Abstract
Abstract: We present a local convergence analysis of an eighth order three step method in order to approximate a locally unique solution of nonlinear equation in a Banach space setting. In an earlier study by Sharma and Arora (2015), the order of convergence was shown using Taylor series expansions and hypotheses up to the fourth order derivative or even higher of the function involved which restrict the applicability of the proposed scheme. However, only first order derivative appears in the proposed scheme. In order to overcome this problem, we proposed the hypotheses up to only the first order derivative. In this way, we not only expand the applicability of the methods but also propose convergence domain. Finally, where earlier studies cannot be applied, a variety of concrete numerical examples are proposed to obtain the solutions of nonlinear equations. Our study does not exhibit this type of problem/restriction.
Introduction
Numerical analysis is a wide-ranging discipline having close connections with mathematics, computer science, engineering and the applied sciences. One of the most basic and earliest problem of numerical analysis concerns with finding efficiently and accurately the approximate locally unique solution x * of the equation of the form F (x) = 0,
Introduction
where F is a Fréchet differentiable operator defined on a convex subset D of X with values in Y , where X and Y are the Banach spaces. Analytical methods for solving such equations are almost non-existent for obtaining the exact numerical values of the required roots. Therefore, it is only possible to obtain approximate solutions and one has to be satisfied with approximate solutions up to any specified degree of accuracy, by relying on numerical methods which are based on iterative procedures. Therefore, researchers worldwide resort to an iterative method and they have proposed a plethora of iterative methods [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] . While, using these iterative methods researchers face the problems of slow convergence, non-convergence, divergence, inefficiency or failure (for detail please see Traub [15] and Petkovic et al. [13] ).
Introduction
The convergence analysis of iterative methods is usually divided into two categories: semi-local and local convergence analysis. The semi-local convergence matter is, based on the information around an initial point, to give criteria ensuring the convergence of iteration procedures. On the other hand, the local convergence is based on the information around a solution, to find estimates of the radii of convergence balls. A very important problem in the study of iterative procedures is the convergence domain. Therefore, it is very important to propose the radius of convergence of the iterative methods.
Introduction
We study the local convergence analysis of three step method defined for each n = 0, 1, 2, . . . by
Introduction
where
Introduction
is any two-point optimal fourth-order scheme. The eighth order of convergence of Scheme (2) was shown in [1] when
Introduction
. That is when [·, ·; F ] is a divided difference of first order of operator F [5, 6] . The local convergence was shown using Taylor series expansions and hypotheses reaching up to the fifth order derivative. The hypotheses on the derivatives of F and H limit the applicability of Scheme (2) . As a motivational example, define function
Introduction
Then, we have that
Introduction
One can easily find that the function F (x) is unbounded on D at the point x = 0. Hence, the results in [1] , cannot apply to show the convergence of Scheme (2) or its special cases requiring hypotheses on the fifth derivative of function F or higher. Notice that, in particular, there is a plethora of iterative methods for approximating solutions of nonlinear equations [1] [2] [3] [4] [5] [6] [7] [8] [10] [11] [12] [13] [14] [15] [16] . These results show that initial guess should be close to the required root for the convergence of the corresponding methods. However, how close an initial guess would be required for the convergence of the corresponding method? These local results give no information on the radius of the ball convergence for the corresponding method. The same technique can be applied to other methods.
Introduction
In the present study we expand the applicability of Scheme (2) using only hypotheses on the first order derivative of function F . We also propose the computable radii of convergence and error bounds based on the Lipschitz constants. We further present the range of initial guess x * that tells us how close the initial guess would be required for granted convergence of the Scheme (2). This problem was not addressed in [1] . The advantages of our approach are similar to the ones already mentioned for Scheme (2) . The rest of the paper is organized as follows: in Section 2, we present the local convergence analysis of Scheme (2) . Section 3 is devoted to the numerical examples which demonstrate our theoretical results. Finally, the conclusion is given in the Section 4.
Local Convergence: One Dimensional Case
In this section, we define some scalar functions and parameters to study the local convergence of Scheme (2) . 
Local Convergence: One Dimensional Case
Suppose that
Local Convergence: One Dimensional Case
Then, we have h 2 (0) = −1 < 0. By Equation (3) and the intermediate value theorem, function h 2 has zeros in the interval (0, l). Further, let r 2 be the smallest such zero. Moreover, define functions g 1 , p and h p in the interval 0,
Local Convergence: One Dimensional Case
and parameter r 1 by
Local Convergence: One Dimensional Case
We have g 1 (r 1 ) = 1 and for each t ∈ [0, r 1 ) : 0 ≤ g 1 (t) < 1. We also get h p (0) = −1 and
Local Convergence: One Dimensional Case
. Denote by r p the smallest zero of function h p on the interval 0,
Local Convergence: One Dimensional Case
Furthermore, define functions q and h q on the interval 0,
Local Convergence: One Dimensional Case
Using h q (0) = −1 < 0 and Equation (3), we deduce that function h q has a smallest zero denoted by r q .
Local Convergence: One Dimensional Case
Finally define functions g 3 and h 3 on the interval [0, min{r p , r q }) by
Local Convergence: One Dimensional Case
Then, we get h 3 (0) = −1 and h 3 (t) → +∞ as t → min{r p , r q }. Denote by r 3 the smallest zero of function h 3 on the interval (0, min{r p , r q }). Define
Local Convergence: One Dimensional Case
Then, we have that
Local Convergence: One Dimensional Case
and
Local Convergence: One Dimensional Case
and
Local Convergence: One Dimensional Case
U (γ, s) andŪ (γ, s) stand, respectively for the open and closed balls in X with center γ ∈ X and radius s > 0.
Local Convergence: One Dimensional Case
Next, we present the local convergence analysis of Scheme (2) using the preceding notations.
Local Convergence: One Dimensional Case
and
Local Convergence: One Dimensional Case
Moreover, suppose that there exist
Local Convergence: One Dimensional Case
where the radius of convergence r is defined by Equation (4) and
Local Convergence: One Dimensional Case
Then, the sequence {x n } generated by Scheme (2) for x 0 ∈ U (x * , r) − {x * } is well defined, remains in U (x * , r) for each n = 0, 1, 2, . . . and converges to x * . Moreover, the following estimates hold
Local Convergence: One Dimensional Case
and
Local Convergence: One Dimensional Case
where the "g" functions are defined by previously. Furthermore, for T ∈ r,
Local Convergence: One Dimensional Case
Proof. We shall show estimates Equations (19)-(21) hold with the help of mathematical induction. By hypotheses x 0 ∈ U (x * , r) − {x * }, Equations (5) and (13), we get that
Local Convergence: One Dimensional Case
It follows from Equation (22) and the Banach Lemma on invertible operators [5, 14] that
Local Convergence: One Dimensional Case
, y 0 is well defined and
Local Convergence: One Dimensional Case
Using the first sub step of Scheme (2) for n = 0, Equations (4), (6), (11) and (23), we get in turn
Local Convergence: One Dimensional Case
which shows Equation (18) for n = 0 and y 0 ∈ U (x * , r). Then, from Equations (3) and (12), we see that Equation (20) follows. Hence, z 0 ∈ U (x * , r). Next, we shall show that
Local Convergence: One Dimensional Case
Using Equations (4), (5), (7), (13), (14) and (24), we get in turn that
Local Convergence: One Dimensional Case
It follows from Equation (25) that
Local Convergence: One Dimensional Case
Similarly, but using Equation (8) instead of Equation (7), we obtain in turn that
Local Convergence: One Dimensional Case
That is
Local Convergence: One Dimensional Case
Hence, x 1 is well defined by the third sub step of Scheme (2) for n = 0. We can write by Equation (11)
Local Convergence: One Dimensional Case
Notice that x * +θ(x 0 −x * )−x * = θ x 0 −x * < r. Hence, we have that x * +θ(x 0 −x * ) ∈ U (x * , r).
Local Convergence: One Dimensional Case
Then, by Equations (17) and (29) we get that
Local Convergence: One Dimensional Case
We also have that by replacing x 0 by z 0 in Equation (30) that
Local Convergence: One Dimensional Case
since z 0 ∈ U (x * , r).
Local Convergence: One Dimensional Case
Then, using the last substep of Scheme (2) for n = 0, Equations (4), (10), (15), (20) (for n = 0), (26), (28), and (31) that
Local Convergence: One Dimensional Case
which shows Equation (21) for n = 0 and x 1 ∈ U (x * , r). By simply replacing x 0 , y 0 , z 0 by x m , y m , z m in the preceding estimates we arrive at Equations (19)-(21). Then, from the estimates x m+1 − x * < x m − x * < r, we conclude that lim m→∞ x k = x * and x m+1 ∈ U (x * , r). Finally, to show the uniqueness part, let y * ∈Ū (x * , T ) be such that F (y
Local Convergence: One Dimensional Case
Equation (14), we get that
Local Convergence: One Dimensional Case
, we conclude that
Local Convergence: One Dimensional Case
(a) In view of Equation (11) and the estimate
Local Convergence: One Dimensional Case
condition Equation (13) can be dropped and M can be replaced by
Local Convergence: One Dimensional Case
).
Local Convergence: One Dimensional Case
(b) The results obtained here can be used for operators F satisfying the autonomous differential equation [5, 6] of the form
Local Convergence: One Dimensional Case
where P is a known continuous operator. Since F (x * ) = P (F (x * )) = P (0), we can apply the results without actually knowing the solution x * . Let as an example F (x) = e x + 2. Then, we can
Local Convergence: One Dimensional Case
(c) The radius r 1 was shown in [5, 6] to be the convergence radius for Newton's method under conditions Equations (11) and (12) . It follows from Equation (4) and the definition of r 1 that the convergence radius r of the Scheme (2) cannot be larger than the convergence radius r 1 of the second order Newton's method. As already noted, r 1 is at least the size of the convergence ball given by Rheinboldt [14] r R = 2 3L .
Local Convergence: One Dimensional Case
In particular, for L 0 < L, we have that r R < r 1 and
Local Convergence: One Dimensional Case
That is our convergence ball r 1 is at most three times larger than Rheinboldt's. The same value for r R given by Traub [15] .
Local Convergence: One Dimensional Case
(d) We shall show that how to define function g 2 and l appearing in condition Equation (3) for the method
Local Convergence: One Dimensional Case
Clearly method (34) is a special case of Scheme (2) . If X = Y = R, then Method (34) reduces to Kung-Traub method [15] . We shall follow the proof of Theorem 1 but first we need to show that [y n , x n ; F ] −1 ∈ L(Y, X). We get that
Local Convergence: One Dimensional Case
As in the case of function p, function h p 0 = p 0 (t) − 1, where p 0 (t) = (K 0 g 1 (t) + K 1 )t has a smallest zero denoted by r p 0 in the interval 0,
Local Convergence: One Dimensional Case
. Set l = r p 0 . Then, we have from the last sub step of Method (34) that
Local Convergence: One Dimensional Case
It follows from Equation (36) that λ = 2 and
Local Convergence: One Dimensional Case
2 . Then, the convergence radius is given by r = min{r 1 , r 2 , r p 0 , r 3 }.
Local Convergence: One Dimensional Case
(37)
Numerical Example and Applications
In this section, we shall check the effectiveness and validity of our theoretical results which we have proposed in Section 2 on the scheme proposed by Sharma and Arora [1] . For this purpose, we shall choose a variety of nonlinear equations which are mentioned in the following examples including motivational example. At this point, we chose the following eighth order methods proposed by Sharma and Arora [1] 
Numerical Example and Applications
and
Numerical Example and Applications
denoted by M 1 , M 2 and M 3 , respectively. The initial guesses x 0 are selected with in the range of convergence domain which gives guarantee for convergence of the iterative methods. Due to the pages limit, all the values of parameters are done for only 5 significant digits and displayed in the Tables 1-3 and examples Equations (1)-(3), although 100 significant digits are available. The considered test examples with corresponding initial guess, radius of convergence and necessary number of iterations (n) for getting the desired accuracy are displayed in Tables 1-3. In addition, we also want to verify the theoretical order of convergence of Methods (38)-(40). Therefore, we calculate the computational order of convergence (COC) [9] approximated by using the following formulas
Numerical Example and Applications
or the approximate computational order of convergence (ACOC) [9] ρ * = ln
Numerical Example and Applications
, for each n = 1, 2, . . .
Numerical Example and Applications
During the current numerical experiments with programming language Mathematica (Version 9), all computations have been done with multiple precision arithmetic, which minimize round-off errors. We use = 10 −200 as a tolerance error. The following stopping criteria are used for computer programs:
Numerical Example and Applications
Further, we use λ = 2 and function g 2 as defined above Equation (37) in all the examples.
Numerical Example and Applications
. We obtain different radius of convergence, COC (ρ) and n in the following Table 1 . Table 1 . Different values of parameters which satisfy Theorem 1. 
Cases
we have that
Cases
Then, for x * = 0, we obtain that L 0 = 7.5, L = 15, M = K = 2 and
Cases
. We obtain different radius of convergence in the following Table 2 . (80 + 16π + (11 + 12 log 2)π
Cases
and our required zero is x * = 1 π
Cases
. We obtain different radius of convergence, COC (ρ) and n in the following Table 3 . 
Conclusions
Most of the time, researchers mentioned that the initial guess should be close to the required root for the granted convergence of their proposed schemes for solving nonlinear equations. However, how close an initial guess would be required to grantee the convergence of the proposed method? We propose the computable radius of convergence and error bound by using Lipschitz conditions in this paper. Further, we also reduce the hypotheses from fourth order derivative of the involved function to only first order derivative. It is worth noticing that Scheme (2) is not changing if we use the conditions of Theorem 1 instead of the stronger conditions proposed by Sharma and Arora (2015) . Moreover, to obtain the error bounds in practice and order of convergence, we can use the computational order of convergence which is defined in numerical Section 3. Therefore,we obtain in practice the order of convergence in a way that avoids the bounds involving estimates higher than the first Fréchet derivative. Finally, on account of the results obtained in Section 3, it can be concluded that the proposed study not only expands the applicability but also gives the computable radius of convergence and error bound of the scheme given by Sharma and Arora (2015) for obtaining simple roots of nonlinear equations.

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