Extended Higher Order Iterative Method for Nonlinear Equations and its Convergence Analysis in Banach Spaces

: In this article, a novel higher order iterative method for solving nonlinear equations is developed. The new iterative method obtained from fifth order Newton-Özban method attains eighth order of convergence by adding a single step with only one additional function evaluation. The method is extended to Banach spaces and its local as well as semi-local convergence analysis is done under generalized continuity conditions. The existence and uniqueness results of solution are also provided along with radii of convergence balls. From the numerical experiments, it can be inferred that the proposed method is more accurate and effective in high precision computations than existing eighth order methods. The computation of error analysis and norm of functions demonstrate that proposed method takes a lead over the considered methods.


Introduction
In numerical analysis, higher order iterative methods have acquired foremost significance for solving nonlinear equations that arise in numerous branches of science and technology [1,2].Various researchers have developed a plethora of iterative methods [3][4][5][6][7][8][9][10][11][12][13] for solving nonlinear equations given in the form ( ) 0, f x = where f : D ⊂ R → R is a continuously differentiable nonlinear function and D is an open interval.Most widely used iterative method is quadratically convergent Newton's method given by 1 ( ) , 0,1, 2, .( ) Numerous applications in fields of chemical speciation, transportation, chemical engineering, electron theory, the geometric theory of relativistic string, queuing models etc. also give rise to innumerable such equations.But most of the time the transformed nonlinear equations can not be solved using analytical approach.Thus, to find the numerical solution of such equations, iterative methods are taken into consideration.To have an efficient approximation and more accuracy in finding the solution of nonlinear equations of the form (1), current trend is to develop higher order iterative methods.Such methods are of utmost importance as a number of applications in multidisciplinary areas need faster convergence.But maintenance of an equilibrium between the convergence order and operational cost is another important issue at the same time.In order to improve the convergence of Newton's method, various higher-order methods have been proposed by researchers worldwide.
(Q 1 ) It is worth noticing that all the aforementioned methods (3)-( 7) are defined on the real line.There are common limitations in the aforementioned works related to the usage of Taylor series to show the convergence of these methods.Moreover, this approach requires assumptions on the existence of higher order derivatives that do not appear on these methods.Let us consider the motivational example.Define the function f : and satisfy m 2 + m 3 = 0.Then, clearly t * = 1 ∈ D, and f (t * ) = 0.However, f '''(t) is not continuous at t * = 0 ∈ D. Hence, the results involving methods (3)-( 7) cannot guarantee their convergence because all of them require existence of f ''' and even higher.However, these methods may converge.Furthermore, there are other limitations: (Q 2 ) There are no computable a priori estimates on ||x k − x * ||.That is we do not know in advance how many iterates should be computed to reach a predecided error tolerance.
(Q 3 ) The choice of the initial point is hard (i.e. a "shot in dark").This problem exists, since no radius of convergence is found for these methods.
(Q 4 ) No isolation of the solution results are given either.
In particular, these concerns exist for our method studied in Section 1.The limitations (Q 1 )-(Q 4 ) restrict the usage of these methods.The novelty of this method is that we address all these concerns positively in Section 3, where the method of Section 2 is extended in the setting of Banach spaces.Notice also that the technique developed in Section 3 is very general.We simply use it on an eighth convergence order method.However, the same technique can be used to extend the applicability of other methods using inverses of linear operators.Hence, this is the motivation and the novelty of this paper.This is the direction of our future research.
The contents of the paper are summarized as under.Section 2 includes the establishment of the eighthorder method and its convergence analysis is discussed.In section 3, the proposed method is extended to Banach spaces and its local and semi-local convergence analysis is provided.In Section 4, numerical examples are figured out to ascertain the theoretical postulates for comparing the proposed methods with the current methods.Section 5 contains the concluding remarks.

Development of eighth order method
In this section, we propose a new iterative method for solving the nonlinear equation of the form (1) from fifthorder Newton-Özban composition given by Grau-Sánchez et al. [19].This method is given as follows: Extension of fifth order method (8) to obtain an eighth order iterative method is done by adding a step in the following manner: ( ) where, k = 0, 1, 2, ... and the initial approximation x 0 is suitably chosen.The foremost aim of our study is to develop a novel and efficient eighth-order iterative method.The convergence analysis of the eighth-order method ( 9) is established in the next theorem.Theorem 2.1 Let f : D ⊂ R → R be a sufficiently differentiable function in an open interval D and x 0 is a close approximation to its simple root x * ∈ D. The iterative method (9) satisfies the following error equation: Substituting ( 14) and ( 16) in the second last substep of ( 9), we obtain Substituting ( 12), ( 14) and (18) in the last substep of (9), we obtain ( ) □

An extension
There are certain limitations with the local convergence analysis of the previous section.
(L 1 ) The analysis is provided only for nonlinear equations defined on the real line.
(L 2 ) The Taylor series expansion technique requires the existence of derivatives such as f ( j ) , j = 2, 3, ..., 7 which are not present on the method.
Let us consider the function f : [−2, 1.5] → R defined by for t ≠ 0 and f (t) = 0 for t = 0.It follows by this definition that f ''' is unbounded on the interval [−2, 1.5], since f ''' is not continuous at t = 0. Notice also that t * = 1 solves the equation f(t) = 0. Therefore, the results of the previous section cannot guarantee the convergence of the method to t * .But the method converges to t * say for t 0 = 0.9.
(L 3 ) There are no error estimates on the distances ||x k − x * || which can be computed a priori.Hence, we do not know in advance how many iterations should be carried out to obtain a desired error tolerance.
(L 4 ) There are no results on uniqueness of the solution x * .
(L 5 ) The more interesting semi-local convergence of the method is not given.
The limitations (L 1 )-(L 5 ) constitute the motivation for writing this section.These problems are positively addressed as follows: (L 1 )' The convergence analysis is carried out for Banach space valued operators.(L 2 )' The convergence conditions involve only f ' which is the only derivative appearing on the method.(L 3 )' Upper bounds on the error distances ||x k − x * || are provided which can be computed in advance.Therefore, we do know the number of iterations required to achieve a certain error tolerance.
(L 4 )' A certain neighborhood of x * is determined with only one solution.
(L 5 )' The semi-local convergence the method is developed based on majorizing sequences [22].Both type of analyses rely on the concept of generalized continuity of the derivative [20,22].
In order to achieve the extensions (L 1 )'-(L 5 )' the method has to be rewritten in a Banach space as follows: R and φ = f , the method (20) reduces to method (9) for solving the equation f(t) = 0.
P q t q t t P t P q t t P q t t The equation q 3 (t) − 1 = 0 has an SS r 3 ∈ T 2 − {0}, where P t P q t t d q t P q t t P q t t Define the function ( ) ( ( ) ).

P q t t P t P t P q t t
The equation q 4 (t) − 1 = 0 has an SS r 4 ∈ T 3 − {0}, where P t P t P q t t d q t P t P q t t P q t t θ θ The parameter r * is shown to be a radius of convergence for the method (20) in Theorem 3.1.However, some more conditions are needed.Let S(x, ρ), S[x, ρ] denote respectively, the open and closed balls in B 1 with center x ∈ B 1 and of radius ρ > 0.
(H 2 ) There exists an invertible linear operator M so that for each x ∈ D ( ) The conditions (H 1 )-(H 4 ) are sufficient for the local convergence analysis of the method (20).Theorem 3.1 Suppose that the conditions (H 1 )-(H 4 ) hold.Then, for x 0 ∈ S(x * , r * ) − {x * }, the sequence {x k } generated by the method (20) exists in S(x * , r * ), stays in S(x * , r * ) and is convergent to x * so that for each k = 0, 1, 2, ...

(
) and ( ) where the functions q i are as given previously, and the radius r * is as defined in (21).
1 with ( ) 0 . Consider the linear operator It follows by (H 2 ) and (37) in turn that ( ) which implies the invertability of the operator .Then, from the identity Next, the semilocal analysis of the method (20) is developed in an analogous way to the local case but the role of x * and the "P" functions is exchanged by x 0 and the "h" functions as follows: (A 1 ) There exists a (CNF) ψ 0 : T → R so that the equation ψ 0 (t) − 1 = 0 has an SS ρ 0 ∈ T − {0}.Set T 4 = [0, ρ 0 ).There exists a function ψ : T 4 → R. Define the sequences {a k }, {b k }, {c k } and {d k } for a 0 = 0, some b 0 ≥ 0 and each k = 0, 1, 2, ... by 0 0 ( ) The sequence {a k } is shown to be majorizing for {x k } in Theorem 3.3.But first a convergence condition for it is needed.
Thus, the sequence {x k } is Cauchy in the Banach space B 1 , and as such it is convergent to some x * ∈ S[x 0 , a * ].By letting k → +∞ in (44), we deduce φ(x * ) = 0.
The uniqueness result result for a solution of the equation ( 19) follows.

Proposition 3.4 Suppose:
There exists a solution x  ∈ S(x 0 , ρ 3 ) of the equation (19).The condition (A 3 ) holds in the ball S(x 0 , ρ 3 ), and there exists ρ 4 ≥ ρ 3 , so that Then, the only solution of the equation ( 19) in the region S 3 is x  .
Proof.Let 3 with ( ) 0 . Define the operator Then, by (A 3 ) and (47), we get ( ) (47) Thus, we conclude again as in the local case that x x =  .Remark 3.5 (i) If all conditions of Theorem 3.3 hold, then take ρ 3 = a * and x * = x  .(ii) A popular choice but not the most flexible is M = φ'(x * ) for the local, and M = φ'(x 0 ) for the semi-local case.

Numerical applications
This section comprises of several numerical examples to show the efficiency of our proposed method for approximating the solution by comparing with existing methods.The comparison is made with eighth order methods given by Neta (8).In Table 1, the considered test functions with initial approximation and the corresponding root are displayed.To compare the computational performance, the number of iteration indexes (k), norms of the functions (| f (x k )|), error between two consecutive iterates ||x k+1 − x k || and computational order of convergence given by formula [27]: Then, the method ( 9) is applied on the test functions given in Table 1.
The numerical experiments are stimulated by using Mathematica 8 on Intel(R) Core(TM) i5 -8250U mCPU @ 1.60 GHz   This definition implies that Thus, the novelty, applicability and theoretical results of the current work are corroborated by numerical experiments.

Conclusion
The construction of higher order iterative methods is at topmost importance in numerical analysis now a days due to its numerous applications in various fields.The present study consists of designing of a new eighth order method which is obtained from a fifth order by method by adding one step for which only one additional function evaluation is required.The method is extended to Banach spaces and is analyzed with local and semi-local convergence using generalized conditions.Based on the numerical findings, it is clear that the proposed method performs the best in terms of accuracy in error and norm of the function in all the considered examples as compared to other existing methods.As already noted in the introduction the methodology developed in Section 3 is applicable on other methods.This is the direction of our research which shall establish our approach further.We start with this paper.That will be an improvement of our work.

1
where the letters B 1 and B 2 denote Banach spaces, D an open and convex set, and φ a continuously differentiable operator in the Fréchet sense.Then, we approximate a solution x * ∈ D of the equation ( ) 0 x ϕ = using the extension of the method for x 0 ∈ D and each k = 0, 1, 2, ... by 1 ( ) ( ),

3 )
For each x, y ∈ S 0
1.80 GHz with 8 GB of RAM running on the Windows 10 Pro version 2017.The comparison results for all considered examples for |x k+1 − x k |, | f (x k )| and ρ are displayed in Tables 2-9 up to the third iteration.For every method, the stopping criterion used is | x k+1 − x k | + | f (x k )| < 10 −100 .It can be observed that proposed method has higher accuracy in numerical values of approximations to the root than the existing methods in all the considered examples.

Table 1 .
Test functions

Table 2 .
Comparison of the performances of methods for f 1

Table 3 .
Comparison of the performances of methods for f 2

Table 4 .
Comparison of the performances of methods for f 3

Table 5 .
Comparison of the performances of methods for f 4

Table 6 .
Comparison of the performances of methods for f 5

Table 7 .
Comparison of the performances of methods for f 6

Table 8 .
Comparison of the performances of methods for f 7

Table 9 .
Comparison of the performances of methods for f 8 The next two examples demonstrate the extension of proposed method in Banach spaces presented in Section 3. Example 4.2 Let B 1 = B 2 = R 3 and D = S(0, 1).Define the operator φ on D for s = (s 1 , s 2 , s 3 ) T by