An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior

Abstract In this paper, we derive and analyze a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the family is also considered, and the methods have convergence order three. Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modified Cauchy methods is obtained by using weight function. We also perform some numerical tests and the comparison with existing optimal fourth-order methods to show the high computational efficiency of the proposed scheme, which confirm our theoretical results. The basins of attraction of this optimal fourth-order family and existing fourth-order methods are presented and compared to illustrate some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, from the fractal graphics, with the increase of the value m of the series in iterative methods, the chaotic behaviors of the methods become more and more complex, which also reflected in some existing fourth-order methods.


Introduction
In this paper, we consider iterative methods to nd a simple root α,i.e, f (α) = and f (α) ≠ , of a nonlinear equation where f : I ⊂ R → R for an open interval I is a scalar function.
Finding the simple root of the nonlinear equation (1) is a common and important problems in numerical analysis of science and engineering, and iterative methods are usually used to approximate a solution of these equations. We know that Newton's method is an important and basic approach for solving nonlinear equations [1,2], and its formulation is given by this method converges quadratically. The classical Cauchy's method [2] is expressed as where This family methods given by (3) is a well-known third-order method. However, the method depends on the second derivatives in computing process, and therefore their practical applications are restricted rigorously.
In this paper, we will improve the family de ned by (3) and obtain third and optimal fourth order family of second-derivative-free variants of Cauchy's methods by using Padé approximant. The rest of the paper is organized as follows: In Section 2, we present a new third order family of modi ed Cauchy method and show the order of convergence of this family; In Section 3, di erent numerical tests con rm the theoretical results, and the new methods are comparable with other known methods and give better results in many cases; In Section 4, based on the family of third-order method, a new optimal fourth-order family of iterative methods is obtained by using weight function; In Section 5, numerical tests and the comparison with the existing optimal fourth-order methods are included to con rm our theoretical results; In Section 6, the basins of attraction of the existing optimal fourth-order methods and our methods are presented and compared to illustrate their performances. Finally, we infer some conclusions.

Development of the third order method and its convergence analysis
In order to avoid the evaluation of the second derivatives f (xn) of Cauchy's method (3), we consider approximating it by the derivative y (xn) of the following second degree Padé approximant: where a , a , a and a are real parameters. We impose the tangency conditions y(xn) = f (xn), y (xn) = f (xn), y(wn) = f (wn), where xn is nth iterate and By using the tangency conditions from (6), we obtain the value of a , a , a , and a is determined in terms of a in the following From (5), we also have Substituting (8) into (9) yields . (10) Using (10) we can approximate We de ne Using L f , µ (xn , wn) instead of L f (xn), we obtain a new one-parameter family of modi ed Cauchy method free from second derivative where µ ∈ R. Similar to the classical Cauchy's method, a square root is required in (13). However, this may cost expensively, even fail in the case − L f , µ (xn , wn) < .In order to avoid the calculation of the square roots, we will derive some forms free from square roots by Taylor approximation [4].
It is easy to know that Taylor approximation of where m > . Using (14) in (13), we can obtain the following form where µ ∈ R.
On the other hand, it is clear that Then, Using (16) in (13), we also can construct a new family of iterative methods as follows: where µ ∈ R, m > . We have the convergence analysis of the methods by (17). where f (α) . Furthermore, we have f (xn) = f (α)[ + c en + c e n + c e n + c e n + O(e n )]. (20) Dividing (19) by (20), From (21), we get Expanding f (wn) in Taylor's Series about α and using (22), we get Since (20), we obtain Because of (19), we get From (23) and (24), we get From (20), (24), (25) and (26), we obtain Furthermore, from (27) we have Since (17) and (28), we have from e n+ = x n+ − α, we have Then the methods de ned by (17) is shown to converge of the order three. Similar to the proof of Theorem 2.1, we can prove that the methods de ned by (12) and (15) are third-order methods.

Some special cases
: If µ = , from (12) and (17) we obtain where m > . For m = , we obtain a third-order method(LM ) For m = , we obtain from (17) a third-order method(LM ) : If µ = , from (12) we obtain For m = , we obtain from (17) a third-order method(LM ) : If µ = − , from (12) we obtain For m = , we obtain from (17) a third-order method(LM ) : If µ = − , for m = , we obtain a third-order method (LM ) from (12) and (17) : If µ = , from (12) and (15) we obtain some iterative methods as follows: For m = , we obtain a third-order method For m = , we obtain a third-order method(LM ) For m = , we obtain a third-order method

Numerical examples of the third order methods
In this section, we present the results of numerical simulations in Table 2 to compare the e ciencies of the methods. The considered methods are Newton method (NM), the method of Weerakoon and Fernando [8] (WF), the method of Potra and Pták (PP) [9], Chebyshev's method (CHM) [11][12], Halley's method (HM) [11], and our new methods ( Table 2 are the number of iterations (IT), the number of function evaluations (NFE) counted as the sum of the number of evaluations of the function itself plus the number of evaluations of the derivative, the absolute residual error of the corresponding function value (|f (xn)|), the computing time (TIME, the unit of time is one second) and the distance of two consecutive approximations δ = |xn − x n− |. All computations were done using Matlab 7.1 environment with a ADM athlon(tm) II X2 250-3.01 GHz based PC. We accept an approximate solution rather than the exact root, depending on the precision ϵ of the computer. We use the following stopping criteria for computer programs: |f (xn)| < ϵ, we used the xed stopping criterion ϵ = − . " − " is divergence. We used the following test functions and display the computed approximate zero x * in Table 1 [13].

Development of the optimal fourth order method and its convergence analysis
Corresponding to the well known Traub's method (see [14]), This scheme (17) with order of convergence three, is not optimal in the sense of Kung-Traub conjecture [14]. In this section, we introduce parametric weight functions and the well-known technique of undetermined coe cients to the family of iterative methods (17) to increase the order of convergence to four. Test functions .
We consider using a weight function H(µ(xn , wn , γ i )) instead of µ in the operator (12), and consider the well-known technique of undetermined coe cients to design an new operator L f ,H,μ (xn , wn) as follows where H(µ(xn , wn , γ i )) is a function of real variable γ i (i = , . . . , ) and µ j (j = , . . . , ) are real parameters. Then, using (43) in (17), we also can construct two new optimal fourth-order family of modi ed Cauchy methods as follows: and where m > . In the following result, we present the conditions that the weight function H(µ(xn , wn , γ i )) and the parameters must satisfy for obtaining two families of iterative methods with fourth-order of convergence, which becoming optimal schemes by Kung-Traub conjecture.
Proof. Let en = xn − α, because of the Taylor series expansions of f (xn) and f (wn), we have Taking into account the expansion of µ(xn , wn , γ i ), and by using Taylor series expansion of H(µ(xn , wn , γ i )) around , we obtain From (20), (24), (25), (26) and (43), we obtain Substituting (50) From e n+ = x n+ − α, we consider that if H( ) = , µ = µ , µ = , Then, we obtain the error equation of (46) in the form: Then the methods de ned by (46) is shown to converge of the order four. Similar to the proof of Theorem 3.1, we can prove that the methods de ned by (43) and (45) are fourthorder methods.

Some special cases
: If we consider the following weight function H = H(µ(xn , wn , γ i )) = , from (46) and (54) we obtain where m > . For m = , we obtain a recently developed fourth-order method by Khattri et al. (KM ) [13] x n+ = xn − + L f ,H ,λ (xn , wn) For m = , we also get the existing optimal fourth-order method by Khattri et al. (KM ) [13] x n+ = xn − + L f ,H ,λ (xn , wn) + L f ,H ,λ (xn , wn) For m = , we obtain the developed fourth-order method by Khattri et al. (KM ) [13], which is given by : Now, we consider the following weight function, which also satis es all the conditions of Theorem 3.1. If H = H(µ(xn , wn , γ i )) = µ(xn ,wn ,γ i ) −µ(xn ,wn ,γ i ) , γ = , γ = γ = γ = and λ = , from (43) and (54), we have For m = , from (46) and (60) we obtain a new fourth-order method (LTM ) For m = , we obtain a new fourth-order method (LTM ) For m = , from (46) and (71) we obtain a new fourth-order method For m = , we obtain a new fourth-order method (LTM ) For m = , from (45) and (71) we obtain a new fourth-order method (LTM ) For m = , we obtain a new fourth-order method Numerical examples of the optimal fourth order methods In this section, the computations were done using Matlab 7.1 environment. We accept an approximate solution rather than the exact root, depending on the precision ϵ of the computer. We use the following stopping criteria for computer programs: |f (xn)| < ϵ, we used the xed stopping criterion ϵ = − .
"−" is divergence. "~" means that it converges to other solutions. We used the test functions and display the computed approximate zero x * in Table 1. From Table 3, it is clear that CM , CM , LTM , SBM, LTM , NSPP and LTM require less number of iterations (IT) and function evaluations (NFE) in the corresponding test function f (x) compared with the other fourth-order methods, especially the method CM performs best in terms of convergence.
In test function f (x), the methods KM , CM and LTM have better performances. The numerical results also show that KM have smaller residual error in the corresponding function |f (xn)| compared with LTM . The existing method CM fails in convergence for the case f (x).
Regarding the results of test function f (x), we claim that our methods and the existing fourth-order methods have almost similar performance.
From the results of the test function f (x), our methods LTM , LTM , LTM , LTM , LTM and the existing method KM , KM , CM , CM , CM require less number of iterations (IT) and function evaluations (NFE) than other methods, which demonstrate that several of our methods converge faster than some existing ones.
In test function f (x), the methods KM , CM , CM and our method LTM have better performances in terms of the speed of convergence. The existing methods KM and SBM fails in convergence for the case f (x). The results show that our fourth-order method LTM can compete with KTM, KM , CM , NSPP, SBM and KM .
In test function f (x), in terms of convergence, the methods KM , SBM, LTM , LTM , LTM perform slightly worse, while the method CM performs the worst.
In test function f (x), we also check the e ectiveness of our methods when we consider the same nonlinear equation with same initial approximation. Then, we nd that the methods KM , CM and our methods LTM , LTM , LTM , LTM , LTM , LTM perform better than KTM, KM , KM , CM , CM , NSPP and SBM in terms of speed of convergence for solving the nonlinear equations. However, in this particular case the method LTM don't perform better than other methods. Consequently, our fourth-order methods can compete with some fourth-order known methods, such as KTM, KM , KM , KM , CM , CM , CM , NSPP and SBM, especially the present methods LTM , LTM , LTM , LTM and LTM perform equal or better than some existing methods in many aspects.

Application to a physical problem
We consider Planck's radiation law problem [21][22] which calculates the energy density within an isothermal blackbody. In the expression of formula (83), λ is the wavelength of the radiation, T is the absolute temperature of the blackbody, B is Boltzmann's constant, P is the Planck's constant and c is the speed of light. In some cases, due to the needs of the application, it is often necessary to determine wavelength λ which corresponds to maximum energy density Φ( λ). To nd the critical points, we use the Chain Rule to di erentiate the function of equation (83), and obtain The function (86) is continuous, and it has a solution X = , which is what we do not interest. We want to obtain positive roots of the nonlinear function, so that requires us to apply iterative method to get  Table 4 are the number of iterations (IT), the number of function evaluations (NFE), the absolute residual error of the corresponding function value (|F(Xn)|), the computing time (TIME, the unit of time is one second) and the distance of two consecutive approximations δ = |Xn − X n− |, where " − " is divergence. We use the following stopping criteria for computer programs: |F(Xn)| < ϵ = − . Note that in Table 4, in terms of iterations number (IT) and function evaluations (NFE), the fourth-order methods have the same performance. KTM, KM , KM , CM , NSPP, SBM, and our methods LTM , LTM , LTM , LTM , LTM , LTM have smaller residual error in the nonlinear function as compared to the other methods of fourth-order. Our methods is slightly better at computing time. Consequently, the roots of F(X) = give the maximum wavelength of radiation λ by means of the following relation: BT . (87)

Basin of attractions
In this section, we study some dynamical properties of the family of iterative methods (45) and (46) based on their basins of attraction when they are applied to the complex polynomial P(z). We investigate the structure of the basins of attraction for comparing convergence and stability of the family of iterative methods. Here we brie y introduce some necessary dynamical concepts and basic results to be used later. Most of them can be found in the classic works such as [20,[23][24][25][26][27][28][29][30]  Let z * f be an attracting xed point of the rational function R. The basin of attraction of the xed point z * The set of points whose orbits tends to an attracting xed point z * f is de ned as the Fatou set, F(R). The complementary set, the Julia J(R), is the closure of the set consisting of its repelling xed points, and establishes the borders between the basins of attraction.
Some known and existing fourth-order methods and our fourth-order methods are considered, they are KTM (76) × points, and we apply the iterative methods starting in very z in the square. The iterative methods can converge to the root or, eventually, diverges. As an illustration, we consider the stopping criterium for convergence to be less than a tolerance ϵ = − and a maximum of iterations. If a sequence {zn} with the residual |P(zn)| < ϵ, generated by the iterative method for the initial guess z within the maximum iteration, then we decide the iterative method converges for z , otherwise we consider the method to be divergent. We take black color for denoting lack of convergence to any of the roots or convergence to the in nity.

Test problem 1. Let
Based on the Figure 1-3, we observe that the method CM is the best method in terms of less chaotic behavior on the boundary points, the methods KTM, CM , NSPP, SBM, LTM and LTM are better. From the three Figures, we nd that, with the increase of the value m of (56), the chaotic behaviors of the methods KM , KM and KM become more and more complex, which the feature of attraction basins is also re ected in our methods LTM , LTM , LTM and LTM . In the next we have taken polynomials of increasing degree.

Test problem 2. Let
i}. We conclude based on Figure 4-6 that the methods CM and LTM outperform all the others, and the methods LTM , KTM and SBM are better in terms of less chaotic behavior than other methods. However, the fractal picture of the method LTM has some non convergent points. The method CM has the most divergence points in Figure 5, so it performs worst in this test problem. Since the value of m is bigger, the method KM has the most complex behavior on the boundary points.
i, . ± . i}. In the fractal pictures from Figure 7-9, it is clear that the methods KTM, CM , NSPP, SBM and our methods LTM , LTM have the largest basins of attraction as compared to the other methods. In addition, although LTM , LTM have a small amount of no convergence points, the two methods have less chaotic behavior on the boundary points than other methods, including the known and existing fourth-order methods KTM, KM , KM , KM , CM , NSPP and SBM. In terms of the dynamical behavior on the boundary points, the method KM is most complex, followed by the methods LTM and KM . From Figures 6(a), the method LTM has more non-convergence point regions than other methods.

Conclusions
In this paper, we have designed and studied a new one-parameter family of modi ed Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the methods was also considered, and the methods have convergence order three. Based on the family of third-order method, a new optimal fourth-order family of iterative methods (in the sense of Kung-Traub's conjecture) is obtained by using weight function. We observed from numerical study that the proposed methods are e cient and demonstrate equal or better performance as compared with other well-known fourth-order methods. Finally, the dynamical analysis of this optimal fourth-order family and existing fourth-order methods have been made on some di erent polynomials, showing some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, the fractal graphics show the chaotic behaviors of our methods become more and more complex with the increase of the value m of the series in iterative methods, which also re ected in the existing fourth-order methods KM , KM and KM .