Several New Third-Order and Fourth-Order Iterative Methods for Solving Nonlinear Equations

In order to find the zeros of nonlinear equations, in this paper, we propose a family of third-order and optimal fourth-order iterative methods. We have also obtained some particular cases of these methods. These methods are constructed through weight function concept. The multivariate case of these methods has also been discussed. The numerical results show that the proposed methods are more efficient than some existing thirdand fourth-order methods.


Introduction
Newton's iterative method is one of the eminent methods for finding roots of a nonlinear equation: Recently, researchers have focused on improving the order of convergence by evaluating additional functions and first derivative of functions.In order to improve the order of convergence and efficiency index, many modified third-order methods have been obtained by using different approaches (see [1][2][3]).Kung and Traub [4] presented a hypothesis on the optimality of the iterative methods by giving 2 −1 as the optimal order.It means that the Newton iteration by two function evaluations per iteration is optimal with 1.414 as the efficiency index.By using the optimality concept, many researchers have tried to construct iterative methods of optimal higher order of convergence.The order of the methods discussed above is three with three function evaluations per full iteration.Clearly its efficiency index is 3 1/3 ≈ 1.442, which is not optimal.Very recently, the concept of weight functions has been used to obtain different classes of third-and fourthorder methods; one can see [5][6][7] and the references therein.This paper is organized as follows.In Section 2, we present a new class of third-order and fourth-order iterative methods by using the concept of weight functions, which includes some existing methods and also provides some new methods.We have extended some of these methods for multivariate case.Finally, we employ some numerical examples and compare the performance of our proposed methods with some existing third-and fourth-order methods.

Methods and Convergence Analysis
First we give some definitions which we will use later.
Definition 1.Let () be a real valued function with a simple root  and let   be a sequence of real numbers that converge towards .The order of convergence  is given by lim where  is the asymptotic error constant and  ∈  + .
Definition 2. Let  be the number of function evaluations of the new method.The efficiency of the new method is measured by the concept of efficiency index [8,9] and defined as where  is the order of convergence of the new method.
International Journal of Engineering Mathematics 2.1.Third-Order Iterative Methods.To improve the order of convergence of Newton's method, some modified methods are given by Grau-Sánchez and Díaz-Barrero in [10], Weerakoon and Fernando in [1], Homeier in [2], Chun and Kim in [3], and so forth.Motivated by these papers, we consider the following two-step iterative method: =   −   (  )   (  ) ,  +1 =   −  ()  (  )   (  ) , where  =   (  )/  (  ) and  is a real constant.Now we find under what conditions it is of order three.
Theorem 3. Let  be a simple root of the function f and let f have sufficient number of continuous derivatives in a neighborhood of .The method (4) has third-order convergence, when the weight function () satisfies the following conditions: Proof.Suppose   =   −  is the error in the th iteration and  ℎ =  (ℎ) ()/ℎ!  (), ℎ ≥ 1. Expanding (  ) and   (  ) around the simple root  with Taylor series, then we have Now it can be easily found that By using (7) in the first step of (4), we obtain At this stage, we expand   (  ) around the root by taking (8) into consideration.We have Furthermore, we have By virtue of ( 10) and ( 4), we get Hence, from ( 11) and ( 4) we obtain the following general equation, which has third-order convergence: This proves the theorem.
Particular Cases.To find different third-order methods we take  = 2/3 in (4).
Remark 4. By taking different values of  and weight function () in (4), one can get a number of third-order iterative methods.

Optimal
Fourth-Order Iterative Methods.The order of convergence of the methods obtained in the previous subsection is three with three function evaluations (one function and two derivatives) per step.Hence its efficiency index is 3 1/3 ≈ 1.442, which is not optimal.To get optimal fourthorder methods we consider where () and () are two real-valued weight functions with  =   (  )/  (  ) and  is a real constant.The weight functions should be chosen in such a way that the order of convergence arrives at optimal level four without using additional function evaluations.The following theorem indicate the required conditions for the weight functions and constant  in (22) to get optimal fourth-order convergence.  (3(1)      ≤ +∞. (23) Proof.Using (6) and putting  = 2/3 in the first step of ( 22), we have Now we expand   (  ) around the root by taking (24) into consideration.Thus, we have Furthermore, we have By virtue of ( 26) and ( 22), we obtain Finally, from ( 27) and ( 22) we can have the following general equation, which reveals the fourth-order convergence: It proves the theorem.
Theorem 7. Let  :  ⊆ R  → R  be sufficiently Frechet differentiable in a convex set , containing a root  of () = 0. Let one suppose that   () is continuous and nonsingular in  and  (0) is close to .Then the sequence { () } ≥0 obtained by the iterative expression (40) converges to  with order three.
(60) Thus, we end the proof of Theorem 7.
International Journal of Engineering Mathematics 7 Table 1: Functions and their roots.
The multivariate case of (33) is given by The following theorem shows that this method has fourthorder convergence.
Theorem 8. Let  :  ⊆ R  → R  be sufficiently Frechet differentiable in a convex set , containing a root  of () = 0. Let one suppose that   () is continuous and nonsingular in  and  (0) is close to .Then the sequence { () } ≥0 obtained by the iterative expression (61) converges to  with order four.
Proof.For the convenience of calculation we replace 2/3 by  and put  1 = 13/16,  2 = 3/8, and  3 = −3/16 in (61).From ( 46) and (50), we have +  ( () 4 ) . ( From the above equation we have +  ( () 4 ) . ( With the help of ( 62) and (63), we can obtain By multiplying (64) to (58), we have where The final error equation of method (61) is given by which confirms the theorem.We consider 1000 digits floating point arithmetic using "SetAccuracy []" command.Here we compare the performance of our proposed methods with some well-established third-order and fourth-order iterative methods.In Table 2, we have represented Huen's method by HN3, our proposed third-order method (15) by M3, fourth-order method (17) of [5] by SL4, fourth-order Jarratt's method by JM4, and proposed fourth-order method by M4.The results are listed in Table 2.

Numerical Testing
An effective way to compare the efficiency of methods is CPU time utilized in the execution of the programme.In present work, the CPU time has been computed using the command "TimeUsed []" in MATHEMATICA.It is well known that the CPU time is not unique and it depends on the specification of the computer.The computer characteristic is Microsoft Windows 8 Intel(R) Core(TM) i5-3210M CPU@ 2.50 GHz with 4.00 GB of RAM, 64-bit operating system throughout this paper.The mean CPU time is calculated by taking the mean of 10 performances of the programme.The mean CPU time (in seconds) for different methods is given in Table 3.

Conclusion
In the present work, we have provided a family of thirdand optimal fourth-order iterative methods which yield some existing as well as many new third-order and fourth-order iterative methods.The multivariate case of these methods has also been considered.The efficiency of our methods is supported by Table 2 and Table 4.

Table 2 :
Comparison of absolute value of the functions by different methods after fourth iteration (TNFE-12).

Table 3 :
Comparison of CPU time (in seconds) between some existing methods and our proposed methods.
In this section, ten different test functions have been considered in Table1for single variate case to illustrate the accuracy of the proposed iterative methods.The root of each nonlinear test function is also listed.All computations presented here have been performed in MATHEMATICA 8.Many streams of science and engineering require very high precision degree of scientific computations.