Abstract
Kung and Traub conjectured that a multipoint iterative scheme without memory based on evaluations of functions has an optimal convergence order . In the paper, we first prove that the two-step fourth-order optimal iterative schemes of the same class have a common feature including a same term in the error equations, resorting on the conjecture of Kung and Traub. Based on the error equations, we derive a constantly weighting algorithm obtained from the combination of two iterative schemes, which converges faster than the departed ones. Then, a new family of fourth-order optimal iterative schemes is developed by using a new weight function technique, which needs three evaluations of functions and whose convergence order is proved to be .
1. Introduction
The most basic problem in engineering and scientific applications is to find the root of a given nonlinear equationwhere and is an interval we are interested in, and we suppose that is a simple solution with and .
The famous Newton method (NM) for iteratively solving equation (1) is given bywhich 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 is itself obtained from other nonlinear ordinary differential equations or partial differential equations. In this situation, it is hard to calculate when we apply the Halley method to solve the nonlinear problem.
Kung and Traub conjectured that a multipoint iteration without memory based on evaluations of functions has an optimal convergence order . It means that the upper bound of the efficiency index (E.I.) = is . For , the NM is one of the second-order optimal iterative schemes; however, with , the Halley method is not the optimal one whose E.I. = 1.44225 is low.
The 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 [2–9]. Many methods to construct the two-step fourth-order optimal schemes were based on the operations of where is obtained from the first Newton step [2, 4–8, 10–14]. Recently, Chicharro et al. [9] proposed a new technique to construct the optimal fourth-order iterative schemes based on the weight function technique.
2. Preliminaries
Before deriving the main results in the next section, we begin with some standard terminologies.
Definition 1. Let the iterative sequence generated from an iterative scheme converge to a simple root . If there exists a positive integer and a real number such thatthen is the order of convergence and is the asymptotic error constant.
Let be the error in the th iterate. Then, the relationis called the error equation of an iterative scheme. For example, for the Newton method, the error equation reads aswhere
Definition 2. (see [10]). An iterative scheme is said to have the optimal order , if where is the number of evaluations of functions (including derivatives).
Definition 3. The efficiency index (E.I.) of an iterative scheme is defined by E.I. = .
Definition 4. The conjecture of Kung and Traub asserted that a multipoint iteration without memory based on evaluations of functions has an optimal order of convergence [11]. It indicates that the upper bound of the efficiency index is .
Definition 5. The iterative schemes are of the same class, if they are of the same order and have the same evaluations of the same functions.
3. Main Results
We begin with the error equation of the NM:where
Refer the papers, for instance, [6, 12, 13].
Throughout 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 , 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 and operated with the same evaluations on . The efficiency index (E.I.) of the above eleven iterative schemes is the same , and they are of the optimal fourth-order iterative schemes with three evaluations of in the sense of Kung and Traub, such that . They belong to the same class with the error equations having a common type:where are different constants for different optimal fourth-order 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 schemewhich is based on the evaluations of , must have the following form of error equation:where is some constant, which may be zero.
Proof. Suppose that equation (25) is not true, such that we havewhere .
The weighting factors , , and are subjected toThen, we consider the weighting average of the error equations in equation (23) with and equation (26) to be zero in :which leads toThe determinant of the coefficient matrix of the linear equations (27) and (29) is because and . From equations (27) and (29), we have the unique solution of . Thus, we can derive a new iterative scheme by a weighting combination of three optimal fourth-order iterative schemes with the solved factors whose convergence order is raised to five. This contradicts the conjecture of Kung and Traub, who asserted that the optimal order for the iterative scheme with is for a multipoint iteration without memory based on evaluations of functions.
Obviously, Theorem 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 2. The following two-step iterative scheme:for solving has fourth-order convergence, where is computed by equation (11), and is a weight function in terms ofwithThe corresponding error equation is
Proof. For the proof of the convergence, we let be a simple solution of , i.e., and . We suppose that is sufficiently close to the exact solution , such thatis a small quantity, and it follows thatBy using the Taylor series, we haveIt immediately leads toFrom equations (11) and (38), we haveFrom equations (40), (37), and (36), it follows thatFrom equations (31) and (42), we haveBecause the least order of the term as shown in equation (41) is two, we only need to expand around zero to the second-order by using equation (43) andInserting equations (11), (39), (44), and (41) into equation (30), we haveThrough some manipulations, we can derivewhich, due to equation (32), can be arranged to that in equation (33).
Theorem 3. (see [12]). The following two-step iterative scheme:for solving has fourth-order convergence, where is computed by equation (11). The error equation reads aswhich 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 derivewhereInserting into equation (50) by taking , we haveInserting equation (52) into equation (33), we can deriveThis ends the proof of this theorem.
Theorem 2 includes those in [9, 17] as special cases. The family developed by Chicharro et al. [9]:with and is a special case because we can deriveAccordingly,and and imply and . For , we have only two constraints, but for , 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 toand hence the error equation of iterative scheme (54) isIn [9], Chicharro et al. derived the error equation as (equation (2) in [9]), which is incorrect to miss the term in the error equation.
The general function of is given bywhere is any integrable function. There are two interesting iterative schemes generated from (COSM) and (SINM):
4. Combinations of Iterative Schemes
In this section, we give some methods to combine the iterative schemes as listed in Table 1, which are special cases of the iterative schemes (47) and (30).
From Table 1, we can observe that there exists a cubic term in the error equation for most iterative schemes. Indeed, this term is a dominant factor to enlarge the error, and thus we can combine two iterative schemes by eliminating this term.
Theorem 4. For the following two-step iterative scheme:ifthen the error equation reads as
Proof. The weighting factors are subjected toWe seek the combination of iterative scheme (47) with two sets of the parameters and and demand the coefficient preceding being zero,Solving equations (66) and (67), we can derive equation (64), and the error equation (48) reduces to that in equation (65).
We cannot exhaust all the combinations of the iterative schemes; however, we list the following two: one is the combination of equations (16) and (19), namely, the KOM:The other one is the combination of equations (12) and (19), namely, the COM:
5. Second Family of Optimal Fourth-Order Iterative Schemes
In Theorem 2, we have derived a new family of optimal fourth-order iterative schemes with the assumption that the -function satisfies and . We can relax the conditions to and derive the following result.
Theorem 5. Suppose that there are two different functions and satisfyingThe following two-step iterative scheme:for solving has fourth-order convergence, where is computed by equation (11), and is defined by equation (31). The corresponding error equation iswhere
Proof. From equations (46) and (70), it follows that the error equations corresponding to and are, respectively,whereWe seek a combination of the two iterative schemes corresponding to and and ask the coefficient preceding to be zero, such that we have to solve and fromwhose solutions are given by equation (74). At the same time, the combined error equation is given bywhich can be arranged to that in equation (73).
The family in equation (72) includes some optimal fourth-order iterative schemes with two parameters and , whose error equation again belongs to the type in equation (23). It can be seen that the functions with are very general, and for this class of iterative schemes, the conjecture of Kung and Traub is also true.
6. Numerical Experiments
In this section, we give numerical tests of the proposed combined iterative schemes. The test examples are given by
The corresponding solutions are, respectively, , , , , and .
In Table 2, for different functions, we list the number of iterations (NI) obtained by the presently developed algorithms, which are compared to the NM, the CM1 in equation (12), the CM2 in equation (15), the KM in equation (16) with , the OM in equation (19), the AM in equation (20), the GM in equation (21), the KOM in equation (68) with , and the COM in equation (69).
7. Conclusions
Employing a new weight function, the nonlinear equations were solved by using a new family of the fourth-order iterative scheme, which is optimal according to the conjecture of Kung and Traub, and it was proven to be of fourth-order convergence with E.I. = 1.5874. Theorem 1 indicated that if one can develop a fourth-order iterative scheme based on the evaluations of whose coefficient preceding is not −1, then the Kung–Traub conjecture would be disproved. We also proposed a combination of two fourth-order iterative schemes of which the dominant term in the error equation is eliminated. Upon comparing some examples to other methods, we found that the combined iterative scheme converges faster. The present iterative scheme was competitive to other optimal fourth-order iterative schemes.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.