk−βk−1|<tol
and
βk<Θk
. In addition, the necessary powers of
B
for
computing Pmk(B)are calculated. Next, the scaling factor s≥0 is provided in Step 17:
s=max0, 1
2log2
βk
Θk.
With those values of mkand s, we guarantee that:
Ef(2−sA)≤hmk(4−sβk)<hmk(Θk)<u, (13)
i.e., the relative forward error of T2mk+1(2−sA)is lower than the unit roundoff u.
Step 18 checks whether the matrices
A
and
B
should be scaled or not. If so, the
algorithm analyses the possibility of reducing the order of the Taylor polynomial, but at
the same time ensuring that Equation
(13)
is verified (Steps 19–23). For this purpose, the

Mathematics 2021,9, 1219 9 of 20
scaling value corresponding to the order of the Taylor polynomial immediately below the
one previously obtained is calculated as well. If both values are identical, the polynomial
order reduction is performed. Once the optimal scaling parameter
s
has been determined,
the matrices Aand Bare scaled (Steps 24–27).
Algorithm 5is an adaptation of Algorithm 4, where the orders in the set
M=
{2, 4, 8, 14, 21}
are used. Steps 1–15 of Algorithm 5are equivalent to Steps 1–16 of
Algorithm 4. Both values
β1
and
β2
are computed in the same way in both algorithms
while values
β3
,
β4
, and
β5
are worked out in Algorithm 5for the polynomial orders
m3=
8,
m4=
14, and
m5=
21, respectively. Steps 16–31 of Algorithm 5correspond to
Steps 17–29 of Algorithm 4.
Algorithm 5:
Given a matrix
A∈Cn×n
, the values
Θ
from Table 2,
M={2, 4, 8, 14, 21}
and a tolerance tol, this algorithm computes the order of Taylor approximation mk∈M
and the scaling factor
s
, together with 2
−sA
and the necessary powers of 4
−sB
for
computing 2−sAPmk(4−sB)from (10), (11) or (12).
1B1=−A2;B2=B2
1
2Compute β1≈
B3

1/3 from B1and B2
3f=0; k=1
4while f=0and k<5do
5k=k+1
6if k<5then
7Compute βk≈kBmk+1k1/(mk+1)from B1and B2
8else
9B3=B1B2
10 Compute β5≈
B22

1/22 from B1and B3
11 end
12 if |βk−βk−1|<tol and βk<Θkthen
13 f=1; s=0
14 end
15 end
16 if f=0then
17 s=max0, l1
2log2(βk/Θk)m
18 end
19 if s>0then
20 s0=max0, l1
2log2(βk−1/Θk−1)m
21 if s0=sthen
22 k=k−1
23 end
24 A=2−sA
25 B1=4−sB1
26 B2=16−sB2
27 if k=5then
28 B3=64−sB3
29 end
30 end
31 m=mk

Mathematics 2021,9, 1219 10 of 20
Table 1.
Values of
Θmk
, 1
≤k≤
9, for polynomial evaluation by means of the Paterson–Stockmeyer
method.
m1=21.1551925093100 ×10−3
m2=42.8530558816082 ×10−2
m3=69.7931623314428 ×10−2
m4=92.3519926145338 ×10−1
m5=12 3.7089935615781 ×10−1
m6=16 5.2612365603423 ×10−1
m7=20 6.5111831924355 ×10−1
m8=25 7.73638541973549 ×10−1
m9=30 8.68708923627294 ×10−1
Table 2. Values of Θmk, 1 ≤k≤5, for polynomial evaluation using the Sastre formulas.
m1=21.1551925093100 ×10−3
m2=42.8530558816082 ×10−2
m3=81.88126704493647 ×10−1
m4=14+4.65700446893510 ×10−1
m5=21+6.84669656651721 ×10−1
3. Numerical Experiments
The following codes have been implemented in MATLAB to test the accuracy and the
efficiency of the different algorithms proposed:
•tanh_expm
: this code corresponds to the implementation of Algorithm 1. For obtaining
m∈{2, 4, 6, 9, 12, 16, 20, 25, 30}
and
s
and computing
e2A
, it uses function
exptaynsv3
(see [26]).
•tanh_tayps
: this development, based on Algorithm 2, incorporates Algorithm 4
for computing
m
and
s
, where
m
takes values in the same set than the
tanh_expm
code. The Paterson–Stockmeyer method is considered to evaluate the Taylor matrix
polynomials.
•tanh_pol
: this function, corresponding to Algorithm 3, employs Algorithm 5in the
m
and
s
calculation, where
m∈{2, 4, 8, 14, 21}
. The Taylor matrix polynomials are
evaluated by means of Sastre formulas.
Three types of matrices with distinct features were used to build a battery of tests
that enabled us to compare the numerical performance of these codes. The MATLAB
Symbolic Math Toolbox with 256 digits of precision was used to compute the “exact” matrix
hyperbolic tangent function using the
vpa
(variable-precision floating-point arithmetic)
function. The test battery featured the following three sets:
(a)
Diagonalizable complex matrices: one hundred diagonalizable 128
×
128 complex
matrices obtained as the result of
A=V·D·V−1
, where
D
is a diagonal matrix (with
real and complex eigenvalues) and matrix
V
is an orthogonal matrix,
V=H/√n
,
being
H
a Hadamard matrix and
n
is the matrix order. As 1-norm, we have that
2.56
≤kAk1≤
256. The matrix hyperbolic tangent was calculated “exactly” as
tanh (A) = V·tanh (D)·VTusing the vpa function.
(b)
Non-diagonalizable complex matrices: one hundred non-diagonalizable 128
×
128
complex matrices computed as
A=V·J·V−1
, where
J
is a Jordan matrix with
complex eigenvalues whose modules are less than 5 and the algebraic multiplicity
is randomly generated between 1 and 4.
V
is an orthogonal random matrix with
elements in the interval
[−
0.5, 0.5
]
. As 1-norm, we obtained that 45.13
≤kAk1≤
51.18.
The “exact” matrix hyperbolic tangent was computed as
tanh (A) = V·tanh (J)·V−1
by means of the vpa function.
(c)
Matrices from the Matrix Computation Toolbox (MCT) [
31
] and from the Eigtool
MATLAB Package (EMP) [
32
]: fifty-three matrices with a dimension lower than or

Mathematics 2021,9, 1219 11 of 20
equal to 128 were chosen because of their highly different and significant characteris-
tics from each other. We decided to scale these matrices so that they had 1-norm not
exceeding 512. As a result, we obtained that 1
≤kAk1≤
489.3. The “exact” matrix
hyperbolic tangent was calculated by using the two following methods together and
the vpa function:
•
Find a matrix
V
and a diagonal matrix
D
so that
A=VDV−1
by using the
MATLAB function eig. In this case, T1=Vtanh(D)V−1.
•
Compute the Taylor approximation of the hyperbolic tangent function (
T2
), with
different polynomial orders (
m
) and scaling parameters (
s
). This procedure is
finished when the obtained result is the same for the distinct values of
m
and
s
in IEEE double precision.
The “exact” matrix hyperbolic tangent is considered only if
kT1−T2k
kT1k<u.
Although MCT and EMP are really comprised of 72 matrices, only 53 matrices of
them, 42 from MCT and 11 from EMP, were considered for our purposes. On the one hand,
matrix 6 from MCT and matrix 10 from EMP were rejected because the relative error made
by some of the codes to be evaluated was greater or equal to unity. This was due to the
ill-conditioning of these matrices for the hyperbolic tangent function. On the other hand,
matrices 4, 12, 17, 18, 23, 35, 40, 46, and 51 from MCT and matrices 7, 9, 16, and 17 from
EMP were not generated because they did not satisfy the described criterion to obtain the
“exact” matrix hyperbolic tangent. Finally, matrices 8, 13, 15, and 18 from EMP were refused
as they are also part of MCT.
For each of the three previously mentioned sets of matrices, one test was respectively
and independently carried out, which indeed corresponds to an experiment to analyse the
numerical properties and to account for the computational cost of the different implemented
codes. All these experiments were run on an HP Pavilion dv8 Notebook PC with an
Intel Core i7 CPU Q720 @1.60 Ghz processor and 6 GB of RAM, using MATLAB R2020b.
First, Table 3shows the percentage of cases in which the normwise relative errors of
tanh_expm
are lower than, greater than, or equal to those of
tanh_tayps
and
tanh_pol
.
These normwise relative errors were obtained as
Er =ktanh(A)−˜
tanh(A)k1
ktanh(A)k1
,
where tanh(A)represents the exact solution and ˜
tanh(A)stands for the approximate one.
Table 3.
The relative error comparison, for the three tests, between
tanh_expm
vs.
tanh_tayps
and
tanh_expm vs tanh_pol.
Test 1 Test 2 Test 3
Er(tanh_expm)<Er(tanh_tayps) 32% 0% 22.64%
Er(tanh_expm)>Er(tanh_tayps) 68% 100% 77.36%
Er(tanh_expm)=Er(tanh_tayps) 0% 0% 0%
Er(tanh_expm)<Er(tanh_pol) 44% 0% 30.19%
Er(tanh_expm)>Er(tanh_pol) 56% 100% 69.81%
Er(tanh_expm)=Er(tanh_pol) 0% 0% 0%
As we can appreciate, from the point of view of the accuracy of the results,
tanh_tayps
outperformed
tanh_expm
in 68% of the matrices for Test 1 and 100% and 77.36% of them

Mathematics 2021,9, 1219 12 of 20
for Tests 2 and 3. On the other hand,
tanh_pol
obtained slightly more modest results with
improvement percentages equal to 56%, 100%, and 69.81% for Tests 1, 2, and 3, respectively.
Table 4reports the computational complexity of the algorithms. This complexity was
expressed as the number of matrix products required to calculate the hyperbolic tangent of
the different matrices that make up each of the test cases. This number of products includes
the number of matrix multiplications and the cost of the systems of linear equations that
were solved in the recovering phase, by all the codes, together with one more in Step 2 of
Algorithm 1by tanh_expm.
The cost of each system of linear equations with
n
right-hand side vectors, where
n
denotes the size of the square coefficient matrices, was taken as 4/3 matrix products. The
cost of other arithmetic operations, such as the sum of matrices or the product of a matrix
by a vector, was not taken into consideration. As can be seen, the lowest computational
cost corresponded to
tanh_expm
, followed by
tanh_pol
and
tanh_tayps
. For example,
tanh_expm
demanded 1810 matrix products to compute the matrices belonging to Test 1,
compared to 1847 by tanh_pol and 2180 by tanh_tayps.
Table 4.
Matrix products (P) corresponding to the
tanh_expm
,
tanh_tayps
, and
tanh_pol
functions
for Tests 1–3.
Test 1 Test 2Test 3
P(tanh_expm) 1810 1500 848
P(tanh_tayps) 2180 1800 1030
P(tanh_pol) 1847 1500 855
Respectively, for the three experiments, Figures 1–3illustrate the normwise relative
errors (a), the performance profiles (b), the ratio of the relative errors (c), the lowest and
highest relative error rate (d), the ratio of the matrix products (e), and the polynomial
orders (f) for our three codes to be evaluated.
Figures 1a, 2a and 3a correspond to the normwise relative error. As they reveal, the
three methods under evaluation were numerically stable for all the matrices that were
computed, and all of them provided very accurate results, where the relative errors incurred
were always less than 10
−11
. The solid line that appears in these figures traces the function
ktanhu
, where
ktanh
(or
cond
) stands for the condition number of the matrix hyperbolic
tangent function [4] (Chapter 3), and urepresents the unit roundoff.
It is clear that the errors incurred by all the codes were usually quite close to this
line for the three experiments and even below it, as was largely the case for Tests 1 and 3.
For the sake of readability in the graphs, normwise relative errors lower than 10
−20
were
plotted with that value in the figures. Notwithstanding, their original quantities were
maintained for the rest of the results.
Figures 1b, 2b and 3b depict the performance profile of the codes. In them, the
α
coordinate on the
x
-axis ranges from 1 to 5 in steps equal to 0.1. For a concrete
α
value, the
p
coordinate on the
y
-axis indicates the probability that the considered algorithm has a
relative error lower than or equal to
α
-times the smallest relative error over all the methods
on the given test.
The implementation of
tanh_tayps
always achieved the results with the highest
accuracy, followed closely by
tanh_pol
. For Tests 1 and 2, Figures 1b and 2b indicate
that the results provided by them are very similar, although the difference in favour of
tanh_tayps
is more remarkable in Test 3. As expected from the percentages given in
Table 3,
tanh_expm
computed the hyperbolic tangent function with the worst accuracy,
most notably for the matrices from Tests 2 and 3.

Mathematics 2021,9, 1219 13 of 20
0 20 40 60 80 100
Matrix
10-16
10-15
10-14
10-13
10-12
10-11
Er
cond*u
tanh_expm
tanh_tayps
tanh_pol
12345
0.2
0.4
0.6
0.8
1
p
tanh_expm
tanh_tayps
tanh_pol
(a) Normwise relative errors. (b) Performance profile.
0 20 40 60 80 100
Matrix
10-1
100
101
102
103
104
Relative error ratio
Er(tanh_tayps)/Er(tanh_expm)
Er(tanh_pol)/Er(tanh_expm)
Lowest relative error rate
27%
44%
29%
Highest relative error rate
51%
22%
27%
tanh_expm
tanh_tayps
tanh_pol
(c) Ratio of relative errors. (d) Lowest and highest relative error rate.
0 20 40 60 80 100
Matrix
0.9
1
1.1
1.2
1.3
1.4
Matrix product ratio
P(tanh_tayps)/P(tanh_expm)
P(tanh_pol)/P(tanh_expm)
0 20 40 60 80 100
Matrix
5
10
15
20
25
30
Approximation polynomial order
M(tanh_expm)
M(tanh_tayps)
M(tanh_pol)
(e) Ratio of matrix products. (f) Polynomial orders.
Figure 1. Experimental results for Test 1.
Precisely, the relationship among the normwise relative errors incurred by the codes
to be examined is displayed in Figures 1c, 2c and 3c. All these ratios are presented in
decreasing order with respect to Er(
tanh_tayps
)/Er(
tanh_expm
). This factor is less than 1
for the great majority of the matrices, which indicates that
tanh_tayps
and
tanh_pol
are
more accurate codes than tanh_expm for estimating the hyperbolic tangent function.
These data are further corroborated by the results shown in Figures 1d, 2d and 3d.
These graphs report the percentages of the matrices, for each of the tests, in which each
code resulted in the lowest or highest normwise relative error among the errors provided
by all of them. Thus,
tanh_tayps
exhibited the smallest relative error in 44% of the matrices
for Test 1 and in 47% of them for Test 3, followed by
tanh_pol
in 29% and 36% of the cases,
respectively. For all the other cases,
tanh_expm
was the most reliable method. For Test 2,
tanh_pol
was the most accurate code in 53% of the matrices, and
tanh_tayps
was the
most accurate in 47% of them. In line with these results,
tanh_expm
was found to be the
approach that led to the largest relative errors in 51% of the cases in Test 1, in 100% of them
in Test 2, and in 64% for Test 3.

Mathematics 2021,9, 1219 14 of 20
0 20 40 60 80 100
Matrix
10-15
10-14
10-13
Er
cond*u
tanh_expm
tanh_tayps
tanh_pol
12345
0
0.2
0.4
0.6
0.8
1
p
tanh_expm
tanh_tayps
tanh_pol
(a) Normwise relative errors. (b) Performance profile.
0 20 40 60 80 100
Matrix
10-1
100
Relative error ratio
Er(tanh_tayps)/Er(tanh_expm)
Er(tanh_pol)/Er(tanh_expm)
Lowest relative error rate
0%
47% 53%
Highest relative error rate
100%
0%
tanh_expm
tanh_tayps
tanh_pol
(c) Ratio of relative errors. (d) Lowest and highest relative error rate.
0 20 40 60 80 100
Matrix
1
1.05
1.1
1.15
1.2
Matrix product ratio
P(tanh_tayps)/P(tanh_expm)
P(tanh_pol)/P(tanh_expm)
0 20 40 60 80 100
Matrix
20
22
24
26
28
30
Approximation polynomial order
M(tanh_expm)
M(tanh_tayps)
M(tanh_pol)
(e) Ratio of matrix products. (f) Polynomial orders.
Figure 2. Experimental results for Test 2.
In contrast, although
tanh_expm
proved to be the most inaccurate code, it is also evi-
dent that its computational cost was usually the lowest one, as Table 4and
Figures 1e, 2e and 3e reported. The ratio between the number of
tanh_tayps
and
tanh_expm
matrix products ranged from 0.82 to 1.43 for Test 1, was equal to 1.20 for Test 2, and ranged
from 0.82 to 1.8 for Test 3. Regarding
tanh_pol
and
tanh_expm
, this quotient varied from
0.82 to 1.13 for Test 1, from 0.68 to 1.2 for Test 3, and was equal to 1 for Test 2.
Table 5lists, in order for Tests 1, 2, and 3, the minimum, maximum, and average
values of the degree of the Taylor polynomials
m
and the scaling parameter
s
employed by
the three codes. Additionally, and in a more detailed way, Figures 1f, 2f and 3f illustrate
the order of the polynomial used in the calculation of each of the matrices that compose
the testbed. The value of
m
allowed to be chosen was between 2 and 30 for
tanh_expm
and
tanh_tayps
and between 2 and 21 for
tanh_pol
. As we can see, the average value
of
m
that was typically used varied from 25 and 30 for
tanh_expm
. It was around 25 for
tanh_tayps, and it ranged from 14 to 21 for tanh_pol.

Mathematics 2021,9, 1219 15 of 20
0 10 20 30 40 50 60
Matrix
10-20
10-15
10-10
Er
cond*u
tanh_expm
tanh_tayps
tanh_pol
2345
0
0.2
0.4
0.6
0.8
1
p
tanh_expm
tanh_tayps
tanh_pol
(a) Normwise relative errors. (b) Performance profile.
0 10 20 30 40 50 60
Matrix
10-20
10-15
10-10
10-5
100
105
Relative error ratio
Er(tanh_tayps)/Er(tanh_expm)
Er(tanh_pol)/Er(tanh_expm)
Lowest relative error rate
16%
47%
36%
Highest relative error rate
64%
17%
19%
tanh_expm
tanh_tayps
tanh_pol
(c) Ratio of relative errors. (d) Lowest and highest relative error rate.
0 10 20 30 40 50 60
Matrix
0.8
1
1.2
1.4
1.6
1.8
Matrix product ratio
P(tanh_tayps)/P(tanh_expm)
P(tanh_pol)/P(tanh_expm)
0 10 20 30 40 50 60
Matrix
0
5
10
15
20
25
30
Approximation polynomial order
M(tanh_expm)
M(tanh_tayps)
M(tanh_pol)
(e) Ratio of matrix products. (f) Polynomial orders.
Figure 3. Experimental results for Test 3.
Table 5.
The minimum, maximum, and average parameters
m
and
s
employed for Tests 1–3, respec-
tively.
m s
Min. Max. Average Min. Max. Average
tanh_expm 16 30 27.41 0 5 3.55
tanh_tayps 9 30 25.09 0 6 4.65
tanh_pol 14 21 15.47 0 6 4.83
tanh_expm 30 30 30.00 2 2 2.00
tanh_tayps 25 25 25.00 3 3 3.00
tanh_pol 21 21 21.00 3 3 3.00
tanh_expm 2 30 26.23 0 8 2.77
tanh_tayps 9 30 24.38 0 9 3.74
tanh_pol 4 21 15.36 0 9 3.87
Having concluded the first part of the experimental results, we continue by comparing
tanh_tayps
and
tanh_pol
, the Taylor series-based codes that returned the best accuracy
in the results. Table 6presents the percentage of cases in which
tanh_tayps
gave place to
relative errors that were lower than, greater than, or equal to those of
tanh_pol
. According

Mathematics 2021,9, 1219 16 of 20
to the exposed values, both methods provided very similar results, and the percentage of
cases in which each method was more accurate than the other was approximately equal to
50% for the different tests.
Table 6. The relative error comparison for the three tests between tanh_tayps vs. tanh_pol.
Test 1 Test 2Test 3
Er(tanh_tayps)<Er(tanh_pol) 56% 47% 50.94%
Er(tanh_tayps)>Er(tanh_pol) 44% 53% 45.28%
Er(tanh_tayps)=Er(tanh_pol) 0% 0% 3.77%
Figures 4–6incorporate the normwise relative errors (a), the ratio of relative errors
(b), and the ratio of matrix products (c) between
tanh_tayps
and
tanh_pol
. The graphs
corresponding to the performance profiles and the polynomial orders are not included
now since the results match with those already shown in the previous figures. All this
information is also complemented by Table 7, which collects, respectively for each test, the
minimum, maximum, and average relative errors incurred by both methods to be analysed,
together with the standard deviation.
0 20 40 60 80 100
Matrix
10-16
10-15
10-14
10-13
10-12
10-11
Er
cond*u
tanh_tayps
tanh_pol
0 20 40 60 80 100
Matrix
10-4
10-2
100
102
104
Relative error ratio
Er(tanh_pol)/Er(tanh_tayps)
(a) Normwise relative errors. (b) Ratio of relative errors.
0 20 40 60 80 100
Matrix
1
1.1
1.2
1.3
1.4
Matrix product ratio
P(tanh_tayps)/P(tanh_pol)
(c) Ratio of matrix products.
Figure 4. Experimental results for Test 1.
As Table 6details, for Tests 1 and 3,
tanh_tayps
slightly improved on
tanh_pol
in
the percentage of matrices in which the relative error committed was lower, although it
is true that the difference between the results provided by the two methods was small in
most cases. However, when such a difference occurred, it was more often in favour of
tanh_tayps than the other way around, in quantitative terms.
With all this, we can also appreciate that the mean relative error and the standard
deviation incurred by
tanh_pol
were lower than those of
tanh_tayps
. For matrices that
were part of Test 2, the numerical results achieved by both methods were almost identical,
and the differences between them were not significant.

Mathematics 2021,9, 1219 17 of 20
0 20 40 60 80 100
Matrix
10-15
10-14
10-13
Er
cond*u
tanh_tayps
tanh_pol
0 20 40 60 80 100
Matrix
100
Relative error ratio
Er(tanh_pol)/Er(tanh_tayps)
(a) Normwise relative errors. (b) Ratio of relative errors.
0 20 40 60 80 100
Matrix
10-1
100
101
Matrix product ratio
P(tanh_tayps)/P(tanh_pol)
(c) Ratio of matrix products.
Figure 5. Experimental results for Test 2.
0 10 20 30 40 50 60
Matrix
10-20
10-15
10-10
Er
cond*u
tanh_tayps
tanh_pol
0 10 20 30 40 50 60
Matrix
10-1
100
101
102
103
Relative error ratio
Er(tanh_pol)/Er(tanh_tayps)
(a) Normwise relative errors. (b) Ratio of relative errors.
0 10 20 30 40 50 60
Matrix
1.1
1.2
1.3
1.4
1.5
Matrix product ratio
P(tanh_tayps)/P(tanh_pol)
(c) Ratio of matrix products.
Figure 6. Experimental results for Test 3.
To conclude the analysis and with regard to the computational cost of both codes, such
as depicted in Figures 4c, 5c and 6c, it simply remains to note that
tanh_tayps
performed
between 1 and 1.43 more matrix products compared with
tanh_pol
for Test 1, 1.2 more for
Test 2, and between 1.06 and 1.5 more for Test 3. Therefore,
tanh_pol
computed the matrix

Mathematics 2021,9, 1219 18 of 20
tangent function with a very similar accuracy in the results compared to
tanh_tayps
but
with a considerably lower computational cost.
Table 7.
The minimum, maximum, and average values and the standard deviation of the relative
errors committed by tanh_tayps and tanh_pol for Tests 1–3.
Minimum Maximum Average Standard
Deviation
tanh_tayps 4.84 ×10−16 6.45 ×10−12 1.22 ×10−13 7.40 ×10−13
tanh_pol 4.55 ×10−16 3.64 ×10−12 8.48 ×10−14 4.69 ×10−13
tanh_tayps 9.71 ×10−16 9.06 ×10−14 1.25 ×10−14 1.41 ×10−14
tanh_pol 9.92 ×10−16 9.35 ×10−14 1.26 ×10−14 1.47 ×10−14
tanh_tayps 1.09 ×10−254 1.10 ×10−10 2.26 ×10−12 1.52 ×10−11
tanh_pol 1.09 ×10−254 1.16 ×10−11 4.10 ×10−13 1.96 ×10−12
4. Conclusions
Two alternative methods to approximate the matrix hyperbolic tangent were ad-
dressed in this work. The first was derived from its own definition and reduced to the
computation of a matrix exponential. The second method deals with its Taylor series ex-
pansion. In this latter approach, two alternatives were developed and differed on how the
evaluation of the matrix polynomials was performed. In addition, we provided algorithms
to determine the scaling factor and the order of the polynomial. As a result, we dealt
with a total of three MATLAB codes (
tanh_expm
,
tanh_tayps
, and
tanh_pol
), which were
evaluated on a complete testbed populated with matrices of three different types.
The Taylor series-based codes reached the most accurate results in the tests, a fact
that is in line with the recommendation suggested in [
33
] of using a Taylor development
against other alternatives whenever possible. However, codes based on Taylor series can be
computationally expensive if the Paterson–Stockmeyer method is employed to evaluate the
polynomial, as we confirmed with the
tanh_tayps
implementation. One idea to reduce
this problem is to use Sastre formulas, as we did in our
tanh_pol
code, resulting in an
efficient alternative that significantly reduced the number of matrix operations without
affecting the accuracy.
The results found in this paper demonstrated that the three codes were stable and
provided acceptable accuracy. However, and without underestimating the other two
codes, the
tanh_pol
implementation proposed here offered the best ratio of accu-
racy/computational cost and proved to be an excellent method for the computation of
the matrix hyperbolic tangent.
Author Contributions:
Conceptualization, E.D.; methodology, E.D., J.I., J.M.A. and J.S.; software, J.I.,
J.M.A. and J.S.; validation, J.M.A.; formal analysis, J.I. and J.S.; investigation, E.D., J.I., J.M.A. and J.S.;
writing—original draft preparation, E.D., J.I., J.M.A. and J.S.; writing—review and editing, P.A.-J. All
authors have read and agreed to the published version of the manuscript.
Funding:
This research was funded by the Spanish Ministerio de Ciencia e Innovación under grant
number TIN2017-89314-P.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.

Mathematics 2021,9, 1219 19 of 20
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