At present, there are a lot of iterative methods for solving nonlinear equations and systems of equations (see    and reference therein). In particular, the derivative-free methods are necessary when the derivative of the function f is unavailable or expensive to obtain. In the last decade, the derivative-free two and three-point methods with better convergence properties were developed (see  -  and references therein). It should be pointed out that most of these methods were proposed mainly for the concrete choice of parameters (see Table 1). Evidently, a systematic theory or an approach for constructing derivative-free methods is still needed. It is therefore of interest and necessity to develop a global theory. The aim of this paper is to fill up the above mentioned gap
Table 1. The derivative-free three-point iterative methods.
and to obtain the wide class of optimal derivative-free three-point methods. The paper is organized as follows. In Section 2, we give the necessary and sufficient conditions for derivative-free three-point iterations to be optimal order eight. We also establish the connection between derivative presence and derivative-free three-point methods. In Section 3, we apply the sufficient convergence conditions to obtain the optimal derivative-free methods which are dependent on parameters in the third-step of considered iterations. We obtain families of optimal derivative-free three-point methods. They include many existing methods as particular cases as well as new methods with the higher order of convergence. In last section, we present the results of numerical experiments that confirm the theoretical conclusion about the convergence order and make comparison with other known methods of the same order of convergence. Finally, numerical results show that new iterative methods can be significant by its high precision and practical use.
2. The Optimal Derivative-Free Three-Point Iterations
Typically, the optimal three-point iterative methods have a form 
in which the parameters and are given by
where , and . In  was proven the following theorem.
Theorem 1. Let the function be sufficiently smooth and have a simple root . Furthermore, let the initial approximation be sufficiently close to . Then, the convergence order of the iterative method (1) is eight if and only if the parameters and satisfy conditions (2) and (3), respectively.
Remark. The second sub-step in (1) can be rewritten as any two-point optimal fourth-order method
where is a real function using the evaluation of and . Each method in has a parameter given by (2) with own and .
Now we consider the derivative-free variant of (1)
Here is any second-order method. Actually, in Formula (4), the fundamental quantities are
Then , for , where is a simple root of . If is any two-point optimal fourth-order method then , therefore . The iteration (4) obtained from (1) replacing by . Due to change (5), the parameters in (4) does not remain as before and we denote them by and . We call the iterations (1) and (4) the derivative presence (DP) and derivative-free (DF) variants respectively. If we use the notations
then we have
DP can be derived from DF by substituting . The following is the main result of our work .
Theorem 2. Let the assumptions of Theorem 1 be fulfilled. Then, the convergence order of the iteration (4) is eight if and only if the parameters and in (4) are given by formulas
The proposed method (4) with parameters given by (7) and (8) is three-point derivative free and optimal in the sense of Kung and Traub. Kung-Traub conjecture  states that the multi-point iterative methods, based on k evaluations, could achieve optimal convergence order . Our proposed method is in concurrence with the conjecture as it needs only four function evaluation per iteration i.e., . Moreover, using ideas in   one can propose more general construction for and as following:
Define as sufficiently smooth functions of . It is easy to show that if and only if , where , . Hence, under the restriction , (4) is optimal if and only if
Those can be easily checked with using (6). For the optimal formula, the remainder term is in (8) because . In this sense, we can say that (4) is optimal if and only if can be written as (7) and (8).
When the Formula (7) leads to (2) and the Formula (8) leads to (3). A query may arise that there exists an optimal (DF) variant (4) for each optimal (DP) variant (1) and vice versa. If yes, how to find its (DF) variant? To respond this we use the connection of formulas (3) and (8). Actually, from (3) and (8) we deduce that
where is obtained replacing by in in (1). From (9) we find that
These relations (9) and (10) give the rule of mutual transition of (DP) and (DF) variants. There exists the one optimal (DP) variant (4) for each optimal (DF) variant (1). The converse does not true. Namely there are several (DF) variants of (DP).
3. Application of Sufficient Convergence Condition to Derive New DF Iterations
Now we give the application of Theorem 2 to construct new iterations. The sufficient convergence conditions (7) and (8) allow us to design new derivative-free optimal methods. Depending on the form of we can obtain different iterations. We consider some special cases.
1) Let in (4) be a form
where , and are smooth enough functions. As regarding the iteration (4) with given by (11) we give the following result.
Theorem 3. The iteration (4) with given by (7) and with given by (11) have the order of convergence eight, if the following conditions hold:
Proof. Using the Taylor expansion of smooth enough functions and we obtain an expression for (11). The comparison of this expression with sufficient condition (8) gives conditions (12).
When in (7) the Theorem 3 leads to a theorem in . That is to say, the similar theorem was proved in  only for special case of :
Therefore, Theorem 3 is more general, than that of . Note that, in  are proposed four variants of that include redundant terms like and
. By neglecting these terms, can be simplified essentially without loss of the order of convergence. When the condition (12) reduced to
It means that the derivative presence variant (1) with parameters given by (7) and (11) has a convergence order eight under conditions (14).
Thukral and Petković considered in  the particular case of (1) with given by (11) and with
In this case and and the condition (14) coincides with that of . They also considered another particular case of (1) with given by (11) and
In this case and the condition (14) leads to that of . The function in (11) can be written as
Due to generating function method  instead of we can take any function H
As a result, we have a family of optimal derivative-free three-point methods (4) with (11), (15), and (16). The constants and can be expressed through and as:
That is we have the iterations (4) with is given by (16) and is given by
Note that the choice of parameter defined by (16) includes almost all the choices listed in Table 1 as particular cases. Thus the family of iterations (4) with (16) and (17) represents a wide class of optimal derivative-free three-point iterations.
2) Let in (4) be a form
where is given by (7) and is sufficient smooth function of and .
Theorem 4. The iteration (4) with given by (7) and given by (18) has the order of convergence eight, if the following conditions hold:
Proof. From (7) and (8) it is clear that
which holds under conditions (19).
The (DP) variant of this iteration is obtained from (4), (7), and (18) when . Note that the similar scheme was considered in .
In some cases, the form
obtained from (18) is useful. Using (20) we obtain
For the iteration (4) with (7) and (21) we can formulate the following:
Theorem 5. The iteration (4) with (7) and (21) has the order of convergence eight, if the following conditions hold:
Proof. If we take (22) into account in the Taylor expansion of function we arrive at (23).
When the conditions (23) take a form
Remark. Obviously, as for one can take any function H given by (16) in the formulas (17) and (21).
Note that in  were obtained some conditions that guarantee order eight of the method (4) with (7) and (21) i.e.,
that does not coincide with (22). Moreover, the terms and seem to be redundant, because it suffices to determine with accuracy .
Note that (DP) methods with (7) and (21) are often used. For example, Kung-Traub’s eighth-order method  has a form (1) with
The Bi-Wu-Ren’s optimal eighth-order method  has a form (1) with
But (27) is not the example for (21).
The Sharma and Arora’s optimal eighth-order method  has a form (1) with
Moreover, we suggest that more general theory for as
3) Let in (4) be a form
that often used in practice, see     . Of course, and given by (7) and (29) satisfy the sufficient conditions (7) and (8). The (DP) variant of (4) with (7) and (29) has a form (1)
In  is proposed the eighth-order iteration (1) with (29) ( ) and special
Our iteration (1) with (2) and (30) is more general than that of .
4) Let in (4) be a form
We shall find the coefficients and such that the iteration (4) with (7) and (32) has the order of convergence eight and state the following:
Theorem 6. The iteration (4) with (7) and (32) has the order of convergence eight, if the following conditions hold:
Proof. Using the following relations
Substituting (35) into (32) and using the sufficient convergence condition (8) we arrive at (33).
Thus, we have a family of optimal three-point (DF) the iteration (4) with (7) and (32) that contains three parameters a, b and c. Now, we consider some particular cases of the iteration (4) with (7) and (32). Let and . Then from (33) we find that
Hence we obtain
where . The sign ≈ in (37) indicates that it holds with accuracy . Now, we consider concrete choice of :
For the choice (38) we have
The iteration (4) with (38) and (36) (or (37)) is converted to one given by Soleymani in  for and one given by Thukral in  for . For the choice the parameter is simplified as
or using we have
Then and we have
The iteration (4) with (41) and (42) coincides with one given by Soleymani in  with
here we can neglect the redundant terms . Let , and . Then from (33) we find that
The Formula (32) is converted to
On the other hand, the direct calculation using relations (34) gives
We choose parameter in (44) such that the expression (44) coincides with the numerator of (43) within accuracy . That is to say, that
As a result, (43) can be rewritten as
Thus, we find a family of optimal (DF) iteration (4) with (7) and (45), that contains some existing iterations as particular cases. Thukral in  proposed eighth-order derivative-free iterations (called ) for some special :
In this case and hence , the given by (45) leads to that of and in . So, the Thukral’s method ( ) are included in our family of (4) with (7) and (45). Thukral in  proposed also Petković type methods ( ). For we get , i.e. . In this case in (45) and our family of method (4) with (45) converted to . For we get
In this case and . Thus, our family of method (4) with (45) converted to . It means that the ( ) methods are also included in our family of (4) with (7) and (45). As stated above for the choice of (41) we have , so (45) is simplified as
Thus, we have optimal (DF) methods
where is defined by (47). This is (DF) variant of Sharma and Sharma’s optimal methods given in   within accuracy . It means that we develop (DF) variant of Sharma and Sharma’s method.
4. Numerical Experiments
In this section, we make some numerical experiments to show the convergence behavior of the presented derivative-free method (4) with parameters and . We also compare them with the ones developed by Soleymani , Thukral   and Sharma et al. . For this purpose, we consider smooth and non-smooth nonlinear functions, which are given as follows:
All computations are performed using the programming package Maple18 with multiple-precision arithmetic and 2500 significant digits. The test functions have been used with stopping criterion , where is a root of and the approximation to . In all examples, we consider that the parameter and that in Chebyshew-Halley’s method.
Nowadays, high order methods are important due to scientific computations in many areas of science and engineering use. For instance, planetary orbit calculation, radiation calculations and many real life problems demand higher precision for desired results  . The first example addresses this situation and we apply the presented methods to solve one such physical problem. In  have considered one of the famous classical physics problem which is known as Planck’s radiation law problem. First nonlinear function arises from this problem.
has two zeros. Obviously, one of the roots is not taken for discussion. Another root is . Now, we give some numerical experiments and compare new methods with some well-known methods for the smooth function using the initial guess . In Table 2 and Table 3, we exhibit computational order of convergence (COC) and absolute error as well as iteration numbers n are displayed. For presented methods and test functions, by using (see, e.g.,   )
we have computed the order of convergence.
From Table 2, we can observe that computed results completely support the
Table 2. Convergence behavior of scheme (4) for .
Table 3. Some particular cases of (4) with (16) and (29).
theory of convergence discussed in previous section. In addition to the comparison of new methods with other methods we include some special cases of proposed family (4) in Table 3.
Table 4 illustrates the number of iterations needed to achieve approximate solution and absolute residual error of the corresponding function using the stopping criterion .
Table 4. Comparisons between different methods.
Furthermore, when the iteration diverges for the considered initial guess , we denote it by “−”.
From Table 4, we see that the convergence behavior of the presented families with different parameters and the iteration number n are the same as for all considered methods.
The result of Table 5 demonstrates that new methods iteration numbers are
Table 5. Comparison of various iterative methods for .
used lesser than other existing methods under condition . However, the dynamic behavior of iterations may depend on the choices of parameters and problems under consideration. In sum, numerical results show that new iterative methods can be significant by its high precision and practical use.
We derive the necessary and sufficient conditions for derivative-free three-point iterations with the optimal order. The use of these conditions allows us to derive the families of optimal derivative-free iterations. We propose the families of optimal derivative-free iterations (4) with given by (16) and given by (17), (29), (32), and (45). Our families include many existing iterations as particular cases, as well as new effective iterations. We reveal redundant terms in well-known methods given in   . Dropping these terms allows us to simplify their algorithms and save computation time.
The authors wish to thank the editor and anonymous referees for their valuable suggestions on the first version of this paper. This work was supported by the Foundation of Science and Technology of Mongolian under grant SST_18/2018.
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