A Novel Geometric Modification to the Newton-Secant Method to Achieve Convergence of Order 1 + √ 2 and Its Dynamics

A geometric modification to the Newton-Secant method to obtain the root of a nonlinear equation is described and analyzed.With the same number of evaluations, the modified method converges faster than Newton’s method and the convergence order of the new method is 1 + √2 ≈ 2.4142. The numerical examples and the dynamical analysis show that the new method is robust and converges to the root in many cases where Newton’s method and other recently published methods fail.


Introduction
One of the most important problems in numerical analysis is to find the solution(s) of the nonlinear equations.Given  0 , a close approximation for the root of a nonlinear equation () = 0, Newton's method [1][2][3], defined as produces a sequence that converges quadratically to a simple root  of () = 0.Many variations of Newton's method were recently published [4][5][6] with different kinds of modification on Newton's method.In particular, in [7], the authors presented a two-step iterative method with memory based on a simple modification on Newton's method with the same number of function and derivative evaluations but with convergence order 1 + √ 2. This method, starting with  0 sufficiently close to the root , is defined as for  ≥ 0 and with  * −1 =  −1 =  0 .Observe that, in the first step, this method realizes three functional evaluations and only two in the other iterations steps.For further explanations of this method we can see [8].

Convergence Analysis
Theorem 1.Let  :  ⊂ R → R be a sufficiently differentiable function and let  ∈  be a simple zero of () = 0 in an open interval , with   () ̸ = 0 on .If  −1 ∈  is sufficiently close to , then the method NSM, as defined by (7), has convergence order equal to 1 + √ 2.

Numerical Examples
In this section we check the effectiveness of the new method (NSM) introduced in this paper and compare this with Newton's classical method (NM) and McDougall's method (MM).All computations are done using ARPREC C++ [9].The iteration is stopped if one of the stopping criteria | +1 −   | < 10 −100 and |( +1 )| < 10 −100 is satisfied.We test the different iterative methods using the following smooth functions that are the same as those used in [4,6,7]: Table 1 presents the number of iterations (), the estimated error of the iterations | +1 −   |, and the number of functional evaluations (NOFE) of the different methods.
The estimated convergence order of the proposed methods is always equal to or better than that of the other iterative methods.The number of iterations is sometimes lower.When the number of iterations is the same, the estimated error of the last iterate is also lower.

Dynamical Analysis
For the study of the concepts on complex dynamics [10,11] we take a rational function  : Ĉ → Ĉ, where Ĉ is the Riemann sphere.For  ∈ C, we define its orbit as the set orb() = , (),  2 (), . ... A point  0 ∈ C is called periodic point with minimal period  if   ( 0 ) =  0 , where  is the smallest integer with this property.A periodic point with minimal period 1 is called fixed point.Moreover, a point  0 is called attracting if and neutral otherwise.The Julia set of a nonlinear map (), denoted by (), is the closure of the set of its repelling periodic points.The complement of () is the Fatou set (), where the basins of attraction of the different roots lie.We use the basins of attraction for comparing the iteration algorithms.The basin of attraction is a method to visually comprehend how an algorithm behaves as a function of the various starting points [12,13].In this section, polynomial and rational functions have been considered, which are the same functions that appear in [14][15][16]: (1)  1 () =  2 − 1; (2)  2 () =  3 − 1; (3)  3 () =  4 − 1; (4)  4 () =  5 − 1; (5)  5 () =  2 − 1/; (6)  6 () = ( 3 − 1)/(2 + 1); For the dynamical analysis of iterative method, we usually consider the region [2, 2] × [2,2] of the complex plane, with 400×400 points, and we apply the iterative method starting in every  −1 in this region.If the sequence generated by iterative method reaches zero  of the function with a tolerance | +1 − | < 10 −3 and a maximum of 100 iterations, we decide that  −1 is in the basin of attraction of these zeros and we paint this point in a color previously selected for this root.In the same basin of attraction, the number of iterations needed to achieve the solution is showed in different colors.Black color denotes lack of convergence to any of the roots (with the maximum of iterations established) or convergence to the infinity.
For example, for the first function,  1 () =  2 −1, we have   () = 2 and the NSM given by ( 7) can be written as Observe that expression (17) can be calculated always for all values  −1 ,   except if  −1 = 0 for initial step that uses Newton's method.Expression (17) gives us the strength of method (7) in the case of function  1 and this is represented in Figure 2(a).
In this section, we observe that the schemes for the new method have a simple boundary of basins.We also found that the new iterative method has no chaotic behavior.Based on figures we also observe that method has no diverging points (black area).Finally, our method has lower number of diverging points and large basins of attraction.

Conclusion
In this paper, we have developed a new iterative method based on a geometric modification of Newton-Secant method to find simple root of nonlinear equations.New proposed method is obtained without adding more evaluations.Numerical and dynamical comparisons have also been presented to show the performance of the new method.From numerical and graphical comparisons, we can conclude that the new method is efficient and robust and gives tough competition to some existing methods.Finally, further research is needed to implement these new iterative methods in solving systems of nonlinear equations.Such implementations may be based on divided difference operator of order 1 or 2 in the sense of [17,18].

Figure 2 :
Figure 2: Basins of attraction for the functions.

Table 1 :
Numerical results for smooth functions.