Extended Convergence of Two Multi-Step Iterative Methods

: Iterative methods which have high convergence order are crucial in computational mathematics since the iterates produce sequences converging to the root of a non-linear equation. A plethora of applications in chemistry and physics require the solution of non-linear equations in abstract spaces iteratively. The derivation of the order of the iterative methods requires expansions using Taylor series formula and higher-order derivatives not present in the method. Thus, these results cannot prove the convergence of the iterative method in these cases when such higher-order derivatives are non-existent. However, these methods may still converge. Our motivation originates from the need to handle these problems. No error estimates are given that are controlled by constants. The process introduced in this paper discusses both the local and the semi-local convergence analysis of two step ﬁfth and multi-step 5 + 3 r order iterative methods obtained using only information from the operators on these methods. Finally, the novelty of our process relates to the fact that the convergence conditions depend only on the functions and operators which are present in the methods. Thus, the applicability is extended to these methods. Numerical applications complement the theory


Introduction
The most commonly recurring problems in engineering, the physical and chemical sciences, computing and applied mathematics can be usually summed up as solving a non-linear equation of the form with G : D ⊆ E 1 → E 2 being differentiable as, per Fréchet, E 1 , E 2 denotes complete normed linear spaces and D is a non-empty, open and convex set. Researchers have attempted for decades to trounce this nonlinearity. From the analytical view, these equations are very challenging to solve. The utilisation of iterative methods (IM) to find the solution of such non-linear equations is predominantly chosen among researchers for this very reason. The most predominantly used IM for solving such nonlinear equations is Newton's method. In recent years, with advancements in science and mathematics, many new higher-order iterative methods for dealing with nonlinear equations have been found and are presently being employed [1][2][3][4][5][6][7][8]. Nevertheless, these results on the convergence of iterative methods that are currently being utilised in the above-mentioned articles are derived by applying high-order derivatives. In addition, no results address the error bounds, convergence radii or the domain in which the solution is unique.
The study of local convergence analysis (LCA) and semi-local analysis (SLA) of an IM permits calculating the radii of the convergence domains, error bounds and a region in which the solution is unique. The work in [9][10][11][12] discusses the results of local and semi-local convergence of different iterative methods. In the above-mentioned articles, important results discussing radii of convergence domains and measurements on error estimates are discussed,thereby expanding the utility of these iterative methods. Outcomes of these type of studies are crucial as they exhibit the difficulty in selecting starting points.
In this article, we establish theorems of convergence for two multi-step IMs with fifth (2) and 5 + 3p (3) order convergence proposed in [8]. The methods are: and . . .
where p is a positive integer. It is worth emphasizing that (2) and (3) are iterative and not analytical methods. That is, a solution denoted by x * is obtained as an approximation using these methods. The iterative methods are more popular than the analytical methods, since in general it is rarely possible to find the closed form of the solution in the latter form.
Motivation : The LCA of the methods (2) and (3) is given in [8]. The order is specified using Taylor's formula and requires the employment of higher-order derivatives not present in the method. Additionally, these works cannot give estimates on the error bounds x i − x * , the radii of convergence domains or the uniqueness domain. To observe the limitations of the Taylor series approach, consider G on D = [−0.5, 1.5] by Then, we can effortlessly observe that since G is unbounded, the conclusions on convergence of (2) and (3) discussed in [8] are not appropriate for this example.
Novelty : The aforementioned disadvantages provide encourage us to introduce convergence theorems providing the domains and hence comparing the domains of convergence of (2) and (3) by considering hypotheses based only on G . This research work also presents important results for the estimation of the error bounds x i − x * and radii of the domain of convergence. Discussions about the exact location and the uniqueness of the root x * are also provided in this work.
The rest of the details of this article can be outlined as follows: Section 2 deals with LCA of the methods (2) and (3). The SLA considered more important than LC and not provided in [8] is also dealt with in this article in Section 3. The convergence outcomes are tested using numerical examples and are given in Section 4. Example 4 deals with a real world application problem. In Example 5, we revisit the motivational example to show that lim n→+∞ x n = x * = 1. Conclusions of this study are given in Section 5.
(ii) There exists a function ϕ:T 0 → R which is NC so that the equation g 0 (t) − 1 = 0 admits a MS r 0 ∈ T 0 − {0}, with the function g 0 :T 0 → R being In applications, the smallest version of the functionφ shall be chosen. Set r = min{r 0 , r 1 }.
The parameter r is the radius of the convergence ball (RC) for the method (2) (see Theorem 1). Set Then, if t ∈ T 2 , it is implied that and 0 ≤ g 1 (t) < 1.
The following conditions justify the introduction of the functions ϕ 0 and ϕ and helps in proving the LC of the method (2). and where (6) gives the formula for the radius r and the functions g 0 and g 1 are previously provided.

Proof. Let us define the linear operator
By utilizing the conditions (ii) and (iii), we attain in turn that

Remark 1.
(1) The parameter ρ 2 can be chosen to be r.
(2) The result of Theorem 1 can immediately be extended to hold for method (3) as follows: Define the following real functions on the interval T 2 Assume that the equations g k (t) − 1 = 0 admits smallest solutions r k ∈ T 2 − {0}. Define the parameterr byr = min{r 0 , r, r k }.
Then, the parameterr is a RC for the method (3).
Proof. By applying Theorem 1, we get in turn that Then, the calculations for the rest of the sub-steps are in turn: where we also used the estimates and By switching z 1 , z 2 by z k−1 , z k in the above calculations we get Moreover, in particular Therefore, we deduce that lim n→+∞ x n = x * and all the iterates {z k (x m )} ⊂ S(x * ,r).

Remark 2.
The conclusions of the solution given in Proposition 1 are also clearly valid for method (3).

Semi-Local Analysis
The convergence in this case uses the concept of a majorizing sequence. Define the scalar sequence for α 0 = 0 and β 0 ≥ 0 and for each i = 0, 1, 2, . . . as follows The sequence {α i } is shown to be majorizing for method (3). We now produce a general convergence result for it.

Remark 3.
(1) The limit point δ * is the unique least upper bound (LUB) for the sequence {α i }.
(2) A possible choice for δ = ρ 0 , where the parameter ρ 0 is given in condition (i) of Section 2.
Next, a region is determined in which the solution is unique.
(iii) There exists q 1 > q such that Then, in the region D 2 , where D 2 = D ∩ S[x 0 , q 1 ], the only solution of (1) is z * .
Values of ψ 0 (β i ) and β i can be found in Table 3.
Here, δ * = 0.0179926. Hence, we can conclude that the sequence {x i } is convergent to some x * ∈ S[x 0 , δ * ]. Example 4. We now discuss a real-world application problem which has wide applications in physical and chemical sciences. At 500 • C and 250 atm, the quartic equation for fractional conversion which depicts the fraction of the nitrogen-hydrogen feed that gets converted to ammonia can be framed as follows 674. We get ψ 0 (t) = ψ(t) = 1.56036t, ρ 0 = 0.640877 and D 2 = D ∩ S[x 0 , ρ 0 ]. Condition (20) is verified in Table 4.
Here, δ * = 0.0229759. Therefore, we can conclude that lim i→∞ x i = x * ∈ S[x 0 , δ * ]. Example 5. We reconsider the numerical example given in the introduction part to emphasize the aspect that our method does not require the existence of higher-order derivatives. Using (2), we obtain the solution x * = 1 after three iterations starting from x 0 = 1.2. Also, we can analyze this solution from the graph of G(t) given in Figure 1. On examining, we find that (A 1 )-(A 4 ) hold if ϕ 0 (t) = ϕ(t) = 96.8 t, ρ 0 = 0.0103306 and D 0 = D ∩ S(x * , ρ 0 ). Values of r andr( for p = 3, 4, 5, 6) are given in Table 5. Error estimates are plotted in Figure 2.

Conclusions
Many applications in chemistry and physics require solving abstract equations by employing an iterative method. That is why a new local analysis based on generalized conditions is established using the first derivative, which is the only one present in current methods. The new approach determines upper bounds on the error distances and the domain containing only one solution. Earlier local convergence theories [8] rely on derivatives which do not appear in the methods. Moreover, they do not give information on the error distances that can be computed, especially a priori. The same is true for the convergence region. The methods are extended further by considering the semi-local case, which is considered more interesting than the local and was not considered in [8]. Thus, the applicability of these methods is increased in different directions. The technique relies on the inverse of the operator on the method. Other than that, it is method-free. That is why it can be employed with the same benefits on other such methods [14][15][16][17]. This will be the direction of our research in the near future.