Some Class of Third-and Fourth-Order Iterative Methods for Solving Nonlinear Equations

The object of the present work is to give the new class of thirdand fourth-order iterative methods for solving nonlinear equations. Our proposed third-order method includes methods of Weerakoon and Fernando (2000), Homeier (2005), and Chun and Kim (2010) as particular cases. The multivariate extension of some of these methods has been also deliberated. Finally, some numerical examples are given to illustrate the performances of our proposed methods by comparing them with some well existing thirdand fourth-order methods. The efficiency of our proposed fourth-order method over some fourth-order methods is also confirmed by basins of attraction.


Introduction
Solving nonlinear equations is one of the most important problems in numerical analysis.To solve nonlinear equations, some iterative methods (such as Secant method and Newton method) are usually used.Throughout this paper, we consider iterative methods to find a simple root  of a nonlinear equation () = 0.It is well known that the order of convergence of the Newton method is two.To improve the efficiency of the Newton method, many modified thirdorder methods have been presented in the literature by using different techniques.Weerakoon and Fernando in [1] obtained a third-order method by approximating the integral in Newton's theorem by trapezoidal rule, Homeier in [2] by using inverse function theorem, Chun and Kim in [3] by using circle of curvature concept, and so forth.Kung and Traub [4] presented a hypothesis on the optimality of the iterative methods by giving 2 −1 as the optimal order.This means that the Newton method involved two function evaluations per iteration and is optimal with 1.414 as the efficiency index.By taking into account the optimality concept, many authors are trying to build iterative methods of optimal (higher) order of convergence.
The convergence order of the above discussed methods is three with three (one function and two derivatives) evaluations per full iteration.Clearly, its efficiency index (3 1/3 ≈ 1.442) is not high (optimal).In recent days, authors are improving these types of nonoptimal order methods to optimal order by using different techniques, such as in [5] by using linear combination of two third-order methods and in [6] by using polynomial approximations.Recently, Soleymani et al. [7] have used two different weight functions in the methods of [1,2] to make them optimal.
This paper is organized as follows: in Section 2, we describe a new class of third-order iterative methods by using the concept of weight function which includes the methods of [1][2][3].After that, order of this class of methods has been accelerated from three to four by introducing one more weight function and without adding more function evaluations.Section 3 is devoted to the extension of some proposed methods to the multivariate case.Finally, we give some numerical examples and the new methods are compared with some existing third-and fourth-order methods.Efficiency of our proposed fourth-order method is shown by basins of attraction.

Methods and Convergence Analysis
Before constructing the methods, here we state the following definitions.
2 Journal of Applied Mathematics 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.
2.1.Third-Order Methods.In this section, we construct a new class of two-step third-order iterative methods.Let us consider the following iterative formula: where  =   (  )/  (  ).The following theorem indicates under what conditions on the weight function in (3) the order of convergence is three.
Theorem 3. Let the function  have sufficient number of continuous derivatives in a neighborhood of  which is a simple root of ; then method (3) has third-order convergence, when the weight function () satisfies the following conditions: Proof.Let   =   −  be the error in the th iterate and  ℎ =  (ℎ) ()/ℎ!, ℎ = 1, 2, 3, . ... We provide Taylor's series expansion of each term involved in (3).By Taylor series expansion around the simple root in the th iteration, we have Furthermore, it can be easily found that By considering this relation, we obtain At this time, we should expand   (  ) around the root by taking into consideration (7).Accordingly, we have Furthermore, we have By virtue of ( 9) and ( 4), we attain Finally, using (10) in the second step of (3), we have the following error equation: which has the third order of convergence.This proves the theorem.

Particular Cases
Case 1.If we take () = 2/(1 + ) in (3), then we get the formula which is the same as established by Weerakoon and Fernando in [1].
Case 3. If we take () = (3 − )/2 in (3), then we get the formula which is the same as established by Chun and Kim in [3].
Case 4. If we take () = 2/(3 − 1) in (3), then we get the formula and its error expression is given by Remark 4. By taking suitable weight function () in (3) which satisfies the conditions of (4), one can get number of third-order methods.

2.2.
Fourth-Order Methods.The convergence order of the previous class of methods is three with three (one function and two derivatives) evaluations per full iteration.Clearly its efficiency index (3 1/3 ≈ 1.442) is not high (optimal) so we use one more weight function in the second step of (3) and introduce a constant in its first step.Thus, we consider where () and () are real-valued weight functions with  =   (  )/  (  ) and  is a real constant.The following theorem indicates under what conditions on the weight functions and constant  in (17) the order of convergence will arrive at the optimal level four.
It is clear that our class of fourth-order iterative methods (17) requires three function evaluations per iteration, that is, one function and two first derivative evaluations.Thus our new class is optimal.Clearly its efficiency index is 4 1/3 = 1.5874 (high).Now by choosing appropriate weight functions in (17) which satisfy the conditions of (18), one can give a number of optimal two-step fourth-order iterative methods.
, and  = 2/3 in ( 17), then we get the following method: and its error equation is given by Remark 6.By choosing the appropriate weight functions in (17) which satisfy the conditions of ( 18), one can get number of fourth-order methods.

Extension to Multivariate Case
In this section, we extend some methods of the previous sections (similarly other methods can also be extended) to the multivariate case.Method (15) for systems of nonlinear equations can be written as where 1 ,  () 2 , . . .,  ()  )]; and   ( () ) is the Jacobian matrix of  at  () .
Let + ∈ R  be any point of the neighborhood of exact root  ∈ R  of the nonlinear system () = 0.If Jacobian matrix   () is nonsingular, then Taylor's series expansion for multivariate case is where   = [  ()] −1 ( () ()/!),  ≥ 2, and where  is an identity matrix.From the above equation, we can find where and so on.Here we denote the error at th iterate by  () ; that is,  () =  () − .Now the order of convergence of method (31) is confirmed by the following theorem.Theorem 7. Let  :  ⊆ R  → R  be sufficiently Frechet differentiable in a convex set  containing a root  of () = 0. Let us suppose that   () is continuous and nonsingular in  and  (0) is close to .Then the sequence { () } ≥0 obtained by the iterative expression (31) converges to  with order three.
(45) Finally, using (45) in the second step of (31), we see that the error expression can be expressed as which shows the theorem.
Similarly, method (29) for systems of nonlinear equations is given by The order of the convergence of this method is confirmed by the following theorem.
Let us suppose that   (Χ) is continuous and nonsingular in  and  (0) is close to .Then the sequence { () } ≥0 obtained by the iterative expression (47) converges to  with order four.

Numerical Testing
In Table 4, numerical tests are given for third-order and fourth-order methods.The test functions have been performed with stopping criterion |( +1 )| ≤ 10 −120 ; in addition, the maximum number of iterations is less than 100.The computational results presented in Table 4 show that the presented method M4 II converges more rapidly than almost all these methods.This new method requires less number of functional evaluations.This means that the new method has better efficiency in computing process as compared to the other methods.For some initial guess, our method gives some bad results as compared to the Newton method but in this case other methods either failed (F)/diverge (D) or gave poor performances as compared to our method M4 II.Also one advantage of our method is that some of the third-order and fourth-order methods failed/diverge for some initial guess but our method does not.So we can conclude that our method M4 II is efficient.

Basin of Attraction
The basin of attraction for complex Newton's method was first initiated by Cayley [19].The basin of attraction is a way to view how an algorithm behaves as a function for the various starting points.It is another way to compare the iterative methods.We consider a rectangle  = [−2, 2] × [−2, 2] ∈ C and assign a colour to each point  0 ∈  according to the root at which the corresponding iterative method starting from  0 converges; for more details one may see [20,21].In this section, the following test functions have been considered for comparison: (i)  We compare our fourth-order method (30) (M4 II) with Jarratt's method (JM4) of [22], (17) (SL4) of [7], and (6) (KH4) of [11].The results of the comparisons are given in Figures 1, 2, 3, 4, and 5.In our first figure, we have performed all the methods to obtain the zeros of the quadratic polynomial  2 − 1.From Figure 1, it is clear that JM4 is best followed by our proposed method M4 II; methods KH4 and SL4 do not perform well.In Figure 2, we have taken a cubic polynomial  3 + 4 2 − 10.In this case again JM4 is best with simplest boundary of basin but our proposed method M4 II also shows the simpler boundaries as compared to methods SL4 and KH4.In Figure 3 we have considered a polynomial  3 − 1.
It can be seen that, as before, JM4 and M4 II are better than SL4 and KH4.In Figures 4 and 5, we have taken polynomials  5 − 1 and  4 − 1, respectively.In both figures, methods JM4 and M4 II dominate the other two methods, that is, SL4 and KH4.One can note that taking tighter conditions in programming code may develop picture with better quality than these.
From Figures 1-5, it is clear that our proposed fourthorder method and Jarratt's method perform well as compared to the other two fourth-order methods.

Conclusion
In this paper, we have proposed a class of third-and fourthorder iterative methods for finding simple roots of nonlinear equations.Our third-order method includes methods of Weerakoon and Fernando [1], Homeier [2], and Chun and Kim [3] as particular cases.We have also extended some of our proposed methods from single to multivariate case.A number of numerical examples are given to illustrate the performances of our methods by comparing them with some well existing third-and fourth-order iterative methods.The efficiency of our fourth-order method over some existing fourth-order methods is also supported by basin of attractions.

Table 1 :
Functions and their roots.

Table 2 :
Comparison of the absolute error occurring in the estimates of the exact root.

Table 3 :
Comparison of the absolute error occurring in the estimates of the exact root.

Table 5 :
Comparison of the norm of the error occurring in the estimates of the exact root with some existing third-order methods.

Table 6 :
Comparison of the norm of the error occurring in the estimates of the exact root with some existing fourth-order methods.