Generalized Iterative Method of Order Four with Divided Differences

: Numerous applications from diverse disciplines are formulated as an equation or system of equations in abstract spaces such as Euclidean multidimensional, Hilbert, or Banach, to mention a few. Researchers worldwide are developing methodologies to handle the solutions of such equations. A plethora of these equations are not differentiable. These methodologies can also be applied to solve differentiable equations. A particular method is utilized as a sample via which the methodology is described. The same methodology can be used on other methods utilizing inverses of linear operators. The problem with existing approaches on the local convergence of iterative methods is the usage of Taylor expansion series. This way, the convergence is shown but by assuming the existence of high-order derivatives which do not appear on the iterative methods. Moreover, bounds on the error distances that can be computed are not available in advance. Furthermore, the isolation of a solution of the equation is not discussed either. These concerns reduce the applicability of iterative methods and constitute the motivation for developing this article. The novelty of this article is that it positively addresses all these concerns under weaker convergence conditions. Finally, the more important and harder to study semi-local analysis of convergence is presented using majorizing scalar sequences. Experiments are further performed to demonstrate the theory.


Introduction
Suppose H is a Fréchet differentiable operator mapping from a Banach space S into S , and D is an open convex subset of S .Deriving a solution λ ∈ D of the nonlinear equation of the form has innumerable applications in multiple disciplines of science and engineering.These kinds of problems are formulated as an equation such as (1) using mathematical modeling [1][2][3][4].These equations may be defined on the real line, the Euclidean space with finite dimensions as a result of the discretization of a boundary value problem, and on a Hilbert or Banach space [2][3][4][5].Such equations can be found in the numerical Section 4. The solution of ( 1) is rarely attainable in analytic form.The iterative methods provide a tool by which to handle non-analytic and complex functions, thereby approximating the solution λ of (1).
In particular, we examine the convergence of the fourth-order method, free from derivatives, developed by Sharma et al. [19], which is defined for all m = 0, 1, 2, . . .by and where G −1 m is inverse of first order divided difference of [x m , u m ; H] of H, and I is identity operator.The fourth order of the method (2) is shown provided that S = R k and by assuming the existence of at least the fifth derivative and utilizing Taylor series expansions.Hence, the application is limited to solving nonlinear Equation (1), where the operator is that many times differentiable.However, the method may converge even if H (5) does not exist.
For instance, consider D to be an interval −5 2 , 2 and the function H is defined on D as Clearly, H (3) (t) is not continuous at t = 0.As a result, lthough the method converges, the convergence of method (2) to the solution t * = 1 π cannot be assured using results in [19].Moreover, notice that the method (2) does not have any derivatives.The aforementioned limitations and the ones listed below constitute the motivation for developing this article.Motivation (C 1 ) A priori upper bounds on x m − λ are not given, λ ∈ D being a solution of the Equation (1).The number of iterations to be performed to reach a predecided error tolerance is not known.(C 2 ) The initial guess x 0 is "shot in dark", and no information is available on the uniqueness of the solution.(C 3 ) There convergence of the method is not assured (although it may converge to λ) if at least H (5) does not exist.(C 4 ) The results are limited to the case only when S = R k .(C 5 ) The semi-local convergence, more interesting than the local convergence, is not given in [19].(C 6 ) The same concerns exist for numerous other methods with no derivatives [17,19].

Novelty
All these limitations are taken up positively in the present article.In particular, the local convergence relies on the general concept on ω-continuity [2,5,9] and uses only information from the operators appearing on the method.Moreover, the semi-local convergence not provided in the studies utilizes majorizing sequences [2,5].
The novelty of the article lies in the fact that the process leading to the aforementioned benefits does not rely on the particular method (2).However, it can be utilized on other methods involving inverses of linear operators in a similar manner.Notice that the development of efficiency and computational benefits have been discussed in [19].So, these aspects of the method do not repeat in the present article.
The article is structured as follows: The local convergence in Section 2 is followed by the semilocal convergence in Section 3. The numerical applications and concluding remarks which appear in Section 4 and Section 5, respectively, complete the article.

Local Analysis
The assumptions required are listed below provided Q = [0, +∞).
(E 1 ) There exist functions g 1 : Q → Q, ϕ 0 : Q × Q → Q which are continuous as well as non-decreasing (FCN) such that the equation ϕ 0 (t, g 1 (t)) − 1 = 0 has a minimal positive solution called P. Define the set Q 0 = [0, P). (E 2 ) There exist FCN g 2 : minimal positive solutions in the interval (0, P) denoted by P i , respectively, where the functions and (E 3 ) There exist an invertible operator M and a solution λ ∈ D of the Equation ( 1) such that for all x ∈ D, and Notice that by the definition of P, (E 1 ), and (E 3 ), Thus, y is well defined, since [x, u, H] −1 exists by a Lemma due to Banach for inverses of linear operators [2,4,5], and The developed notation and the assumptions (E 1 )-(E 5 ) are required in the main local result of this section for method (2).Theorem 1.Under the assumptions (E 1 )-(E 5 ) and provided that x 0 ∈ B[λ, P * ) − {λ}, the sequence {x m } is convergent to the solution λ of (1).

Proof. By applying the assumptions
The iterate y 0 exists by method (2, step one, from which The assumptions (E 2 ) and (E 4 ), the formula (3), and the estimates ( 4) and ( 5) lead to Hence, the iterate y 0 ∈ B(λ, P * ).Notice also that by the invertibility of G 0 , the iterate x 1 exists by the method (2) step two and Therefore, by (3), ( 4), (6), and (7), it follows that where the following calculations are also employed: where From the preceding calculations, if repeated for x m , y m , and x m+1 in place of x 0 , y 0 , and x 1 , respectively, the induction for the estimates resulting in x m+1 ∈ B(λ, P * ) as well as lim m→+∞ x m = λ.
Remark 1.The selection of the real functions g 1 and g 2 can be specialized further due to the calculations: Thus, a possible choice for the function g 1 is Similarly, we obtain and consequently, a choice for the function g 2 can be where h 1 is as previously written in (E 2 ).
The functions can be further specified if the linear operator M is precised.A popular choice is M = H (λ).However, in this case, although there are no derivatives on the method (2), it cannot be used to solve non-differentiable equations under previous assumptions, since we assume λ to be a simple solution (i.e., H (λ) is invertible).Thus, M should be chosen so that functions "ϕ" are as tight as possible but not M = H (λ) in the case of non-differentiable equations.
The isolation of the solution domain is specified in the next result.

Proposition 1. Assume:
There exists a solution ζ ∈ B(λ, P) of the equation H(x) = 0 for some P > 0. The condition (E 2 ) and (E 3 ) are validated on the ball B(λ, P), and there exists P ≥ P such that ϕ 3 (P ) < 1.
Then, the equation H(x) = 0 is uniquely solvable by λ in the domain Hence, we conclude ζ = λ.
A possible choice for P = P * .

Semi-Local Analysis
The mission of λ, "ϕ", g 1 , and g 2 functions is exchanged by the initial point x 0 and the "ψ", g 3 , and g 4 functions as defined below.
The semi-local analysis of method (2) follows in the next result.
Theorem 2. Under the Assumptions (T 1 )-(T 4 ), the sequence {x m } is convergent to some solution λ ∈ B[x 0 , q * ] given by method (1) so that Proof.As in the local analysis, we obtain, in turn, the estimates and, by induction, where we also used Moreover, by the first substep, Consequently, we obtain and Thus, the sequence {x m } is fundamental in Banach space S .So, there exists λ = lim m→+∞ x m ∈ B(λ, q * ).By sending m → +∞ in (14) it follows H(λ) = 0.Then, from the estimation Remark 2. The functions g 3 and g 4 can be determined analogously to the functions g 1 and g 2 as follows: Assume that there exist FCN ψ 3 : for all x ∈ Q 1 .Then, we can choose The next result determines the isolation of a solution region.

Proposition 2. Assume the following:
There exists a solution ξ ∈ B(x 0 , υ) of solution H(x) = 0 for some υ > 0. The first condition in (T 3 ) is validated on the ball B(x 0 , υ), and there exists ≥ υ such that where Then, the only solution of the equation H Then, we obtain it, in turn, by the conditions thus, [ξ, Y ; H] −1 ∈ L(B).However, the identity leads to a contradiction, and the divided difference [ξ, Y ; H] cannot be defined.Therefore, we conclude that Y = ξ.
Remark 3. (1) The point q can also be replaced by q * in condition (T 3 ).

Numerical Examples
A local example is given first.The solution in the second example is obtained by using method (2) to solve a non-differentiable equation.

Concluding Remarks
A methodology is provided that proves the convergence of iterative method (2) under weaker conditions than before [19].In particular, this methodology uses only conditions involving the operator on the method, in contrast to earlier approaches using the fifth derivative, not on the method.Moreover, the upper error bounds on the distances that are computable become available, i.e., we can use them to tell in advance how many iterations must be performed in order to obtain a pre-decided error tolerance.Such information is not available in [19] and related studies on other methods [6][7][8][9][10][11][12][13][14][15][16]19].Furthermore, a computable ball, also not given before, is defined, inside which there is only one solution to the equation.Finally, the more difficult and important semi-local analysis not dealt with in [19] or similar studies on other iterative methods [5,17,18] is presented, where the convergence is shown using real majorizing sequences.The same methodology can be applied to other methods utilizing the inverses of linear operators.Further, numerical experiments are performed that demonstrate the theoretical part.In our future research, the methodology shall extend the applicability of multipoint and multi-step methods [2][3][4]17].

Example 1 .
Let S = R × R × R and D = B[λ, 1].Consider the mapping on the ball D given for ρ