A New Family of Fourth-Order Optimal Iterative Schemes and Remark on Kung and Traub’s Conjecture

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Introduction
e most basic problem in engineering and scientific applications is to find the root of a given nonlinear equation where f ∈ C(I, R) and I ⊂ R is an interval we are interested in, and we suppose that r ∈ I is a simple solution with f(r) � 0 and f ′ (r) ≠ 0. e famous Newton method (NM) for iteratively solving equation (1) is given by which is quadratically convergent. Due to its simplicity and rapid convergence, the Newton method is still the first choice to solve equation (1). An extension of the NM to a third-order iterative scheme was made by Halley [1]: For the engineering design of the vibrating modes of an elastic system, sometimes we may need to know the eigenvalues of a large-size square matrix, which results in a highly nonlinear and high-order polynomial equation. More often, the function f(x) is itself obtained from other nonlinear ordinary differential equations or partial differential equations. In this situation, it is hard to calculate f ″ (x) when we apply the Halley method to solve the nonlinear problem.
Kung and Traub conjectured that a multipoint iteration without memory based on m evaluations of functions has an optimal convergence order p � 2 m− 1 . It means that the upper bound of the efficiency index (E.I.) � p (1/m) is 2 (1− 1/m) < 2. For m � 2, the NM is one of the second-order optimal iterative schemes; however, with m � 3, the Halley method is not the optimal one whose E.I. � 1.44225 is low. e pioneering work of Newton has inspired a lot of studies to solve nonlinear equations, whereby different fourth-order iterative methods were developed for more quickly and stably solving nonlinear equations [2][3][4][5][6][7][8][9]. Many methods to construct the two-step fourth-order optimal schemes were based on the operations of [f(x n ), f ′ (x n ), f(y n )] where y n is obtained from the first Newton step [2,[4][5][6][7][8][10][11][12][13][14]. Recently, Chicharro et al. [9] proposed a new technique to construct the optimal fourth-order iterative schemes based on the weight function technique.

Preliminaries
Before deriving the main results in the next section, we begin with some standard terminologies. Definition 1. Let the iterative sequence x n generated from an iterative scheme converge to a simple root r. If there exists a positive integer p and a real number C such that then p is the order of convergence and C is the asymptotic error constant. Let e n � x n − r be the error in the nth iterate. en, the relation is called the error equation of an iterative scheme. For example, for the Newton method, the error equation reads as where Definition 2 (see [10]). An iterative scheme is said to have the optimal order p, if p � 2 m− 1 where m is the number of evaluations of functions (including derivatives).  e conjecture of Kung and Traub asserted that a multipoint iteration without memory based on m evaluations of functions has an optimal order p � 2 m− 1 of convergence [11]. It indicates that the upper bound of the efficiency index is 2 (1− 1/m) < 2.

Definition 5.
e iterative schemes are of the same class, if they are of the same order p and have the same m evaluations of the same functions.

Main Results
We begin with the error equation of the NM: where Refer the papers, for instance, [6,12,13]. roughout of the paper, we fix the following notation: which is the first step of many two-step iterative schemes. We summarize some fourth-order optimal iterative schemes which were modified from the NM by Chun [14]: by Chun [4]: by King [5]: where c ∈ R, by Chun and Ham [2]: by Kuo et al. [8]: by Ostrowski [15]: by Maheshwari et al. [16]: and by Ghanbari [12]: It is interesting that the iterative schemes (12)-(22) are of the same class because they have same convergence order p � 4 and operated with the same evaluations on e efficiency index (E.I.) of the above eleven iterative schemes is the same � 4 3 √ � 1.5874, and they are of the optimal fourth-order iterative schemes with three evaluations of [f(x n ), f ′ (x n ), f(y n )] in the sense of Kung and Traub, such that p � 2 m− 1 � 4. ey belong to the same class with the error equations having a common type: where a i are different constants for different optimal fourthorder iterative schemes, which may be zero. Can we raise the order to five by a suitable combination of these iterative schemes? Later, we will reply to this problem.

Theorem 1.
If the conjecture of Kung and Traub is true, then the two-step optimal fourth-order iterative scheme , must have the following form of error equation: where a 0 is some constant, which may be zero.
Proof. Suppose that equation (25) is not true, such that we have where b 0 ≠ 1. e weighting factors w 1 , w 2 , and w 3 are subjected to en, we consider the weighting average of the error equations in equation (23) with i � 1, 2 and equation (26) to be zero in e 4 n : which leads to a 1 w 1 + a 2 w 2 + a 0 w 3 � 0, e determinant of the coefficient matrix of the linear equations (27) and (29) is (b 0 − 1)(a 2 − a 1 ) ≠ 0 because b 0 ≠ 1 and a 1 ≠ a 2 . From equations (27) and (29), we have the unique solution of (w 1 , w 2 , w 3 ). us, we can derive a new iterative scheme by a weighting combination of three optimal fourth-order iterative schemes with the solved factors (w 1 , w 2 , w 3 ) whose convergence order is raised to five. is contradicts the conjecture of Kung and Traub, who asserted that the optimal order for the iterative scheme with m � 3 is 2 m− 1 � 4 for a multipoint iteration without memory based on m evaluations of functions.
Obviously, eorem 1 demonstrates that we cannot raise the convergence order to five by a weighting combination of any three optimal fourth-order convergence iterative schemes.
□ Theorem 3 (see [12]). e following two-step iterative scheme: for solving f(x) � 0 has fourth-order convergence, where y n is computed by equation (11). e error equation reads as which is not supplied in [12].
Proof. It is easy to check that the weight function in iterative scheme (47): satisfies equation (32); hence, iterative scheme (47) is a special case of iterative scheme (30). We can derive where Inserting Inserting equation (52) into equation (33), we can derive is ends the proof of this theorem. eorem 2 includes those in [9,17] as special cases. e family developed by Chicharro et al. [9]: Journal of Mathematics 5 with G(0) � G ′ (0) � 1 and G ″ (0) � 4 is a special case because we can derive Accordingly, and H(0) � 1 and H ′ (0) � 2 imply G(0) � G ′ (0) � 1 and G ″ (0) � 4. For H, we have only two constraints, but for G, there are three constraints. Hence, iterative scheme (30) is more general than the iterative scheme (54). Moreover, a further differential of the last term in equation (56), leads to and hence the error equation of iterative scheme (54) is In