Further development on Traub’s method for solving system of nonlinear equations and ODE’s

The foremost objective of this work is to propose a eighth and sixteenth order scheme for handling a nonlinear equation. The eighth order method uses three evaluations of the function and one assessment of the ﬁrst derivative and sixteenth order method uses four evaluations of the function and one appraisal of the ﬁrst derivative. Kung-Traub conjecture is satisﬁed, theoretical analysis of the methods are presented and numerical examples are added to conﬁrm the order of convergence. The performance and eﬃciency of our iteration methods are compared with the equivalent existing methods on some standard academic problems. We tested projectile motion problem, Planck’s radiation law problem as an application. The basins of attraction are also given to demonstrate their dynamical behavior in the complex plane. Further, we attempt to proposed a sixteenth order iterative method for solving system of nonlinear equation with four functional evaluation, namely two F and two F (cid:48) and only one inverse of Jacobian. The theoretical proof of the method is given and numerical examples are included to conﬁrm the convergence order of the presented methods. We apply the new scheme to ﬁnd solution on 1-D bratu problem. The performance and eﬃciency of our iteration methods are compared.


Introduction
Tackling nonlinear equations could be a common and critical issue in science and engineering [9]. The boundary value problems in dynamic hypothesis of gasses, flexibility and other connected regions are generally diminished to solving single variable nonlinear equation. Thus, the issue of approximating a arrangement of the nonlinear equation is vital. Iterative strategies are one among the numerical strategies for finding the roots of such equations. Analytic strategies for fathoming such conditions are nearly nonexistent and consequently to get approximate solution by numerical strategies is based on iterative methods. With the progression of computers, the issue of solving nonlinear condition by numerical strategies has picked up more significance than some time recently. Famous Mathematicians who have contributed for the solution of equations are Cauchy, Chebyshev, Euler, Fourier, Gauss, Lagrange, Laguerre and Newton [43]. Here, we consider the problem of locating simple zeros and its denoted by x * of a equation f (x) = 0, where f (x) is a sufficiently continuously differentiable function. The Newton-Raphson method (N M ) is the most widely used algorithm for locating or finding simple zeros, which works with an initial points at x 0 near to the approximate root and obtaining a sequence of successive iterates {x n } ∞ 0 converging quadratically to simple roots. It is is given by , n = 0, 1, 2, 3... .
In the recent years many researchers are working with this problems in order to improve the convergence order and efficiency of N M method, in terms of additional functional evaluations, derivatives, and addition step [38]. Traub [42] proposed a two step variant of Newton's method (T M ) having convergence order three by evaluating two functions, one derivative is given by The huge survey of the literature dealing with these methods of improved order and efficiency are in [37] and references therein. Ezquerro et al. [21], Halley [23], Ostrowski's square root method [37] are well known cubic order iterative methods which needs the three functional evaluation of f , f and f per iteration. These methods are not an optimal method in the sense of Kung and Traub [30]. The conjectured says that the order of convergence of any multi-point without memory method with d function evaluations cannot exceed the bound 2 d−1 , the optimal order. Thus the optimal order for three evaluations per iteration would be four, four evaluations per iteration would be eight, and so on.
To obtain optimality case, researchers are developed and analysed further, some examples of optimal fourth order multi-point methods without memory which requires three evaluations per iteration are Ostrowski's method [37], King's family of methods [28], which contains Ostrowski's method as a special case. Methods of Chun et al. [11,12], Cordero et al. [16], Kou et al. [29], Jarratt [25,26] are seen to be efficient when compared to classical N M method. Some examples of optimal eighth order method without memory which requires four function evaluations per iteration are Liu et al [31] called as LW M , Sharma et al [40] called as SAM , Cordero et al [13] called as CF GT , Cordero et al [19] called as CT V , and Neta et al [35] called as N CS respectively given below (3) . (4) In this work, we contribute a little more in the theory of iterative methods by developing an optimal formula of order four from third order iterative method (2) with weight function for computing simple roots of a nonlinear equation which uses three function evaluations. Also, we develop a class of optimal eighth order methods from proposed fourth order method by using finite difference techniques. On the other hand, we analyze the behavior of eighth order method in the complex plane. Several authors have used these techniques on different iterative methods, viz Curry et al. [20] and Vrscay and Gilbert [44,45] described the dynamical behavior of some well-known iterative methods. The complex analysis of various other known iterative methods, such as King's and Chebyshev-Halley's families, Jarratt method have also been analyzed by various researchers, example, see [2,3,10,18,33,22]. Further, we develop an optimal sixteenth order method from proposed a member of optimal eighth order method with five functional evaluation by using finite difference techniques. This paper is organized as follows. In section 2, a class of optimal eighth-order and sixteenth-order and its proof of convergence are stated and proved for scalar equations. We compare the presented methods with some previously available eighth order methods on some test functions in section 3. In section 4, the proposed eighth order methods are studied in the complex plane using basins of attraction. Some real world problems are discussed in section 5 where new eighth and sixteenth order methods are applied on this problem. In section 6, we further developed a sixteenth order method for solving system of nonlinear equation and and its proof of convergence are stated. Also, we tested the performance of the proposed method with some academic problems. Section 7 gives concluding remarks.

Development of methods and its Convergence
Traub's method (2) having convergence order is three with three function evaluations per full iteration and having EI = 1.442 and it is not an optimal method. In this section, we propose new eighth and sixteenth order iterative method without memory namely three and fourth-step methods respectively. To improve the convergence order and efficiency of the Traub's method with three function evaluations, we used a weight function in the second step. To get an optimal eighth and sixteenth order iterative method, we used a divided difference technique in the third and fourth step. First, we trying to get an optimal fourth-order method in the following way Next, we stated the convergence proof for fourth order, which can easily proved with help of Mathematica.
In this section, our main aim is to develop an eighth and sixteenth-order method without memory. Here, we develop the class of optimal eighth order method with help of fourth order method (8) with following expressions where this method have convergence order eight with 5 function evaluations and it is not an optimal. To obtain an optimal scheme, so we estimate f (w n ) by the following polynomial which satisfies these condition Let us define the divided differences By using above conditions on equation (12), we get system of four linear equations with four unknowns a 0 , a 1 , a 2 and a 3 . From To find a 2 and a 3 , we solve the following equations: Solving above equations by using divided difference, we have Further, using eq. (13), we have the estimation Finally, we obtain a new class of optimal eighth order method where a 2 and a 3 are given in (13). Proof. Letẽ n = y n − x * ,ê n = w n − x * and c j = , j = 2, 3, 4, .... Expansion of f (x n ) and f (x n ) around x * by using Taylor's series, we have f (x n ) = f (x * ) e n + c 2 e 2 n + c 3 e 3 n + c 4 e 4 n + c 5 e 5 n + c 6 e 6 n + c 7 e 7 n + c 8 e 8 n + O(e 9 n )) (15) and f (x n ) = f (x * ) 1 + 2c 2 e n + 3c 3 e 2 n + 4c 4 e 3 n + 5c 5 e 4 n + 6c 6 e 5 n + 7c 7 e 6 n + 8c 8 e 7 n + 9c 9 e 8 n + O(e 9 n ) Thus,ẽ n = c 2 e 2 n + − 2c 2 2 + 2c 3 e 3 n + 4c 3 2 − 7c 2 c 3 + 3c 4 e 4 + − 8c 4 Expanding f (y n ) about x * by using Taylor's series, we have Also, expanding f (w n ) about x * by using Taylor's series, we have Substituting equations (33)- (19) in the third step of (14) and simplifying, we obtain This reveals that the proposed family of methods attains eighth-order convergence. The efficiency of the method (14) is EI = 1.682.
By choice of any value of H (0) in (9), we getting a new eighth order iterative method. Some members of the class (14) are as follows. Proposed method (V T M 1): By choosing H (0) = 0, we obtain as This method has the following error equation This method has the following error equation e n+1 = c 2 2 c 2 2 + c 3 c 3 2 + c 2 c 3 − c 4 e 8 n + O(e 9 n ). Note that based on the analysis done on numerical results, we find that V T M 2 is marginally better or equal than other ones, therefore we are considering V T M 2 to develop further to sixteenth order method, namely V T M 4. In the similar way, we are trying to get a new sixteenth order iterative method (V T M 4) as following way The above method is having convergence order is 16 with six evaluations. However, this is not an optimal method. To get an optimal, we estimate f (z n ) by the following polynomial which satisfies these conditions Using the above conditions on equation (24), we obtain system of five linear equations with five we solve the following equations: Thus by using divided differences, the above equations reduced to Solving the equation (25), we have Using equation (26), we have Finally, we obtain a new optimal sixteenth order method (V T M 4) where b 2 , b 3 and b 4 are given in (26). The efficiency of the method (27) is EI = 1.741.
The following theorem is given without proof, which can be worked out with the help of Mathematica.

Numerical examples
Here, we testing the performance and effectiveness of new methods, classical Newton's method (N M ), and these eighth order methods (3)- (7). Let us consider the following test functions to test, Numerical results are carried out in the Matlab software and we have used the following stopping criteria for satisfying the iterative process error = |x N − x N −1 | < , where = 10 −50 and N is the number of iterations required for convergence. The computational order of convergence is given by ( [17]) From Table 1, we observe that V T M 1, V T M 2, V T M 3, and V T M 4 converges with lesser number of iterations or with least error than compared methods (3)- (7). We conform that the theoretical order of convergence and computational order of convergence are approximately equal. Note that the initial guess are near to the root, then we will get converge with least iteration and error. Concluding that the proposed method V T M 4 having good efficiency in all the test function as compared to other methods. Hence, the V T M 4 can be considered competent enough to existing other compared methods.
Remark 1: We are trying to compare the test function with "f zero" command in Matlab software and the results are given in table 2. Here N 1 is the number of iterations to converge the interval containing the root and f (x n ) is the error after converging N number of iterations. For the f zero command, the zeros are consider to be location where the function actually cuts, not just meet the x-axis. It is observed that the new methods converge with a lesser number of iteration and total function evaluations than the f zero solver. Also, we conclude that the Newton-type iterative methods are better than f zero command.

Projectile Motion Problem
We consider the classical projectile problem in which a shot is propelled from a tower of tallness h > 0, with beginning speed v and at an point θ with regard to the horizontal onto a slope, which is characterized by the function ω, called the affect function which is subordinate on the horizontal distance, x. We wish to discover the ideal dispatch point θ m which maximizes the even distance. In our calculations, we disregard air resistances. The path function y = P (x) that depicts the movement of the shot is given by  When the shot hits the slope, there's a value of x for which P (x) = ω(x) for each value of x. We wish to discover the value of theta that maximize x (See more [27]).

Planck's Radiation Law Problem
We consider the following Planck's radiation law problem found in [7]: which calculates the vitality thickness inside an isothermal blackbody. Here, λ is the wavelength of the radiation, T is the supreme temperature of the blackbody, k is Boltzmann's steady, h is the Planck's consistent and c is the speed of light. Assume, we would like to decide wavelength λ which compares to greatest vitality thickness ϕ(λ).    Table 4 appears that the method V T M 4 is converging with least number of iteration with least error than other compared methods. Hence, the method V T M 4 can be considered competent sufficient to existing other compared methods. Also, we accommodate that the hypothetical arrange of convergence and computational arrange of merging are roughly break even with.

Basins of attraction
The consider on basin of attractions of the rational function related to an iterative procedure gives basic information around blending and strength of the procedure. To start with, we grant underneath a number of essential definitions of rational function in organize to think about capacities inside the complex space with complex zeros as found in [3,39].
The Fatou set is characterized as the set of focuses whose circles tend to an pulling in settled point z 0 . The closure of the set comprising of repulsing settled points is called Julia set which is nothing but the complement of Fatou set, which builds up the borders between the basins of attraction. That suggests, the basin of fascination of any settled point incorporates a put to the Fatou set and the boundaries of these basins of fascination have a place to the Julia set.
Consider a square region having the boundaries R × R = [−3, 3] × [−3, 3] of 90000 lattice points. These points are gotten from 300 columns and 300 lines which see just like the pixels of a computer show and this speak to a locale of the complex plane. The iterative strategy endeavors a zero z * j of the condition with a condition |f (z (k) )| < 1e − 4 and a most extreme of 100 iteration, we conclude that z (0) is within the basin of attraction of this zero. In the event that the iterative strategy beginning in z (0) comes to a zero in N iterations (N ≤ 100), at that point we check this point z (0) with colors in case |z (N ) − z * j | < 1e − 4. On the off chance that N > 50, we conclude that the starting point has diverged and we assign a dark blue color. Let N D be number of diverging points and we check the number of beginning points which converge in 1, 2, 3, 4, 5 or over 5 iterations. In this way, we recognize the basin attractors by distinctive colors for diverse roots and diverse behaviors like converging or diverging. We analyze the basins of attraction for new eighth order methods and a few equivalent methods for the three polynomials p 1 (z) = z 2 − 1, p 2 (z) = z 3 − 1 and p 3 (z) = z 4 − 1. The roots for polynomials of p 1 (z) are given by α 1 = 1, α 2 = −1. The polynomiographs of p 1 (z) are displayed in Fig. 1, though Table 5 presents the number of meeting and wandering mesh points for each iterative method. We observe that the proposed methods V T M 1, V T M 2, and V T M 3 has no chaotic behaviour, no divergent (N D ) points, and have less mean (µ) number of iteration then other compared methods.

Polynomiographs of
The roots for polynomials of p 2 (z) are given by α 1 = 1, α 2 = −0.5000 − 0.8660i and α 3 = −0.5000 + 0.8660i. The polynomiographs of p 2 (z) are displayed in Fig. 2, though Table 6 presents the number of meeting and wandering mesh points for each iterative method. We observe that the proposed methods V T M 1, V T M 2, and V T M 3 has less chaotic behaviour, no divergent points, and have less mean number of iteration then other compared methods.

Polynomiographs of p 3 (z) = z 4 − 1
The roots for polynomials of p 3 (z) are given by α 1 = 1, α 2 = −1, α 3 = i and α 4 = −i. The polynomiographs of p 3 (z) are displayed in Fig. 3, though Table 7 presents the number of meeting and wandering mesh points for each iterative method. We observe that the proposed methods V T M 1, V T M 2, and V T M 3 has less chaotic behaviour, less divergent points, and have less mean number of iteration then other compared methods. Remark 2: We acclimate that a point z 0 containing to the Julia set at whatever point the elements in a neighborhood of point displays touchy on the conditions based. Hence, adjacent introductory conditions driving to the marginally diverse behavior afterward in a few number of iterations. Subsequently, a few compared       Table 7: Results of the polynomials p 3 (z) = z 4 − 1 methods are getting many divergent initial points. The boundaries of the basins of attraction are Julia set of the iteration function. Note that the proposed methods are less chaotic and more reliable than the other compared methods.

Further Development
In this section, we are considering third order method (2) to develop new sixteenth order iterative methods by using weight function for solving system of nonlinear equations, whereas the method required only two function, two derivative and only one inverse of Jacobian needed per cycle. Hence the new method having the sense that, Kung-Traub conjecture is fails in scalar equations. Recently, the similar work done by Ahmad [1], Babajee [4], Babajee and Madhu [5] and they proved their methods are fails in Kung-Traub conjecture for quadratic equations.
Let us modified the method (2) for solving system of nonlinear equation as given below where , s ≥ 1, α i 's are constants, and I is the identity matrix. Here, the system of nonlinear equations Let us define Using the notations in [14], it is noted that c 2 e (k) ∈ L(R n ). The error at the (k + 1)th iteration is e (k+1) = L(e (k) ) p + O (e (k) ) p+1 , where L is a p-linear function L ∈ L(R n × · · · × R n , R n ), is called the error equation and p is the order of convergence. Observe that (e (k) ) p is (e (k) , e (k) , · · · , e (k) ).
This following theorem can be proved with help of Mathematica software.

Numerical examples
This section deals with numerical comparisons in the Matlab computer code rounding to 1000 significant digits. The criteria to stopping used for the iterative process The approximated computational order of convergence p c given by (see [17]) .
Test Problem 1 (TP1) We consider the following nonlinear system: Whose positive root is given by We use x (0) = (1, 2) T as initial vector.
In table 8, we investigated the number of iterations N required to converge to the solutions, the total number of function evaluations n total , the total number of inverse evaluations n inv , computational order of convergence p c and the residual minimum error err min . The n total is counted as sum of the total number of function evaluations in F and F at point x k . For example, here we shall calculate n total for the T P 3. In this case, requires four function evaluations in F and twelve function evaluations in F . Per iteration N M method uses one F and one F , which implies that n total in 8 iterations is 128. Here, we can observe that the computational order of convergence is supports the theoretical order of convergence. The proposed method (31) requires less iterations than other compared methods. Also, the proposed method requires less n total and n inv than other compared methods in all the test problems.

Application on One-dimensional Bratu Problem
The 1-D Bratu problem [8] is given by with the boundary value conditions U (0) = U (1) = 0. The one-dimensional Bratu problem has bifurcated, two known, actual solutions for the values of λ < λ c , no solution for λ > λ c , and one solution for λ = λ c . The value of λ c is 8(η 2 −1), where η is the fixed point of function coth (x). The exact solution to the problem (38) is where θ is a constant to be find, that satisfies the boundary value conditions and is fastidiously chosen and assuming the answer of the problem (38). Similar way as in [36], we trying to show the way to get the crucial worth of λ. Substitute the equation (39) in the equation (38), simplify and collocate at the point x = 1 2 between in the interval. Selected some other point, but low order of approximations are possibly to being better think the collocation factors are dispensed extremely similarly for the duration of the region. Then, we have Differentiating equation (40) with respect to θ and setting dλ dθ = 0, the critical value λ c satisfies By eliminating λ from equations (40) and (41), we have the value of θ c for the critical λ c satisfying for θ c = 4.798714560 can be found by using an iterative method. Then, we will get λ c = 3.513830720 from (40). The one dimensional problem by using standard finite difference scheme is given below is calculated correct to fourteen decimal places. Let N λ be the mean of iteration number for the 350 λ's.  Table 9 give the results for the 1-D Bratu problem, where N denoting number of iterations for convergence. The proposed method (31) is the most efficient method among the other compared methods because it has the lowest mean number of iteration and its converging many initial points are 2 and 3 iterations. Thus, the proposed method is converging faster than other compared methods.

Conclusions
This paper has developed a classes of optimal eighth order methods and sixteenth order method without memory to solve nonlinear scalar equations numerically. The advantage of the proposed methods were high efficiency index, which do not require second derivative, high accuracy in numerical examples and also consistency with the conjecture of Kung-Traub. We have tested some examples using the proposed schemes and some known schemes, which illustrate the superiority of the proposed methods. Also, Projectile motion problem and Planck's radiation law problem are used to validate our proposed methods. The results obtained are interesting and encouraging for the new method. We have also compared the basins of attraction of various eighth order methods in the complex plane.
Further, we have developed sixteenth order iterative method with four functional evaluation namely two F and two F for solving system of quadratic equation. We have tested some examples using the proposed schemes and some known schemes, which illustrate the superiority of the proposed methods. Also, we test new method and some existing methods on the 1-D bratu problem and the results obtained are interesting and encouraging for the new methods. Concluding that the proposed method having good efficiency in all the test function as compared to other methods. Hence, the new method can be considered competent enough to existing other compared methods.

Declarations
Availability of data and materials: Not applicable.
Competing interests: The authors declare no competing of interest.