Abstract
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 . 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.
1. Introduction
One of the most important problems in numerical analysis is to find the solution(s) of the nonlinear equations. Given , a close approximation for the root of a nonlinear equation , Newton’s method [1–3], defined asproduces a sequence that converges quadratically to a simple root of . Many variations of Newton’s method were recently published [4–6] 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 . This method, starting with sufficiently close to the root , is defined asfor and with . 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].
2. Development of the Method
Suppose that is a root of a nonlinear equation , where is a scalar function for an open interval . Consider that is close to and ; then we defineNewton’s method.
Now, as can be seen in Figure 1(a), if is a convex function, we have the following inequality:then using the two points and , we have the following line equation:thus, we can define
(a) Convex case
(b) Concave case
In this way, given we define, , andwith , which uses two functional evaluations.
In the case of Figure 1(b), the concave case, we have the inequality and analogous to the case of convex function we have the same result.
3. Convergence Analysis
Theorem 1. Let be a sufficiently differentiable function and let be a simple zero of in an open interval , with on . If is sufficiently close to , then the method NSM, as defined by (7), has convergence order equal to .
Proof. We consider the following equalities,and the following developments of Taylor’s polynomial around the root :where , , and for some . Using (7) and (8), we haveUsing (9), we have, in the denominator, , whereAnd, in the numerator, we have , whereSimplifying and dividing, numerator and denominator by , and considering that the quotient has order at least , we haveNow, suppose that is asymptotic to , with . Consequently, by expressing (13) in terms of , we obtainIn order to satisfy the previous asymptotic equation, it is evident that has to be a positive root of , that is, , the convergence order of the method. Moreover, the asymptotic constant can be calculated by
4. 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 and 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 , 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.
5. Dynamical Analysis
For the study of the concepts on complex dynamics [10, 11] we take a rational function , where is the Riemann sphere. For , we define its orbit as the set . A point is called periodic point with minimal period if , where is the smallest integer with this property. A periodic point with minimal period is called fixed point. Moreover, a point is called attracting if , repelling 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 [14–16]:(1);(2);(3);(4);(5);(6);(7);(8).
For the dynamical analysis of iterative method, we usually consider the region of the complex plane, with points, and we apply the iterative method starting in every in this region. If the sequence generated by iterative method reaches zero of the function with a tolerance and a maximum of iterations, we decide that 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, , we have and the NSM given by (7) can be written as
Observe that expression (17) can be calculated always for all values except if for initial step that uses Newton’s method. Expression (17) gives us the strength of method (7) in the case of function and this is represented in Figure 2(a).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
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.
6. 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].
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.