A Comparison Study of the Classical and Modern Results of Semi-Local Convergence of Newton-Kantorovich Iterations

: There are a plethora of semi-local convergence results for Newton’s method (NM). These results rely on the Newton–Kantorovich criterion. However, this condition may not be satisﬁed even in the case of scalar equations. For this reason, we ﬁrst present a comparative study of established classical and modern results. Moreover, using recurrent functions and at least as small constants or majorant functions, a ﬁner convergence analysis for NM can be provided. The new constants and functions are specializations of earlier ones; hence, no new conditions are required to show convergence of NM. The technique is useful on other iterative methods as well. Numerical examples complement the theoretical results.


Introduction
The concern of this study is to solve the following nonlinear equation using the celebrated Newton's method (NM) in the following form Here, F : Ω ⊂ B 1 −→ B 2 is differentiable according to Fréchet and operates between Banach spaces B 1 and B 2 , whereas subset Ω = ∅. Kantorovich provided the semi-local convergence analysis of NM utilizing Banach's contraction mapping principle. In particular, he presented two different proofs using majorant functions or recurrent relations [1]. The Newton-Kantorovich Theorem is that no assumption as to the solution is made, while at the same time the existence of the solution x * is established. Numerous researchers have used this theorem, both in applications and as a theoretical tool [2][3][4][5][6][7][8][9][10]. While the convergence criteria may not hold, NM may converge.
Assume that there exist constants η ≥ 0, L 1 > 0, and a point z ∈ Ω such that the Lipschitz condition and hold for each u, v ∈ Ω. The main convergence criterion is provided by There are even simple scalar equations where criterion (5) is not satisfied. Indeed, let Ω = U[z, 1 − q], q ∈ (0, 1 2 ), F(t) = t 3 − q, and set z = x 0 = 1. Then, conditions (3) and (4) are satisfied provided that L 1 = 2(2 − q) and η = 1−q 3 . However, then condition (5) is not satisfied for any q ∈ (0, 1 2 ), as 2L 1 η > 1. Hence, there is a need to replace (5) with a weaker criterion without adding conditions. The same extension is useful in the Hölder case or when L 1 is replaced by a majorant function. The last two cases are not considered in [11]. Moreover, the Lipschitz case is extended further than in [11] (see the end of Section 4). This is the motivation for presenting new results that both extend the convergence region and provide more precise error estimates and better knowledge as to the location of the solution. In this way, the use of NM can be extended. The novelty of the present article is that these benefits require no additional conditions. The same technique used here can be applied to extend other iterative methods along the same lines.
Majorizing scalar sequences for NM are provided in Section 2. The semi-local convergence of NM appears in Section 3. A discussions is provided in Section 4, Examples can be found in Section 5, and Conclusions in Section 6.

Majorizing Sequences
Real functions are utilized to develop sequences that are shown to be majorizing for NM in Section 3. Let S = [0, ∞). Assume: There exist functionsψ τ : S −→ S, ψ 1 : S −→ S non-decreasing and continuous such that function 1] continuous and non-decreasing.
Let η ≥ 0 be a given constant. Define scalar sequence {t n } for t 0 = 0, t 1 = η by Next, results appear on the convergence of sequence {t n }. and Then, sequence {t n } is such that t n ≤ t n+1 , n = 0, 1, 2, . . . and lim n−→∞ t n = t * , where t * is the unique least upper bound of sequence {t n } satisfying t * ∈ [0, b 0 ].
(ii), although the conditions of Lemma 1 can be verified by constructing sequence {t n } in advance. Next, convergence conditions which are stronger than (7)-(9) are provided, which are easier to verify. Let K 0 , K, L 0 , L 1 be non-negative constants and p ∈ [0, 1]. Define functions and ψ τ (t) = L(θt) p .
Proof. Induction is utilized to show Inequality (18) holds if m = 1 per the definition of δ 0 and δ. Then, it follows per (11) that Assume estimates (19) and (20) hold for all values of m ≤ n. It follows by the induction hypotheses that and Hence, per estimates (21) and (22), assertion (18) holds if The recurrent functions are related, as (25)

Case of Condition (H1): Define function
Per estimates (23) and (26), it follows that and which is true per condition (16). The induction under (H1) is terminated.

Case of Condition (H2): It follows that estimate (24) holds, as
That is, the induction is completed under condition (H2) as well. Hence, under either condition (H1) or (H2), sequence {t m } is non-decreasing and bounded from above. Hence, it converges to t * .
It follows that f p (0) − L 1+p < 0 and f p (1) = L 0 > 0. Hence, the intermediate value theorem ensures that the equation f p (t) = 0 has zeros in (0, 1). Denote by γ p the smallest such zero. Notice that The parameter δ is preferred in closed form. This is possible in certain cases. Lipschitz Case: Set p = 1. Then, the choice for δ = γ 1 .
In the case that p ∈ (0, 1), the zero γ p is not in closed form; instead (33) Indeed, because γ p solves equation f p (t) = 0, then By solving the left hand side inequality for η, The right hand side inequality in condition (34) is solved as follows: To consider the rest of the cases, define function ϕ : S −→ S by .

Convergence of NM
The notation U(w, ρ), U[w, ρ] refers to the open and closed balls with radius ρ > 0 and center w ∈ X, respectively. The semi-local convergence of NM is presented under conditions (A). The functions ψ are provided in Section 2. Assume: Next, the main semi-local convergence result for NM is developed. Proof. Estimates and ∀i = 0, 1, 2, . . . shall be shown by induction.
Next, we present a uniqueness of the solution result . (ii) There exists r ≥ r 0 such that Let Ω 1 = U[x 0 , r] ∩ Ω. Then, the only solution of equation F(x) = 0 is x * .

Discussion and Conclusions
There are a plethora of results on NM. The sufficient convergence criteria in these studies are weakened by the new technique without adding conditions (see the numerical Section). Notice that with the developed technique there is an at least as tight subset Ω 0 of Ω containing the iterates. Therefore, the Lipschitz-Hölder majorant functions are at least as tight as the ones provided in earlier studies, leading to the advantages stated in the introduction. Below, we provide several examples.
and Q(0) = 0, Q has an at least as small zero r Q . Hence, the new convergence criterion is at least as weak. This is the contribution in the case of majorant functions. In particular, in the case where ω and ψ 1 are constant functions, such as which is weaker, as L ≤ L 1 and L 1 is the Lipschitz constant on Ω.
(a) Lipschitz Case: Convergence criteria already in the literature are listed below in the left column, and the new criteria in the right column. [11,14],T 3 : η ≤ 1 where and It follows from these definitions that Consequently, although not necessarily vice versa unless L 0 = L = L 1 . Parameter L 1 is the Lipschitz constant on Ω used by Kantorovich and others [1,7]. Another comparison between convergence criteria can be made as follows. Notice that Similarly, These limits show by how many times at most the applicability of NM can be extended. Moreover, under the new approach this applicability is extended even further. Indeed, notice that Similarly, Clearly, the new convergence criterion can be arbitrarily many times weaker than the previous ones. This technique of replacing constants with smaller ones leads to both weaker criteria and more precise majorizing sequences. Indeed, consider again iterations {t n } and {t n } with the choice of functions ω and ψ 1 obtained by (62) and (63), respectively. It follows that 0 ≤ t n ≤t n , then 0 ≤ t n+1 − t n ≤t n+1 −t n and s 1 = lim n→∞ t n ≤ lim n→∞t n = s 2 , The constant r 2 was reported by Kantorovich. Hence, tighter error estimates on x n+1 − x n and x * − x n become possible if L < L 1 . Moreover, the information on the location of the solution is more accurate if s 1 < s 2 , which is possible for L < L 1 . Concerning the uniqueness of the solution balls, u(x 0 , r 3 ) ⊂ u(x 0 , r 4 ) where Thus, the uniqueness of the solution ball is extended. The constant r 3 is provided by Kantorovich. Furthermore, if we choose ψ 1 (t) = L 0 t and assume η ≤ 1 2L 0 , set . Then, r 4 < r 5 if L 0 < L, and the solution x * is unique in the ball u(x 0 , r 5 ) (c) It turns out that the results can be weakened even more if Lipschitz or Hölder constants are replaced by constants at least as small in two different ways, namely: (i) Noting that convergence conditions should not hold for all x, y ∈ Ω 0 , only for x ∈ Ω 0 and y = x − F (x) −1 F(x), the proofs of the results proceed in this weaker setting. However, the corresponding constant is at least as tight. Denote such a constant per L 2 ; it follows that L 2 ≤ L.
(ii) Consider ball Ω 2 = U(x 1 , b 0 − η) ⊂ Ω for b 0 > η and have Ω 2 replace Ω 0 in condition (A3). Then, as Ω 2 ⊂ Ω 0 ,, the constant is again at least as small. Denote such a constant per L 3 . It follows that L 3 ≤ L. The rest of the convergence results found in the literature can be immediately weakened under this technique as well.
Finally, it is worth noticing that the new constants L 0 , L, L 2 , L 3 are special cases of L 1 . Hence, the benefits under the new technique are obtained without additional computational effort or conditions, which represents one contribution of this paper.

Numerical Experiments
In the first example, the new constants are smaller than those of earlier studies.

Example 1. Define scalar function
for t 0 = 0 where λ j and j = 0, 1, 2, 3 are real parameters. It follows by this definition that for λ 3 sufficiently large and for λ 2 sufficiently small L 0 L 1 can be arbitrarily small. In particular, L 0 L 1 −→ 0.
In the last two examples, Kantorovich criterion (66) is not satisfied; however, the new criterion (66) holds. Example 2. Let B 1 = B 2 = R, Ω = [q, 2 − q] for q ∈ M = (0, 0.5), and t 0 = 1. Define real function F on Ω as Then, the parameters are η = 1 3 (1 − q), K 0 = K = L 0 = 3 − q, and L 1 = 2(2 − q). Moreover, one obtains Similarly, when solving for q in new criteria (67)-(69), the intervals S 2 , S 3 , S 4 are extended. This is due to the fact that L < L 1 for all q in M. Hence, the range of values q is extended in the new approach. Notice in particular that the Newton-Kantorovich criterion (66) is not satisfied for any q ∈ (0, 0.5).

Conclusions
A comparison study between results on the semi-local convergence of NM is provided. The technique uses recurrent functions. In this way, the new sufficient convergence criteria are weaker in the Lipschitz, Hölder, and more general cases, as they rely on smaller constants. Other benefits include more precise error bounds and uniqueness of the solution results. The new constants are special cases of earlier ones. The technique is very general, rendering it useful to extend the usage of other iterative methods.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.