Improved Convergence Ball and Error Analysis of Müller’s Method

We present an improved convergence analysis of Müller’s method for solving nonlinear equation under conditions that the divided differences of order one of the involved function satisfy the Lipschitz conditions. Our result improves the earlier work in literature. Numerical examples are presented to illustrate the theoretical results.


Introduction
In this study we are concerned with the convergence analysis of Müller's method which is used to solve the following equation Many problems in Computational Sciences and other disciplines can be brought in a form like (1.1) using mathematical modelling [1]. The solutions of these equations can be rarely be found in closed form. That is why most solution methods for these equations are usually iterative.
The study about convergence of iterative procedures is normally centered on two types: semi-local and local convergence analysis. The semi-local convergence matter is, based on the information around an initial point, to give conditions ensuring the convergence of the iterative procedure. While the local analysis is based on the information around a solution, to find estimates of the radii of convergence balls.
The famous Müller's method is defined in [2] by where, and f [·, ·], f [·, ·, ·] are divided differences of order one and two, respectively (see [3]). The sign in the denominator of (1.2) is chosen so as to give the larger value.
Müller's method is widely used [2,4]. It is a free-deravative method and has a convergence order 1.839 . . . under reasonable conditions [2]. Xie [5] established a semilocal convergence theorem of the method under bounded third and fourth derivatives. Bi et.al [6] presented a new semilocal convergence theorem of the method under γ-condition. Wu et.al [7] gave the convergence ball and error analysis of the method under the hypotheses that the second-order and third-order derivative of function f are bounded. In this paper, we provide a new estimate on the radius of convergence ball of Müller's method under the weaker conditions than the coresponding conditions in [7]. In fact, we assume that f is differentiable in D, f ′ (x ⋆ ) = 0, and the following Lipschitz conditions are true: where, K > 0 and K ⋆ > 0 are constants, and x ⋆ ∈ D is a solution of (1.1).
The paper is organized as follows: Section 2 contains the convergence ball analysis of method (1.2). The numerical examples including favorable comparisons with earlier study [7] are presented in the concluding Section 3.

Improved convergence ball analysis of method (1.2)
We present the local convergence of method (1.2) in this section. Denote U (x, r) as an open ball around x with radius r. We have: Then, the sequence {x n } generated by Müller's method (1.2) starting from any three distinct points is well defined, and converges to x ⋆ . Moreover, the following estimates hold: where, and {F n } is Fibonacci sequence, and is defined by F −1 = 1, F 0 = 1 and F n+1 = F n + F n−1 for any n = 0, 1, 2, . . .
Following now an inductive procedure on n = 0, 1, 2, . . ., we have x n ∈ U (x ⋆ , R ′ ), and It is easy to see that the following ralation holds:

Remark 2.2 (a)
It is easy to see that the condition (1.4) used in Theorem 2.1 is weaker than bounded conditions of the second-order and third-order derivative of function f used in [7]. In fact, suppose that f is twice differentiable on D, and then, for any x, y, u, v ∈ D, we have (b) Note that a semilocal convergence theorem for Müller's method is given in [6] under the γcondition of order two which is a condition of third-order derivative of function f , see [6]. However, we don't use any information of f ′′′ in our theorem.

Remark 2.3
Notice that from (1.4) and (1.5) holds in general and K K⋆ can be arbitrarily large [1]. Hence, the condition K ⋆ ≤ K in Theorem 2.1 involves no loss of generality. If we use only condition (1.4) in Theorem 2.1, we can establish a similar theorem by replacng (2.1) and (2.3) by

Numerical examples
In this section, we present some examples.
Then, we have 3) and Then, we have It is obvious that x ⋆ = 0, f ′ (x ⋆ ) = 1 and for any x, y, u, v ∈ D, we have