Extending the Applicability of Stirling’s Method

: Stirling’s method is considered as an alternative to Newton’s method when the latter fails to converge to a solution of a nonlinear equation. Both methods converge quadratically under similar convergence criteria and require the same computational effort. However, Stirling’s method has shortcomings too. In particular, contractive conditions are assumed to show convergence. However, these conditions limit its applicability. The novelty of our paper lies in the fact that our convergence criteria do not require contractive conditions. Hence, we extend its applicability of Stirling’s method. Numerical examples illustrate our new ﬁndings.


Introduction
In this work we deal with finding a fixed point x * of the equation where F is a Fréchet-differentiable operator defined on a convex subset D of a Banach space X with values into itself. By I we denote the identity linear operator in L(X, X). The symbol L(X, X) stands for the space of bounded linear operators from X into X. Many applications from different areas, including education, reduce to dealing with Equation (1) utilizing mathematical modelling . However, the solution x * is found in closed form only in rare cases. This problem leads to the usage of methods that are iterative in nature.
We study Stirling's method given for all n = 0, 1, 2, . . . by where x 0 ∈ D. Further we will introduce an operator Γ(x) ∈ L(X, X) such that Γ(x) = (I − F (F(x))) −1 with x ∈ D, and denote Γ 0 = Γ(x 0 ) for use in later Sections. This method converges quadratically as Newton's method does, and also requires the same computational effort (see details in [1,22]). It is considered to be a useful alternative in cases where Newton's method fails to converge (see such examples in [22]). However, the usage of Stirling's method has a drawback, since the convergence criteria require contractions. We have detected some other problems listed in Remarks 3 and 4. These drawbacks limit the applicability of Stirling's method.
In order to extend its applicability, we do not use contractive conditions in our semi-local as well as the local convergence results.
The rest of the work is structured as follows. Section 2 includes the semi-local convergence analysis. Section 3 contains the local analysis. The numerical results are given in Section 4.

Semi-Local Convergence Analysis
Let L 0 > 0, L > 0 and γ ≥ 0. Consider a real sequence {t n } as Next, we study the convergence of sequence {t n } by developing relevant lemmas and theorems. where Then, sequence {t n } generated for t 0 = 0 by (4) is increasing, converges to its unique least upper bound t * , so that where Proof. It is convenient to first simplify sequence {t n }. Define sequence {α n } by α n = 1 − L 0 t n . Then, by (4) we can write α 0 = 1, We have by (4) that δθ 1 < 1 and 0 < θ 2 < θ 1 . Suppose that 0 < θ k < θ k−1 and δθ k < 1. Then, we get in turn that and δθ k+1 < δθ k < 1.
In what follows the set denoted by U(x, a) is a ball with center x ∈ X and of radius a > 0. To simplify, the notation, by || || in this work, we denote the operator norm or the norm on the Banach space. The semi-local convergence analysis is based on the conditions (C): Hypotheses of Lemmas 1 and 2 hold with We suppose from now on that the conditions (C) hold. Next, the semi-local convergence result is given for Stirling's method (2).
That is, operator F must be a contraction on D. Moreover, the convergence of Stirling's method was shown in [22] under (C 2 ), D 0 = D and a ∈ (0, 1 3 ]. However, in the present study no such assumption is made. Hence, the applicability of Stirling's method (2) is extended. Notice also that we can have a 0 ≤ a, b 0 ≤ b and c can be chosen as b = cb 1 . (b) Estimate (4) is similar to the sufficient convergence Kantorovich-type criteria for the semi-local convergence of Newton's method given by us in [4]. However, the constantsb 0 andb are the center-Lipschitz and Lipschitz constants for operator F (see also part (e)). (c) If set D 0 is switched by D 1 = D ∩ U(x 1 , r 0 − ||x 0 − F(x 0 )||), since D 1 ⊆ D and the iterates remain in D 1 the results can be improved even further. The corresponding constants to b and b 1 will be at least as small. (d) In view of the proof of Theorem 1, scalar sequence {s n } defined by is also a majorizing sequence for Stirling's method (2), where s n+1 − s n ≤ t n+1 − t n and s * = lim n→∞ s n ≤ t * .
Consider, itemsc,γ,L 0 ,L,L 1 ,Γ 0 ,b 0 ,b,b 1 ,r 0 ,D 0 andh, corresponding to c, γ, L 0 , L, L 1 , Γ 0 ,b 0 , b, b 1 , r 0 , D 0 and h respectively as ||I − F (x 0 )|| ≤c, The scalar sequencet n is defined as Then, Stirling's method sufficient convergence criteria, error bounds and information on the uniqueness of the solution are better than Newton's method when the "bar" constants and sets are smaller than the non bar constants. Similar favorable comparison can be made in the local convergence case that follows.

Local Convergence
The conditions (H) are used in the local convergence analysis of Stirling's method (2): F : D ⊂ X → X is a Fréchet differentiable operator, and there exists x * ∈ D such that Γ * = (I − F (x * )) −1 ∈ L(X, X) and F(x * ) = x * .
Proof. We shall show using mathematical induction that sequence {x n } is well defined, remains in U(x * , R) and converges to x * so that (16) is satisfied. We have by (H 1 ) and (H 2 ) for x ∈ U(x * , R) that Hence, Γ(x) ∈ L(X, X) and In particular, (18) holds for x = x 0 , which shows that x 1 is well defined by Stirling's method for n = 0. We can write by (H 1 ) that We get in turn by (H 2 ) and (H 3 ) Then, by (18)- (20), we get that also so (16) holds for n = 0 and x 1 ∈ U(x * , R). Switch x 0 by x k in the preceding estimates, we arrive at (16). In view of the estimate ||x k+1 − x * || < ||x k − x * || < R, we conclude that lim k→∞ x k = x * and x k+1 ∈ U(x * , R). Let x 0 = x * in (15) to show the uniqueness part.

Remark 2.
The local results in the literature use (C 2 )' and D * 0 = D. But (H 2 ) is weaker than (C 2 )'. Hence, we extend the applicability of Stirling's method (2) in the local case too.

Numerical Example with Concluding Remarks
In the next example, we compare Stirling's method with Newton's method.
Clearly, the quadratic polynomial joins smoothly with the linear parts.
(I) Semilocal case (i). If we choose x 0 = 3, we see that x 1 = y 1 = x * = 0. Moreover, the semi-local convergence criteria of Theorem 1 are satisfied (with γ = 0, a 0 = 1 3 and c = 4 3 ). (II) Local convergence criteria of Theorem 2 (with µ = 1 3 , since the derivative of the quadratic polynomial satisfies 1 3 |2x − 7| ≤ 1 3 for all x ∈ [3,4]). (III) In Tables 1 and 2 we present some cases in which Stirling's method stands better than Newton's one. In the current study, we have successfully demonstrated our claims on Stirling's method by focusing on very classic problems, but in the future we will consider studying other complex problems such us solving symmetric ordinary differential equations with a more favorable theory.
Author Contributions: All authors have equally contributed to this work. All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest:
The authors have no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.