Extending King’s Method for Finding Solutions of Equations

: King’s method applies to solve scalar equations. The local analysis is established under conditions including the ﬁfth derivative. However, the only derivative in this method is the ﬁrst. Earlier studies apply to equations containing at least ﬁve times differentiable functions. Consequently, these articles provide no information that can be used to solve equations involving functions that are less than ﬁve times differentiable, although King’s method may converge. That is why the new analysis uses only the operators and their ﬁrst derivatives which appear in King’s method. The article contains the semi-local analysis for complex plane-valued functions not presented before. Numerical applications complement the theory.


Introduction
In this article, the function F : Ω ⊂ T −→ T is differentiable, where T = R or T = C and Ω is an open nonempty set.
The nonlinear equation is studied in this article. An analytic form of a solution x * is preferred. However, this form is not always available. So, mostly iterative solution methods have been applied to approximate the solution x * . In particular, King's [1] fourth-order method (KM) has been used; where γ ∈ T is a parameter and A n = F(u n ) + (γ − 2)F(v n ). This definition gives µ (s) = 6 log s 2 + 60s 2 − 24s + 22.
This is the case, since Taylor series requiring derivatives of high order (not in KM) are utilized in the analysis for convergence. This is a common observation for other methods, such as Traub's, Jarratt's, and the Kung-Traub method to mention some [2,3,[5][6][7][8][9][10]. On the top of these concerns, some other problems exist with earlier studies. No computable data are provided for distances u n+1 − u n or u n − x * or the uniqueness and location of solution x * .
The next four sections include semi-local analysis, local analysis, the experiment, and conclusions, respectively.
Then, the following assertions hold t n ≤ s n ≤ t n+1 (5) and lim where t * is the unique least upper bound of sequence {t n }.
Another result is given for the sequence {t n } using stronger conditions but which are easier to verify than (4). However, first, we need to introduce some concepts. Let Develop polynomials defined on the interval [0, 1) as n (t) = 2(L 3 + L 4 t n+1 η)t n+1 η − 2(L 3 + L 4 t n η)t n−1 η + L 0 (1 + t), 1 (t), and f (2) Moreover, set g 2 (t) = g 2 (t). Notice that polynomials g 1 and g 2 are independent of n. In particular, say Then, condition g 1 (t) ≥ 0 needed in the next Lemma holds if The left side of this estimate is a positive multiple of η. However, the right side of it is positive but independent of η. So, this estimate certainly holds for sufficiently small η. The same observation is made for polynomial g 2 and condition g 2 (t) ≥ 0.
An auxiliary result connects these polynomials.

Lemma 2.
The following items hold: Proof. By the definition of these polynomials, we get in turn: n (t)t n η; (iii) This estimate follows immediately from the first two; (iv) It follows similarly from the definition of polynomials g 2 and f (2) n , since t ∈ [0, 1).

Proof. Mathematical induction is used to show
and These estimates are true for m = 0 by (7) or (8) and the definition of sequence {t m }. and Then, Evidently, (12) holds if It can be shown instead from Lemma 2 that However, by (15) and (20), Then, (21) holds by (10) and (22). Moreover, instead of (13), we can show and hold. Indeed, (24) holds if However, this holds because of the choice of β 2 and (9). Moreover, estimate (25) holds if which is true by the choice of β 3 and (9). Then, (23) holds if or However, this holds by (11). By sequence {t m }, (12) and (13), the estimate (14) also holds. Therefore, the induction for estimates (12)-(14) is terminated. Hence, {t m } is bounded by t * * , which is non-decreasing. Hence, it converges to t * .

Proof. Let
Then, by (H2) and (40), we obtain in turn that
The local convergence of KM uses conditions (C). Suppose that there is where d n = u n − x * , and the functions g 1 , g 2 were previously defined.
A uniqueness of the solution result follows next.
Next, the fourth-order convergence is shown using only the first derivative. Suppose: and hold for all x, y, z ∈ Ω, for some constants ω > 0 and ω 0 > 0. Further, suppose Let   (2) is convergent to x * with order four, i.e., Proof. The first substep of (2) and (56) gives Therefore, (55) and (56) give

Numerical Example
We verify convergence criteria using KM.

Example 3.
The example used in the introduction gives ω = ω 0 = 96.6629073. Then, for γ = 2, the radius is Recall that it was shown in the Introduction that earlier articles cannot be used to solve this problem. The method used is a specialization of KM for γ = 2.

Conclusions
In this article, the extension of KM is presented. The convergence of KM has been shown by assuming the existence of a fifth derivative which was not considered before. This observation holds true for other high-convergence order methods such as Traub's and Jarratt's method. Other such methods can be found in [1][2][3][4][5][6][7][8] and the references therein. Therefore, these results cannot assure convergence. However, these methods may converge. Other concerns involve the absence of error estimates or uniqueness results that can be computed. This is our motivation for presenting a convergence analysis based on the first derivative used in KM. The generality of the technique allows its usage in other methods mentioned previously. This can be a fruitful direction of future research.