Generalized conformable fractional Newton-type method for solving nonlinear systems

In a recent paper, a conformable fractional Newton-type method was proposed for solving nonlinear equations. This method involves a lower computational cost compared to other fractional iterative methods. Indeed, the theoretical order of convergence is held in practice, and it presents a better numerical behaviour than fractional Newton-type methods formerly proposed, even compared to classical Newton-Raphson method. In this work, we design a generalization of this method for solving nonlinear systems by using a new conformable fractional Jacobian matrix, and a suitable conformable Taylor power series; and it is compared with classical Newton’s scheme. The necessary concepts and results are stated in order to design this method. Convergence analysis is made and a quadratic order of convergence is obtained, as in classical Newton’s method. Numerical tests are made, and the Approximated Computational Order of Convergence (ACOC) supports the theory. Also, the proposed scheme shows good stability properties observed by means of convergence planes.


Introduction
Fractional calculus is a generalization of classical calculus, and many properties from this are held. Many problems in real life can be described by using mathematical tools form fractional calculus, because of the higher degree of freedom compared to classical calculus tools [1,2].
In order to find the solutionx ∈ R of a nonlinear function f (x) = 0, where f : I ⊆ R −→ R is a continuous function in I ∈ R, some fractional Newton-type methods for solving nonlinear equations were proposed in recent years by using the Riemann-Liouville, Caputo and conformable fractional derivatives (see [3][4][5]). Our goal is to design a conformable vectorial Newton-type method, and make a comparison with the classical vectorial Newton method in terms of convergence analysis and numerical stability.
Let us firstly introduce some preliminary concepts related to scalar conformable derivative. The left conformable fractional derivative of a function f : [a, ∞) −→ R, starting from a, of order α ∈ (0, 1], α, a, x ∈ R, a < x, is defined as (see [11]) If that limit exists, f is said to be α-differentiable. If f is differentiable, (T a α f )(x) = (x − a) 1−α f (x). If f is α-differentiable in (a, b), for some b ∈ R, (T a α f )(a) = lim x→a + (T a α f )(x). It is also easy to see that T a α C = 0, being C a constant. The conformable fractional derivative is the most natural definition of fractional derivatives and involves a low computational cost, because it does not require the evaluation of special functions, such as Gamma or Mittag-Leffler functions.
Recently, a fractional Newton-type method by using conformable derivative has been designed for solving nonlinear equations in [5] with the following iterative expression: Where (T a α f )(x k ) is the left conformable fractional derivative of order α, α ∈ (0, 1], starting at a, a < x k , ∀k. When α = 1, the classical Newton-Raphson method is obtained. In [5], the quadratic convergence of this method by using a suitable conformable Taylor series (see [6]) is stated by the next result. Theorem 1 ([5]) Let f : I ⊆ R −→ R be a continuous function in the interval I ∈ R containing the zerox of f (x). Let (T a α f )(x) be the conformable fractional derivative of f (x) starting from a, with order α, for any α ∈ (0, 1]. Let us suppose that (T a α f )(x) is continuous and not null atx. If an initial approximation x 0 is sufficiently close tox, then the local order of convergence of the conformable fractional Newton-type method is at least 2, where 0 < α ≤ 1, and the error equation is where for j = 2, 3, 4, . . .

Remark 1
It can be shown that, by using the conformable product and chain rules stated in [11], the asymptotic constant of the error equation can be expressed as (3) α(x − a) α−1 C 2 = α(x − a) α−1 1 2α (T a α f ) (2) being for j = 2, 3, 4, . . . , which is the classical asymptotical error constant. In this case, j = 2. It can also be proven that the error equation of iterative scheme (2) by using the classical Taylor Series is: So, (4) and (5) show that error equation obtained by both Taylor series (the classical one, and that provided in [6]) is the same.

Remark 2
As predicted by Traub, since conformable Newton-type method proposed in [5] and the classical one have the same order of convergence, the asymptotical error constant of conformable Newton-type method equals the asymptotical error constant of classical one, plus some value described in [7] (Theorem 2-8).
In both error equations, (3) and (5), when α = 1, we obtain the error equation of classical Newton's method. In this work, we are going to use both Taylor series to make the convergence analysis, in this case, for a vector valued function.
That method proposed in [5], as seen in Theorem 1, can be only used to solve scalar nonlinear problems. In order to design a conformable vectorial Newton's method to find the solutionx ∈ R n of a nonlinear system F (x) =0, with coordinate functions f 1 , . . . , f n , where F : D ⊆ R n −→ R n is a sufficiently Fréchetdifferentiable function in an open convex set D, we have to state the existing concepts and results which will be necessary.
First, for the analysis of the convergence of nonlinear systems by using the classical Taylor Series, we can find in [8,9] the following notation: It can be observed that: From properties above, we can use the following notation: In [10], we can find a definition of conformable partial derivative as shown next: Definition 2 Let f be a function in n variables, x 1 , . . . , x n , the conformable partial derivative of f of order α ∈ (0, 1] in x i > a = 0 is defined as: In [10] is also defined the conformable Jacobian matrix as: Definition 3 Let f , g be functions in 2 variables x and y, and their respective partial derivatives exist and are continuous, where x > a 1 and y > a 2 , being a = (a 1 , a 2 ) = (0, 0) =0, then the conformable Jacobian matrix is given by: This can be directly extended to higher dimensions and, as it will be seen in the next section, a can be considered not null.
Another necessary concept, the Hadamard product, can be found in [12]: In next section, the new concepts and results needed to design a vectorial conformable Newton-type method are stated.
In this manuscript, the design and convergence analysis of the proposed method are made in Section 3, the numerical tests and numerical stability are discussed in Section 4, and the conclusions are given in Section 5.

New concepts and results
Regarding that, in (6), x i ∈ (0, ∞), we can define the conformable partial derivative in x i ∈ (a, ∞) as follows: Definition 5 Let f be a function in n variables, x 1 , . . . , x n , the conformable partial derivative of f of order 0 In the case This derivative is linear, and the product, quotient and chain rules are satisfied, likewise to conformable derivative given in [11]. In next result, a relation between classical partial derivative and conformable partial derivative is stated: We can also define the conformable Jacobian matrix for x 1 ∈ (a 1 , ∞) and x 2 ∈ (a 2 , ∞), where x = (x 1 , x 2 ) and a = (a 1 , a 2 ): Definition 6 Let f and g be coordinate functions of a vector valued function F : R 2 −→ R 2 in variables x 1 > a 1 and x 2 > a 2 , where x = (x 1 , x 2 ) and a = (a 1 , a 2 ), such that their respective partial derivatives exist and are continuous. Then, the conformable Jacobian matrix is given by This can be directly extended to higher dimensions.
To analyze the convergence of nonlinear systems by using a conformable Taylor Series, we can use the following notation analogous to Definition 1: It can be observed that: From properties above, we can use the following notation: To define a conformable Taylor series for a vector valued function, we proceed in a similar way as in Theorem 4.1 from [11].
Theorem 3 Let us suppose that F : R n −→ R n is an infinitely α-differentiable vector valued function, for some α ∈ (0, 1], around a point t 0 ∈ R n . Then, F has the conformable Taylor power series where F α(k) t 0 (t 0 ) means the mapping of conformable derivative k times.
If we map the conformable derivative once to F , and then we evaluate at t 0 , we If we map the conformable derivative twice to F , and then we evaluate at t 0 , we obtain F α (2) t 0 (t 0 ) = 2K 2 α 2 , so, K 2 = F α (2) t 0 (t 0 ) 2α 2 . Proceeding by induction, we have So, (11) is obtained.
Thus, F (t) in (11) may be written as . . As it may be seen, the conformable derivatives start at t 0 , which is the value where derivatives are being also evaluated. This is a problem to be avoided in order to define a conformable Newton-type iterative method.
Proceeding as in [6] (Theorem 4.1), we can obtain a new Taylor series by using Theorem 3 , where the conformable derivatives start at some point a = (a 1 , . . . , a n ) ∈ R n different from another point b = (b 1 , . . . , b n ) ∈ R n where they are evaluated: Then, F has the conformable Taylor power series being the Hadamard power.
Proof Let us denote by t 0 = a in (11), Evaluating (14) at b, isolating F (a), we get If we map the conformable derivative once and twice to F , starting at a, we obtain, respectively, and Replacing all derivatives evaluated at a in (14), with all derivatives evaluated at b in (16), (17) and (18) we obtain (13), which can be written as and the proof is finished.

Remark 4
With these expressions, we can write the Taylor power series expansion of F around the solutionx, being the conformable Jacobian matrix F α(1) a (x) not singular, as shown next: Remark 5 By using Definition 7, Theorem 2 and Hadamard power, we obtain and respectively, for a vector valued function F , being F (x) the classical Jacobian matrix. Note that, in (21) x → a + means that Moreover, in order to make the convergence analysis of our main proposal, another concept must be introduced.
Theorem 5 Let x, y ∈ R n , r ∈ R, and be the Hadamard power/product. The Newton's binomial theorem for vector values and fractional power is given by being the generalized binomial coefficient (see [13]) Proof Since Hadamard power/product is an element-wise power/product, the proof is analogous to classical one.
In next section, we deduce the conformable Newton-type method for solving nonlinear systems.

Design and convergence analysis
As we proceeded in [5], let us regard the approximation of function F through the Taylor power series (13) up to order one evaluated at the solutionx, as follows: Multiplying both sides of (25), by α F From (x − a) α , we isolatex, sō Regarding iterates x (k) and x (k+1) are approximations of solutionx, we obtain the conformable Newton-type method for nonlinear systems: (28) Next, convergence analysis of conformable Newton-type method (28) is made by using the conformable Taylor series (13), and the classical one.
In next result, the quadratic convergence of vectorial Newton-type method (28) by using the conformable Taylor series (13) is proven.
be the conformable Jacobian matrix of F starting at a ∈ R n , of order α, for any α ∈ (0, 1]. Let us suppose that F α (1) a (x) is continuous and not singular atx. If an initial approximation x (0) ∈ R n is sufficiently close tox, then the local order of convergence of conformable vectorial Newton's method is at least 2, and the error equation is being Proof By using Definition 7, Theorem 4, and regarding x (k) = e (k) +x, the conformable Taylor power series expansion of F (x) aroundx is Regarding (20), and using again Definition 7 and Theorem 5, the conformable Jacobian matrix of F x (k) can be expressed as We can set the Taylor power series expansion of F Solving for X 2 , Thus, Then, Using once again Theorem 5, Finally, And this completes the proof.
As in (4), it can be shown that, by using the product and chain rules, and considering (20), in error (29), . , which is the classical asymptotical error constant for a vector valued function F , and F is the classical Jacobian matrix. For this case, j = 2. In next result, the quadratic convergence of conformable Newton-type method (28) by using the the classical Taylor series can be proven: initial approximation x (0) ∈ R n is sufficiently close tox, then the local order of convergence of conformable vectorial Newton's method is at least 2, and the error equation is being Remark 6 It is confirmed that error equations given in (29) and (30) are the same.
In next section, some numerical tests with some nonlinear systems of equations are made. We remark that, in all tests, a comparison with classical Newton-Raphson's method (when α = 1) is made. Also, the dependence on initial estimates of both methods is analyzed by using the convergence plane.
To make a comparison to each of all test vector valued functions, we have used the same initial estimation for each table, and α ∈ (0, 1]. From each table, two figures with error curves are provided in order to visualize the error committed ( x (k+1) − x (k) ) versus number of iterations for different values of α; firstly, it is shown a figure for all the able values of α, then, it is shown a figure for some values of α in order to distinguish each curve from others. In the latter case, the curves chosen correspond to values of α with fewer iterations, or to an arbitrary choice when the number of iterations is the same. For each case, the corresponding curve to α = 1 is always chosen if possible to visualize both methods, the classical one (when α = 1) and the proposed in this paper, in the same figure.
In Table 1, we observe for F 1 (x, y) that classical Newton's method (when α = 1) does not find any solution in 500 iterations, whereas conformable vectorial Newton's  Table 1 procedure converges. We can also observe that ACOC may be even slightly greater than 2 when α = 1. We have to remark that a complex root is found with real initial estimate and different values of α when conformable vectorial Newton's method is used. In Figs. 1 and 2, error curve for classical Newton's procedure (when α = 1) is not provided because no solution was found in this case, whereas we can see that error curves stop erratic behaviour in later iterations.
In Table 2, we can see for F 1 (x, y) with a different initial estimation that, classical vectorial Newton's scheme and conformable vectorial Newton's method have a Table 2 Results for F 1 (x, y) =0 with initial estimation being a = (a 1 , a 2 ) = (−10, −10). It can be seen in Tables 3 and 4 for F 2 (x, y) that classical Newton's method and conformable vectorial Newton's scheme have a similar behaviour as in amount Table 4 Results for F 2 (x, y) =0 with initial estimation    Table 5 being a = (a 1 , a 2 ) = (−10, −10). We can see in Table 5 for F 3 (x, y) that conformable vectorial Newton's procedure requires less iterations than classical Newton's method for lower values of α. It can also be observed that ACOC may be slightly greater than 2 for lower values of α. In Figs. 9 and 10, errors are decreasing in each iteration.
In Table 6, we can see for F 3 (x, y) that conformable vectorial and classical Newton's method require the same amount of iterations, and ACOC is around 2 in all cases. Again, the errors are decreasing in each iteration in Figs. 11 and 12.   Table 6  Table 7 Results for F 4 (x, y) =0 with initial estimation The fourth test vector valued function is F 4 (x, y) = (x 2 +y 2 −4, e x +y −1) T with real rootsx 1 ≈ (−1.8163, 0.8374) T andx 2 ≈ (1.0042, −1.7296) T . The conformable Jacobian matrix of F 4 (x, y) is being a = (a 1 , a 2 ) = (−10, −10).  Table 7 We observe in Table 7 for F 4 (x, y) that again, conformable vectorial Newton's scheme requires less iterations than classical Newton's method for all values of α. It can also be seen that ACOC is around 2. We can see in Figs. 13 and 14 that errors are decreasing with iterations.
In Table 12, we can observe for F 6 (x) that classical Newton's method does not find any solution in 500 iterations, whereas conformable one converges for most of values of α. We can see that ACOC is around two, but much greater than 2 when α = 0.1. No results are shown when α = 0.7 because conformable Jacobian matrix becomes singular. Again, in Figs. 23 and 24, we can observe that error curves stop erratic behaviour in later iterations.
In Tables 11 and 12, some errors are zero because double precision arithmetic is used. A value very close to zero could be observed if a variable precision arithmetic be used.

Numerical stability
In this section, we study the stability of conformable vectorial Newton's method tested above. In that sense, we analyze the dependence on initial estimates by means of convergence planes, which is defined in [15], and used in [3][4][5]. Only two dimensions can be visualized in convergence planes, so we are going to provide them for vector valued functions F 1 , F 2 , F 3 and F 4 .  Table 12 For the construction of convergence planes, we consider from initial estimates (x 0 , y 0 ), the points x 0 in horizontal axis, and values of α ∈ (0, 1] in vertical axis. Each one of 8 planes in each figure is representing a different value of y 0 from initial estimates (x 0 , y 0 ). Each color represents a different solution found, and it is painted in black when no solution is found after 500 iterations. Each plane is made with a 400 × 400 grid, with a maximum of 500 iterations, and tolerance 0.001.
In Fig. 25, we can see for F 1 (x, y) that in (e), (f), (g) and (h) almost 100% of convergence is obtained, whereas in (a), (b), (c) and (d) it is obtained around 86% of convergence. For each case, this method converges to all roots, even to complex root with real initial estimate.
In Fig. 26, for F 2 (x, y) almost 100% of convergence is obtained for each plane. In (a), (b), (c) and (d) this method converges to 2 of 4 roots, and in (e), (f), (g) and (h) this method converges to the other 2 roots.
In Fig. 27, for F 3 (x, y) we can observe that between 77% and 98% of convergence is obtained. For each plane, this method converges to both real roots.
In Fig. 28, for F 4 (x, y) we can see that 100% of convergence is obtained in some cases, and almost in other cases. For (a), (b), (c), (d) and (e) this method converges to both real roots, and for (f), (g) and (h) converges to one root.
We can also observe, in general, it is possible to find several solutions with the same initial estimate by choosing distinct values for α.

Conclusion
In this work, the first conformable fractional Newton-type iterative scheme for solving nonlinear systems was designed. Also, we have introduced all the analytical tools required to construct this method. The convergence analysis was made, and the quadratic convergence is held as in classical Newton's method for nonlinear systems. It was concluded that, by using the conformable Taylor series introduced in this work, and the classical one, the same error equation is obtained in both versions (the conformable scalar method in [5], and the conformable vectorial method proposed in this work). Numerical tests were made, error curves were provided, and the dependence on initial estimates was analyzed, supporting the theory. We could observe that the conformable vectorial Newton-type method presents, in some cases, a better numerical behaviour than classical one in terms of amount of iterations, ACOC, and wideness of basins of attractions of the roots. We also could observe that complex roots may be obtained with real initial estimates, and several roots may be obtained with the same initial estimate by choosing different values of α. Code availability Not applicable.

Declarations
Ethics approval Not applicable.

Consent for publication Not applicable.
Human and animal ethics Not applicable.

Conflict of interest The authors declare no competing interests.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4. 0/.