\(^\ast \)Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA, e-mail: iargyros@cameron.edu.

\(^{\ast \ast }\)Department of Mathematical and Computational Sciences, NIT Karnataka, India-575 025, e-mail: sgeorge@nitk.ac.in.

1 Introduction

Let \(F:D\subseteq S\rightarrow S\) is a nonlinear function, \(D\) is a convex subset of \(S\) and \(S\) is \(\mathbb {R}\) or \(\mathbb {C}.\) Consider the problem of approximating a locally unique solution \(x^\ast \) of equation

Newton-like methods are famous for finding solution of (1), these methods are usually studied based on: semi-local and local convergence [ 3 , 4 , 20 , 21 , 22 , 24 , 26 ] .

Third order methods such as Euler’s, Halley’s, super Halley’s, Chebyshev’s [ 1 ] – [ 28 ] require the evaluation of the second derivative \(F''\) at each step, which in general is very expensive. That is why many authors have used higher order multipoint methods [ 1 ] – [ 28 ] . In this paper, we study the local convergence of fourth order method defined for each \(n=0,1,2,\ldots \) by

where \(x_0\) is an initial point. Method (2) was studied by Cordero et.al. in [ 13 ] . In particular the fourth order of convergence was shown under hypotheses reaching up to the fourth derivative of function \(F.\) Notice that method (2) involves three functional evaluations. Therefore the efficiencyindex \(EI=p^{\frac{1}{m}}\) where \(p\) is theorder of convergence and \(m\) is the number of functional evaluations per step gives \(EI=4^{\frac{1}{3}}=1.5874.\) Kung and Traub conjecture [ 28 ] that the order of convergence of any multipoint method without memory cannot exceed the bound \(2^{m-1}\) (called the optimal order). Thus, the optimal order for a method with three function evaluations per step should be four.

Other single and multi-point methods can be found in [ 2 , 3 , 20 , 25 ] and the references therein. The local convergence of the preceding methods has been shown under hypotheses up to the fourth derivative (or even higher). These hypotheses restrict the applicability of these methods. As a motivational example, let us define function \(f\) on \(D=[-\frac{1}{2},\frac{5}{2}]\) by

Choose \(x^\ast =1.\) We have that

Then, obviously, function \(f'''\) is unbounded on \(D.\) In the present paper we only use hypotheses on the first Fréchet derivative. This way we expand the applicability of method (2).

The rest of the paper is organized as follows: Section 2 contains the local convergence analysis of methods (2). The numerical examples are presented in the concluding Section 3.

2 Local convergence analysis

We present the local convergence analysis of method (2) in this section. Let \(L_0 {\gt}0, L {\gt}0\) and \(M\geq 1.\) It is convenient for the local convergence analysis of method (2) to introduce some functions and parameters. Define functions \(g_1,\, g_2,\, h_2,\, g_3,\, h_3\) on the interval \([0, \tfrac {1}{L_0})\) by

and parameter

We have that \(h_2(0)=-1{\lt}0\) and \(h_2(r_A)=\tfrac {M}{1-L_0r_A} {\gt} 0,\) since \(\tfrac {Lr_A}{2(1-L_0r_A)} =1\) and \(1-L_0r_A {\gt} 0.\) It then follows from the intermediate value theorem that function \(h_2\) has zeros in the interval \((0, r_A).\) Denote by \(r_2\) the smallest such zero. Similarly, we have that \(h_3(0)=-1 {\lt} 0\) and \(h_3(r_2) {\gt} 0.\) Denote by \(r\) the smallest zero of function \(h_3\) in the interval \((0, r_2).\) Then, for each \(t\in [0, r)\) we have

and

Denote by \(U(v,\rho ), \bar{U}(v,\rho )\) the open and closed balls in \(S\), respectively, with center \(v\in S\) and of radius \(\rho {\gt}0.\) Next, we show the following local convergence result for method (2) using the preceding notation.

It follows from (13) and the Banach Lemma on invertible functions [ 3 , 4 , 19 , 20 , 22 , 23 ] that \(F'(x_0)\neq 0\) and

Hence, \(y_0\) and \(x_0\) are well defined. We also get from (2), (2), (7) and (14) that

which shows (10) for \(n=0\) and \(y_0\in U(x^\ast , r).\) We can write

Then, by (8), (17) we obtain that

where we also used \(|x^\ast +\theta (x_0-x^\ast )-x^\ast |=\theta |x_0-x^\ast | {\lt} r.\) That is \(x^\ast +\theta (x_0-x^\ast )-x^\ast \in U(x^\ast , r).\) We also have that

Then, by the second substep of method (2) for \(n=0,\) (3), (5) and (19) we get that

which shows (11) for \(n=0\) and \(z_0\in U(x^\ast , r).\) Next, we show that \(F(x_0) \neq 0.\) Using (6), we get that

it follows from (21) that \(F(x_0)\neq 0\) and

Hence \(x_1\) is well defined. Then, using the last substep of method (2) for \(n=0,\) (4), (14), (18), (19), (20) and (22) we get in turn that

which shows (12) for \(n=0\) and \(x_1\in U(x^\ast , r).\) By simply replacing \(x_0, y_0, z_0,\, x_1\) by \(x_k, y_k, z_k,\, x_{k+1}\) in the preceding estimates we arrive at estimates (10)–(12). Using the estimate \(|x_{k+1}-x^\ast | {\lt} |x_k-x^\ast | {\lt} r,\) we deduce that \(x_{k+1}\in U(x^\ast , r)\) and \(\lim _{k\rightarrow \infty }x_k=x^\ast .\) To show the uniqueness part, let \(Q=\int _0^1F'(y^\ast +\theta (x^\ast -y^\ast )d\theta \) for some \(y^\ast \in \bar{U}(x^\ast , T)\) with \(F(y^\ast )=0.\) Using (6) we get that

It follows from (23) and the Banach Lemma on invertible functions that \(Q\) is invertible. Finally, from the identity \(0=F(x^\ast )-F(y^\ast )=Q(x^\ast -y^\ast ),\) we deduce that \(x^\ast =y^\ast .\)

3 Numerical Examples

We present numerical examples in this section.