Fractional pseudo-Newton method and its use in the solution of a nonlinear system that allows the construction of a hybrid solar receiver

Fractional pseudo-Newton method and its use in the
solution of a nonlinear system that allows the
construction of a hybrid solar receiver
A. Torres-Hernandez ,a, F. Brambila-Paz †,b, P. M. Rodrigo ‡,c,d, and E. De-la-Vega §,c
aDepartment of Physics, Faculty of Science - UNAM, Mexico
bDepartment of Mathematics, Faculty of Science - UNAM, Mexico
cFaculty of Engineering, Universidad Panamericana - Aguascalientes, Mexico
dCentre for Advanced Studies on Energy and Environment (CEAEMA), University of Jaén, Spain.
Abstract
The following document presents a possible solution and a brief stability analysis for a nonlinear system,
which is obtained by studying the possibility of building a hybrid solar receiver; It is necessary to mention that
the solution of the aforementioned system is relatively difficult to obtain through iterative methods since the
system is apparently unstable. To find this possible solution is used a novel numerical method valid for one and
several variables, which using the fractional derivative, allows us to find solutions for some nonlinear systems in
the complex space using real initial conditions, this method is also valid for linear systems. The method described
above has an order of convergence (at least) linear, but it is easy to implement and it is not necessary to invert
some matrix for solving nonlinear systems and linear systems.
Keywords: Iteration Function, Order of Convergence, Fractional Derivative, Parallel Chord Method, Hybrid
Solar Receiver.
1. Introduction
A classic problem of common interest in Physics, Mathematics and Engineering is to find the zeros of a function
f : Ω ⊂ Rn → Rn, that is,
{ξ ∈ Ω : f (ξ) = 0},
this problem often arises as a consequence of wanting to solve other problems, for instance, if we want to
determine the eigenvalues of a matrix or want to build a box with a given volume but with a minimal surface;
in the first example, we need to find the zeros (or roots) of the characteristic polynomial of the matrix, while in
the second one we need to find the zeros of the gradient of a function that relates the surface of the box with its
volume.
Although finding the zeros of a function may seem like a simple problem, in general, it involves solving nonlin-
ear equations and numerical methods are needed to try to determine the solutions to these problems; it should be
noted that when using numerical methods, the word “determine” should be interpreted as to approach a solution
with a degree of precision desired. The numerical methods mentioned above are usually of the iterative type and
work as follows: suppose we have a function f : Ω ⊂ Rn → Rn and we search a value ξ ∈ Rn such that f (ξ) = 0,
then we may start by giving an initial value x0 ∈ Rn and then calculate a value xi close to the searched value ξ
using an iteration function Φ : Rn → Rn as follows
Email address: anthony.torres@ciencias.unam.mx; Corresponding author; ORCID: 0000-0001-6496-9505
†Email address: fernandobrambila@gmail.com; ORCID: 0000-0001-7896-6460
‡Email address: prodrigo@up.edu.mx; ORCID: 0000-0003-0100-6124
§Email address: evega@up.edu.mx; ORCID: 0000-0001-9491-6957
DOI : 10.5121/mathsj.2020.7201
Applied Mathematics and Sciences: AnInternational Journal (MathSJ)Vol.7, No.2, June 2020
1

xi+1 := Φ(xi), i = 0,1,2,··· . (1)
When it is assumed that the iteration function Φ is continuous around ξ and that the sequence {xi}∞
i=0 converges
to ξ, it holds that
ξ = lim
i→∞
xi+1 = lim
i→∞
Φ(xi) = Φ lim
i→∞
xi = Φ(ξ), (2)
the previous result is the reason why the method given in (1) is called fixed point method.
In the last section of this document, we study the nonlinear system that describes a hybrid solar panel, which
consists of a photovoltaic-thermoelectric generator, and we will proceed to find a possible solution for this system
using the fractional pseudo-Newton method, because the apparent instability of the system makes the classic
Newton’s method not the most suitable to solve it.
2. Previous works
2.1. Historical background of fractional calculus
The question that led to the emergence of a new branch of mathematical analysis known as fractional calculus,
was asked by L’Hôpital in 1695 in a letter to Leibniz, as a consequence of the notation dnf (x)/dxn; perhaps it
was a game of symbols that which prompted L’Hôpital to ask Leibniz: “What happens if n = 1/2?”, Leibniz
replied in a letter, almost prophetically: “··· is an apparent paradox from which, one day, useful consequences
will be drawn [1].” Subsequently, the question became: may the order n of the derivative be any number: rational,
irrational or complex? Because the question was answered affirmatively, the name of the fractional calculus has
become an incorrect name and it would be more correct to call it arbitrary order integration and differentiation.
The concepts of arbitrary order differentiation and integration are not new. Interest in these subjects was evi-
dent almost in tandem with the emergence of conventional calculus (differentiation and integration of the integer
order), the first systematic studies were written in the early and mid-19th century by Liouville (1832), Riemann
(1953), and Holmgrem (1864), although Euler (1730), Lagrange (1772), and other authors made contributions
even earlier [1].
When a function does not have integer derivative the notion of weak derivative is required. The weak deriva-
tives give rise to generalized functions or distributions, which are often used in quantum mechanics. It is impor-
tant to mention that there are functions that do not have weak derivative but have fractional derivative, such as
the Cantor function [2].
Caputo in 1967, developed the first application of fractional calculus related to diffusion processes, in what
he named as anomalous diffusion equation [3]. There is practically no branch of classical analysis that remains
exempt from fractional calculus.
2.2. Fractional Iterative Methods
Although the interest in fractional calculus has mainly focused on the study and development of techniques to
solve systems of differential equations of non-integer order [3–7], over the years, iterative methods have also
been developed that use the properties of fractional derivatives to solve algebraic equation systems [8–12]. These
methods in general may be called fractional iterative methods.
It should be mentioned that depending on the nature of the definition of fractional derivative used, fractional
iterative methods have the particularity that they may be used of local form [8] or of global form [12]. These
methods also allow searching for complex roots for polynomials using only real initial conditions.
Although using an iterative method that uses fractional derivatives seems to require an unnecessary effort,
considering that a more natural option would be Newton’s method [13], it should be noted that Newton’s method
may only be used in differentiable functions, while fractional derivatives may be used in a larger number of
functions [2]. Some differences between Newton’s method and two fractional methods are listed in the Table 1
2

Classic Newton Fractional Newton Fractional Pseudo-Newton
Can it find the complex zeros
of a polynomial using
real initial conditions?
No Yes Yes
Can it find multiple zeros
of a function using a
single initial condition?
No Yes Yes
It can be used if the function
is not differentiable?
No Yes Yes
For a space of dimension N
are needed
N × N classic
partial derivatives
N × N fractional
partial derivatives
N fractional
partial derivatives
Is it recommended to solve systems
where the partial derivatives are
analytically difficult to obtain?
Not recommended Not recommended Is recommended
Table 1: Some differences between the classical Newton method and two fractional iterative methods.
2.3. Introduction of a Hybrid Solar Receiver
Concentrator photovoltaic (CPV) systems represent a technological success in solar energy applications because of
the high conversion efficiencies commercially achieved. These systems use optical devices to concentrate the sun-
light onto small highly efficient multi-junction (MJ) solar cells. Efficiency records in a laboratory of 47.1%, 43.4%,
and 38.9% at cell, mono-module, and module level respectively have been shown [14,15], while commercial CPV
modules have a mean efficiency of 30.0% [16]. In spite of these high efficiencies, CPV systems have a higher lev-
elized cost of energy (LCOE) than traditional photovoltaic (PV) systems [17, 18]. Among the strategies that are
being investigated to make CPV systems more competitive, the increase of the concentration factor or the increase
of efficiency are considered.
An increase of efficiency can be achieved by recovering part of the waste heat generated in the solar cells.
Among the strategies to get this, the hybridization with thermoelectric generators (TEG) is being proposed. Ther-
moelectric (TE) materials such as Bi2Te3 can operate under the Seebeck effect to transform a heat flux to elec-
tricity [19]. Hybrid CPV-TEG systems are in the research stage and several laboratory prototypes have been re-
ported [20–22], although currently they are far from the expected benefits.
3. Fractional Pseudo-Newton Method
3.1. Order of Convergence
Before continuing it is necessary to have the following definition [23]
Definition 3.1. Let Φ : Ω ⊂ Rn → Rn be an iteration function with a fixed point ξ ∈ Ω. Then the method (1) is called
(locally) convergent of (at least) order p (p ≥ 1), if exists δ > 0 and exists a non-negative constant C (with C < 1 if
p = 1) such that for any initial value x0 ∈ B(ξ;δ) it holds that
xk+1 − ξ ≤ C xk − ξ p
, k = 0,1,2,··· , (3)
where C is called convergence factor.
The order of convergence is usually related to the speed at which the sequence generated by (1) converges. For
the particular cases p = 1 or p = 2 it is said that the method has (at least) linear convergence or (at least) quadratic
convergence, respectively. The following theorem [23, 24], allows characterizing the order of convergence of an
iteration function Φ with its derivatives
Theorem 3.2. Let Φ : Ω ⊂ Rn → Rn be an iteration function with a fixed point ξ ∈ Ω. Assuming that Φ is p-times
differentiable in ξ for some p ∈ N, and furthermore
Φ(k)(ξ) = 0, ∀k ≤ p − 1, if p ≥ 2
Φ(1)(ξ) < 1, if p = 1
, (4)
then Φ is (locally) convergent of (at least) order p.
3

The previous theorem is usually very useful to generate a fixed point method with an order of convergence
desired, an order of convergence that is usually appreciated in iterative methods is the (at least) quadratic order.
If we have a function f : Ω ⊂ Rn → Rn and we search a value ξ ∈ Ω such that f (ξ) = 0, we may build an iteration
function Φ in general form as [25]
Φ(x) = x − A(x)f (x), (5)
with A(x) := [A]jk(x) a matrix, where [A]jk : Rn → R (1 ≤ j,k ≤ n). Notice that the matrix A(x) is determined
according to the order of convergence desired. Denoting by det(A) the determinant of the matrix A, is possible to
demonstrate that any matrix A(x) that fulfill the following condition [12]
lim
x→ξ
A(x) = f (1)
(ξ)
−1
, det f (1)(ξ) 0, (6)
where f (1) is the Jacobian matrix of the function f [26], guarantees that Φ(1)(ξ) = 0. As a consequence, exists
δ > 0 such that the iteration function Φ given by (5), converges (locally) with an order of convergence (at least)
quadratic in B(ξ;δ).
3.2. Fractional Derivative of Riemann-Liouville
One of the key pieces in the study of fractional calculus is the iterated integral, which is defined as follows [4]
Definition 3.3. Let L1
loc(a,b), the space of locally integrable functions in the interval (a,b). If f is a function such that
f ∈ L1
loc(a,∞), then the n-th iterated integral of the function f is given by [4]
aIn
x f (x) = aIx aIn−1
x f (x) =
1
(n − 1)!
x
a
(x − t)n−1
f (t)dt, (7)
where
aIxf (x) :=
x
a
f (t)dt.
Considerate that (n − 1)! = Γ (n) , a generalization of (7) may be obtained for an arbitrary order α > 0
aIα
x f (x) =
1
Γ (α)
x
a
(x − t)α−1
f (t)dt, (8)
the equations (8) correspond to the definitions of (right) fractional integral of Riemann-Liouville. Fractional
integrals satisfy the semigroup property, which is given in the following proposition [4]
Proposition 3.4. Let f be a function. If f ∈ L1
loc(a,∞), then the fractional integrals of f satisfy that
aIα
x aI
β
x f (x) = aI
α+β
x f (x), α,β > 0. (9)
From the previous result, and considering that the operator d/dx is the inverse operator to the left of the
operator aIx, any integral α-th of a function f ∈ L1
loc(a,∞) may be written as
aIα
x f (x) =
dn
dxn aIn
x (aIα
x f (x)) =
dn
dxn
(aIn+α
x f (x)). (10)
With the previous results, we can built the operator fractional derivative of Riemann-Liouville, as follows
[3,4]
4

aDα
x f (x) :=



aI−α
x f (x), if α < 0
dn
dxn
(aIn−α
x f (x)), if α ≥ 0
, (11)
where n = α + 1. Applying the operator (11) with a = 0 and α ∈ R Z to the function xµ, with µ > −1, we
obtain that
0Dα
x xµ
=
Γ (µ + 1)
Γ (µ − α + 1)
xµ−α
. (12)
3.3. Iteration Function of Fractional Pseudo-Newton Method
Let f a function, with f : Ω ⊂ R → R. We can consider the problem of finding a value ξ ∈ R such that f (ξ)=0. A
first approach to value ξ is by a linear approximation of the function f in a valor xi ≈ ξ, that is,
f (x) ≈ f (xi) + f (1)
(xi)(x − xi), (13)
then, assuming that ξ is a zero of f , from the previous expression we have that
ξ ≈ xi − f (1)
(xi)
−1
f (xi),
as consequence, a sequence {xi}∞
i=0 that approximates the value ξ may be generated using the iteration function
xi+1 := Φ(xi) = xi − f (1)
(xi)
−1
f (xi), i = 0,1,2,··· ,
which corresponds to well-known Newton’s method [13]. However, the equation (13) is not the only way to
generate a linear approximation to the function f in the point xi, in general it may be taken as
f (x) ≈ f (xi) + m(x − xi), (14)
where m is any constant value of a slope, that allows the approximation (14) to the function f to be valid. The
previous equation allows to obtain the following iteration function
xi+1 := Φ(xi) = xi − m−1
f (xi), i = 0,1,2,··· , (15)
which originates the parallel chord method [13]. The iteration function (15) can be generalized to larger
dimensions as follows
xi+1 := Φ(xi) = xi − m−1
In f (xi), i = 0,1,2··· , (16)
where In corresponds to the identity matrix of n × n. It should be noted that the idea behind the parallel chord
method in several variables is just to apply (15) component by component.
Before continuing it is necessary to mention that for some definitions of the fractional derivative, it is satisfied
that the derivative of the order α of a constant is different from zero, that is,
∂α
k c :=
∂α
∂[x]α
k
c 0, c = constant, (17)
where ∂α
k denotes any fractional derivative applied only in the component k, that does not cancel the constants
and that satisfies the following continuity relation with respect to the order α of the derivative
5

lim
α→1
∂α
k c = ∂kc. (18)
Using as a basis the idea of (16), and considering any fractional derivative that satisfies the conditions (17) and
(18), we can define the fractional pseudo-Newton method as follows
xi+1 := Φ(α,xi) = xi − P ,β(xi)f (xi), i = 0,1,2··· , (19)
with α ∈ [0,2] Z, where P ,β(xi) is a matrix evaluated in the value xi, which is given as follows
P ,β(xi) := [P ,β]jk(xi) = ∂
β(α,[xi]k)
k δjk + δjk
xi
, (20)
where
∂
β(α,[xi]k)
k δjk :=
∂β(α,[xi]k)
∂[x]
β(α,[xi]k)
k
δjk, 1 ≤ j,k ≤ n, (21)
with δjk the Kronecker delta, a positive constant 1, and β(α,[xi]k) defined as follows
β(α,[xi]k) :=
α, if |[xi]k| 0
1, if |[xi]k| = 0
, (22)
It should be mentioned that the value α = 1 in (22), is taken to avoid the discontinuity that is generated when
using the fractional derivative of constants in the value x = 0.
Since in the fractional pseudo-Newton method, the matrix P ,β(xi) does not satisfy the condition (6), any se-
quence {xi}∞
i=0 generated by the iteration function (19) has at most one order of convergence (at least) linear.
3.3.1. Some Examples
Example 3.5. Let the function:
f (x) =
1
2
sin(x1x2) −
x2
4π
−
x1
2
, 1 −
1
4π
e2x1 − e +
e
π
x2 − 2ex1
T
,
then the value x0 = (1.03,1.03)T is chosen to use the iteration function given by (19), and using the fractional deriva-
tive given by (12), we obtain the results of the Table 2
αm
mξ1
mξ2
mξ − m−1ξ 2
f (mξ) 2 Rm
1 0.78562 1.03499277 − 0.53982128i 5.41860852 + 4.04164098i 5.62354e − 6 8.38442e − 5 66
2 0.78987 0.29945564 2.83683317 1.09600e − 5 9.63537e − 5 88
3 0.82596 −0.26054499 0.62286899 5.66073e − 5 9.87374e − 5 140
4 0.82671 −0.1561964 − 1.02056003i 2.26280132 − 5.71855964i 4.32875e − 6 9.51178e − 5 194
5 0.83158 1.03499697 + 0.53981525i 5.41862187 − 4.04161017i 3.94775e − 6 8.80344e − 5 84
6 0.85861 1.16151359 − 0.69659512i 8.27130854 + 6.3096935i 2.14707e − 6 9.38721e − 5 164
7 1.15911 1.48131686 −8.38362876 1.20669e − 6 9.56674e − 5 191
8 1.24977 −1.10844524 + 0.10906317i −4.18608959 + 0.66029327i 3.71508e − 6 9.81146e − 5 164
9 1.25662 −1.10844605 − 0.10906368i −4.18608629 − 0.66029181i 3.69483e − 6 9.66271e − 5 170
10 1.26128 1.33741853 −4.14026671 1.89913e − 5 8.51053e − 5 67
Table 2: Results obtained using the iterative method (19) with = −3.
6

f (x) = −3.6x3 x3
1x2 + 1 − 3.6cos x2
2 + 10.8,−1.6x1 x1 + x3
2x3 − 1.6sinh(x3) + 6.4,−4x2 x1x3
3 + 1 − 4cosh(x1) + 24
T
,
then the value x0 = (1.12,1.12,1.12)T is chosen to use the iteration function given by (19), and using the fractional
derivative given by (12), we obtain the results of the Table 3
αm
mξ1
mξ2
mξ3
mξ − m−1ξ 2
f (mξ) 2 Rm
1 0.96743 0.38147704 + 1.10471108i −0.43686196 − 1.3473184i −0.38512615 − 1.4903386i 2.41936e − 6 7.07364e − 5 57
2 0.96745 −0.78311553 + 0.96791081i −0.58263802 + 1.2592471i 0.18175185 − 1.49135484i 3.85644e − 6 8.58385e − 5 37
3 0.96766 0.71500126 − 1.02632085i 0.53575431 − 1.314774i 0.45273307 − 1.35710557i 3.01511e − 6 8.34643e − 5 41
4 0.9677 −0.34118928 + 1.19432023i 0.37199268 − 1.40125985i −0.63215137 + 1.3074313i 2.72698e − 6 8.69377e − 5 49
5 0.96796 0.71500155 + 1.0263218i 0.53575489 + 1.31477495i 0.45273303 + 1.35710453i 2.52069e − 6 7.11216e − 5 34
6 0.97142 −0.34118945 − 1.19432007i 0.37199303 + 1.40125973i −0.63215109 − 1.3074314i 2.32465e − 6 8.66652e − 5 61
7 0.9718 0.38147878 − 1.10471296i −0.43686073 + 1.34732029i −0.3851262 + 1.49033466i 2.38466e − 6 8.53472e − 5 50
8 0.97365 −0.78311508 − 0.96791138i −0.58263753 − 1.2592475i 0.18175161 + 1.49135503i 1.99078e − 6 7.57517e − 5 51
9 1.03148 1.34508926 −1.29220278 −1.44485467 3.68616e − 6 9.58451e − 5 59
10 1.04155 −1.43241693 1.27535274 −1.11183615 4.06891e − 6 8.95830e − 5 48
Table 3: Results obtained using the iterative method (19) with = e − 3.
When we work with a linear system of the form
Ax = b,
it is possible to solve it using the method (19) considering the function
f (x) = Ax − b.
f (x) = (5x1 − 4x2 + 3x3 − 18,2x1 + 5x2 − 6x3 − 24,−2x1 + 7x2 + 12x3 − 30)T
,
then the value x0 = (0.64,0.64,0.64)T is chosen to use the iteration function given by (19), and using the fractional
derivative given by (12), we obtain the results of the Table 4
αm
mξ1
mξ2
mξ3
mξ − m−1ξ 2
f (mξ) 2 Rm
1 0.90162 5.97144261 3.88571164 1.22857594 2.55098e − 6 9.53968e − 5 65
4. Equations of a Hybrid Solar Receiver
Considering the notation
X = (x,y,z,v,w)T
:= (Tcell,Thot,Tcold,ηcell,ηT EG)T
,
it is possible to deﬁne the following system of equations that corresponds to the combination of a solar photo-
voltaic system with a thermoelectric generator system [27,28], which is named as a hybrid solar receiver [29]
7




x = y + a1 · a2(1 − v)
y = z + a1 · a3(1 − v)(1 − w)
z = a4 + a1 · a5(1 − v)(1 − w)
v = a6x + a7
w = (a8 − 1) 1 −
z + a9
y + a9
a8 +
z + a9
y + a9
−1
, (23)
whose deduction and some details about the difficulty in finding its solution may be found in the reference [30].
The ai’s in the previous systems are constants defined by the following expressions



a2 = rcell + rsol + Acell


rcop + rcer
AT EG
+
rintercon
0.5 · f ∗ · AT EG b · f ∗ +
√
AT EG


a5 = Acell


rintercon
0.5 · f ∗ · AT EG b · f ∗ +
√
AT EG
+
rcer
AT EG
+ Rheat exch


a1 = ηopt · Cg · DNI, a3 =
Acell · l
f ∗ · AT EG · kT EG
, a4 = Tair
a6 = −ηcell,ref · γcell, a7 = ηcell,ref (1 + 25 · γcell), a8 =
√
1 + ZT
a9 = 273.15
,
with the following particular values [30]



ηopt = 0.85, rintercon = 2.331e − 7, Tair = 20
Cg = 800, Acell = 9e − 6, Rheat exch = 0.5
DNI = 900, AT EG = 5.04e − 5, ηcell,ref = 0.43
rcell = 3e − 6, f ∗ = 0.7, γcell = 4.6e − 4
rsol = 1.603e − 6, b = 5e − 4, ZT = 1
rcop = 7.5e − 7, l = 5e − 4, rcer = 8e − 6
kT EG = 1.5
.
Using the system of equations (23), it is possible to define a function f : Ω ⊂ R5 → R5, that is,
f (X) :=


−x + y + a1 · a2(1 − v)
−y + z + a1 · a3(1 − v)(1 − w)
−z + a4 + a1 · a5(1 − v)(1 − w)
−v + a6x + a7
−w + (a8 − 1) 1 −
z + a9
y + a9
a8 +
z + a9
y + a9
−1


, (24)
then, solving the system (23) is equivalent to finding a value Xξ for the function (24) such that f (Xξ) = 0.
4.1. Generating an Initial Condition
Without loss of generality, we may suppose that we have a function f : Ω ⊂ R → R with a simple root ξ, that is,
f (x) = (x − ξ)g(x), g(ξ) 0,
it should be noted that in Ω it is possible to find pairs of points xa and xb, with xa xb, such that
f (xa) · f (xb) ≤ 0,
8

as a consequence
xξ ∈ [xa,xb] or xξ ∈ [xb,xa] with f (xξ) = 0.
Hence, one way to approach the value xξ, is to generate a set of N pseudorandom numbers {xi}N
i=1, with xi <
xj ∀i < j and xi ∈ Ω ∀i ≥ 1, with the intention of forming intervals [xi,xj] to evaluate the function f at its ends
until finding one interval where it holds that
f (xi) · f (xj) ≤ 0,
then, it is possible to take an initial condition x0 ∈ [xi,xj] for use the iterative method (19).
4.2. Inspecting the System of Equations
Assuming that the system (23) has a solution, then it is possible to find two values Xa and Xb, with [Xa]k
[Xb]k ∀k ≥ 1, such that
[f ]k(Xa) · [f ]k(Xb) ≤ 0, ∀k ≥ 1, (25)
in consequence
[Xξ]k ∈ [[Xa]k,[Xb]k] or [Xξ]k ∈ [[Xb]k,[Xa]k], ∀k ≥ 1. (26)
To determine if the system (23) has a solution, we may take the following values
Xa = (53,51,22,0,0)T
⇒ f (Xa) ≈ (1.831,23.041,1.686,0.424,0.016)T
,
Xb = (54,52,23,1,1)T
⇒ f (Xb) ≈ (−2,−29,−3,−0.576,−0.984)T
,
because the condition (25) is satisfied, it is possible to guarantee that there is a value Xξ that satisfies the
condition (26) with Xξ a zero of (24).
Although we have determined that the system (23) has a solution, we need have in mind that, since the system
is nonlinear iterative methods as (19) are needed to try to solve it. Using iterative methods does not guarantee that
the value Xξ may be found. However, it is possible to find approximate values XN given as follows
XN = Xξ + δξ, δξ = δξ(N), δξ < 1. (27)
Considering (27), it is necessary to give a general idea to determine if the function (24) is stable with respect
to the values XN .
Definition 4.1. Let f : Ω ⊂ Rn → Rn. If Xξ is a zero of the function f , we say that the function is stable with respect to
the value Xξ, if when doing Xξ → Xξ + δξ, with δξ < 1, it holds that
f (Xξ + δξ) = δf < 1. (28)
The condition (28), implies that for a function f to be stable, it is necessary that a slight perturbation δξ in its
solutions does not generate a great perturbation δf in its images. To try to analyze the stability of the function
(24), we may consider the following values
XN1
= (53.8,51.6,22.1,0.3,0.1)T
⇒ f (XN1
) ≈ 3.332,
XN2
= (53.8,51.6,22.1,0.4,0.1)T
⇒ f (XN2
) ≈ 1.408,
XN3
= (53.8,51.6,22.1,0.5,0.1)T
⇒ f (XN3
) ≈ 6.105,
9

although XN2
is not a zero of (24), it may be observed that it is a value close to the solution Xξ. Taking the
canonical vector ê4 = (0,0,0,1,0)T and subtracting 1.408 in all the expressions on the right side, we obtain that
XN1
= XN2
− 0.1 · ê4 ⇒ f (XN1
) − 1.408 ≈ 1.924,
XN2
= XN2
+ 0.0 · ê4 ⇒ f (XN2
) − 1.408 ≈ 0,
XN3
= XN2
+ 0.1 · ê4 ⇒ f (XN3
) − 1.408 ≈ 4.697,
the above helps us to visualize that a small perturbation δξ near the solution Xξ is producing a large perturba-
tion δf near its image, reason why we can say that at first instance the function (24) is unstable with respect to the
values XNk
. The mentioned above may be visualized in the Figure 1.
Figure 1: Graphs of the components [f ]k(X(v)) and f (X(v)) 2 with respect to different values of v.
4.3. Finding a Solution
Considering that the function (24) is unstable, it is necessary to give an initial condition X0 very close to the
solution Xξ to be able to successfully use the iterative method (19), taking X0 = XN2
we obtain that
f (X0) ≈ (0.098,−1.398,−0.11,0.024,−0.084)T
, (29)
then, taking the fractional derivative given by (12), the fractional pseudo-Newton method was implemented
in the function (24) to generate a sequence {Xi}N
i=0 that approaches the solution Xξ. In consequence, the results
shown in the Table 5 were obtained.
α x y z v w
1 1.02934 53.80159759 51.59708283 22.09436105 0.4243031 0.01524
||XN − XN−1||2 ||f (XN )||2 N
2.04578e − 3 4.98732e − 3 606
Using the values of the Table 5, we obtain that
f (XN ) ≈ (0.001,0.0,−0.005,−0.0,0.001)T
, (30)
the previous result allows us to get an idea of how close the solution XN , obtained by the method (19), is to the
solution Xξ of the system (23).
10

5. Conclusions
The fractional pseudo-Newton method to solve the problem of the need to invert a matrix in each iteration that
is present in other methods. However, this method, may has at most an order of convergence (at least) linear, and
hence, a speed of convergence relatively slow. As a consequence, it is necessary to use a larger number of positive
values α ∈ [0,2] Z and a greater number of iterations, then we require a longer runtime to find solutions, so it
may be considered as a method slow and costly, but it is easy to implement.
This fractional iterative method may solve some nonlinear systems and linear systems and is efficient to find
multiple solutions, both real and complex, using real initial conditions. It should be mentioned that this method
is extremely recommended in systems that have infinite solutions or a large number of them.
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