Multipoint Iterative Methods for Finding All the Simple Zeros in an Interval

Two new families of multipoint without memory iterative methods with eighthand sixteenth-orders are constructed using the symbolic software Mathematica.The key idea in constructing such methods is based on producing some generic suitable functions to reduce the functional evaluations and increase the order of convergence along the computational efficiency. Again by applying Mathematica, we design a hybrid algorithm to capture all the simple real solutions of nonlinear equations in an interval. The application of the new schemes in producing fractal pictures is also furnished.


Introduction
In this work, we are concerned with the numerical solution of nonlinear equations with application.As usual, let the scalar function () be sufficiently differentiable in the open interval  and have a zero.That is,  exists such that () = 0.If   () ̸ = 0, we categorize the solution as a simple one, while if   () =   () = ⋅ ⋅ ⋅ =  (−1) () = 0 and  () () ̸ = 0, then the solution is multiple.The application of such schemes, from finding Moore-Penrose inverse to calculating the -adic inverse of a number to the module , has attracted the researchers for constructing multipoint schemes possessing a much more reliable computational efficiency; see, for example, [1][2][3][4].
Basically in the literature (see, e.g., [5,6]), two important features determine the choice of an iteration method for solving the equation () = 0 as follows: first the total number of iterations and second the computational cost.The former is measured by the order of convergence and the latter by the necessary number of evaluations of the function  and its derivatives per full cycle.These two characteristics are linked by the concept of computational efficiency index, defined by Traub in [7] as   √ wherein  is the order and  is the number of functional evaluations per iteration by considering that both function and derivative evaluations have the same computational cost.It should be remarked that, based on the still unproven Kung-Traub conjecture [8], an iterative method without memory with  (functional) evaluation in each iteration has the maximum order of convergence 2 (−1) (called optimal order).The aim of this work is threefold.First is to apply the programming package Mathematica so as to obtain some new multipoint iterative methods of optimal orders eight and sixteen, including three and four steps, respectively.
The second is the application of the new schemes in computer graphics.In fact, each iterative fixed-point type method produces unique basins of attraction and fractal behavior, which results in nice pictures, useful in arts as discussed fully by Kalantari in [9].
The only drawback of such iterative schemes is in the choice of the initial guess (seed) to start the process and guarantee the convergence.In this paper, an algorithm will be constructed to provide a list of simple real initial approximations and then to obtain all the simple real zeros up to any desired tolerance, when high precision computing is needed.This would be the third goal of the present work.

Journal of Applied Mathematics
The rest of the paper unfolds the contents as follows.Section 2 gives a generic three-step family of methods, while Section 3 is devoted to propose a new general family of fourstep optimal methods along its analysis of convergence.The fractal behaviors for one of the new methods in producing nice self-similar pictures in arts will be given in Section 4. In Section 5, we remind of the approach of Wagon [10] for extracting a list of initial approximations for simple real zeros along some notes and a hybrid algorithm for computing all the simple real zeros.Some numerical examples will also be pointed out therein.Section 6 concludes the paper.

A New Optimal Eighth-Order Method with Two Generic Functions
At first in this section, we aim at constructing an optimal eighth-order method without memory, which is used as the first three steps in constructing our four-step optimal sixteenth-order family in the next section.
To this end, consider the following three-point method (without the index ): The first two steps are optimal since that is Ostrowski's fourth-order two-step iterative method, since it reaches the order four using three functional evaluations.But, the third step is not good enough.This method performs five functional evaluations per cycle, which does not coincide with Kung and Traub conjecture for iterations without memory.In fact, functional evaluations must be reduced such that it consumes four instead of five.
Note that we omit the index  for simplicity only.Substituting (2) into (1), we have Although the iteration (3) uses four functional evaluations per full cycle, generally it does not have convergence order eight.It is necessary to find some suitable conditions on the generic functions (, ) and () so that iteration (3) attains an optimal eighth order.The approach of undetermined coefficients, Maclaurin series and Mathematica [11] are our means.For simplicity, we consider the following truncated Maclaurin series: where  , = ( + (, )/    )| (0,0) for ,  = 0, 1, 2, . .., and wherein (2) Its error equation reads Moreover, we obtain From ( 9) and (10), follows Also, ( ).We now need   = (  )/(  ),   = (  )/(  ), and   = (  )/(  ).Then, while It is remarked that although we have not presented higher order terms in the above expressions, one can obtain them simply by the aid of a system of computation such as Mathematica.In the third step of (3), we need to consider Taylor's series of (  ,   ) and (  ) about (0, 0) and 0, respectively.To our purposes, it suffices to use the following: So, we get the following general error equation if  is subtracted from both sides of (3): where Moreover, note that the first two steps are optimal because it is Ostrowski's method.Therefore, we find some necessary conditions on the generic functions (  ,   ) and (  ) in such a way that all the coefficients of    in the general error equation become zero for  = 4, 5, 6, 7. To this end, we proceed as follows.
Vanishing the coefficient of  7  states that, by considering the above conditions for the weight functions (a.k.a.generic functions), we then could have an error equation of the form ).This error reveals that we have constructed an optimal eighthorder scheme and the proof is completed.
At this time, we introduce specific sets of generic functions, satisfying obtained conditions (6) and also applying them to (3) for constructing concrete methods.For example, we could have which satisfy the conditions of Theorem 1.Thus, one iteration scheme can be defined as We could also have And also () = (2 + )/(2 − ).

A New Optimal Sixteenth-Order Family
This section is concerned with construction of a new sixteenth-order family.We add Newton's step to the obtained concrete method (18) in the previous section as follows: Although the first three steps are optimal and iteration (19) has convergence order sixteen, the whole procedure is not optimal since it is not satisfying Kung and Traub conjecture.The iteration (19) reaches order 16 with 6 functional evaluations.To obtain an optimal method, we must reduce one functional evaluation.To this end, our goal is to approximate   () in terms of the other existing function values, that is, (), (), (), (), and   (), that is similar to the case of Section 2. Finding this approximation sounds easy at first, but when we tried to do, it was not simple at all.In fact, after trying many generic functions we had succeeded.
Note that although we have performed all the developments by considering :  ⊆  → , all the theorems could be extended and deduced if the function  is defined in the complex plane as :  ⊆  → , or having a complex zero.In such case a complex seed (initial guess) is needed for converging.

Application in Art
Kalantari in [9] coined the term polynomiography as a clear application of fixed-point iterative methods in producing beautiful fractal pictures, which are in use in computer graphics and art.Polynomiography is defined to be the art and science of visualization in approximation of the zeros of complex polynomials, via fractal and nonfractal images created using the mathematical convergence properties of iteration functions.
As is known according to the Fundamental Theorem of Algebra, a polynomial of degree , with real or complex coefficients, has  zeros (roots) which may or may not be distinct.
In this section, using machine precision and the computer programming package Mathematica 8 (see, e.g., [16]), we produce some of such beautiful fractals obtained from our new methods.In fact, an iteration function is a mapping of the plane into itself; that is, given any point in the plane, it is a rule that provides another point in the plane.This section is necessary in this paper to show how the new schemes could be considered in polynomiography.
Consider a polynomial () and a fixed natural number  ≥ 2. The basins of attraction of a root of () with respect to the iteration function   () are regions in the complex plane such that, given an initial point  0 within them, the corresponding sequence  +1 =   (  ),  = 0, 1, . .., will converge to that root (see for more details [17,18]).
It turns out that the boundary of the basins of attraction of any of the polynomial roots is the same set.This boundary is known as the Julia set and its complement is known as the Fatou set.
From now on, a complex rectangle  = [−3, 3] × [−3, 3] ⊆  will be considered and we subsequently assign a color to each complex point  0 ∈  according to the root, at which the corresponding method starting from  0 converges.We assign a color to each attraction basin of a root.But, we further make the color lighter or darker according to the number of iterations needed to reach the root with the fixed precision required 10 −2 .
An important aspect of the work of Kalantari which emerged in his Carpet design ([9]) is in applying iterations functions for higher order polynomials with various styles of coloring.We herein use six different polynomials with different coloring forms in Mathematica 8. Toward such an option, we used three different colorings as indicated in Figure 1.The results of basins of attraction (by ( 18)), which provide beautiful pictures, are given in Figures 2, 3, and 4, via coloring described in Figure 1.

All the Simple Real Zeros
The fixed-point type iterative methods might diverge if the conditions of the main theorems given in Sections 2 and 3 fail.Actually, an important advantage of multipoint optimal schemes, as the ones in Sections 2 and 3, is that their convergence could not be achieved without having a robust approximation of the position of the zeros.This sometimes is referred to as the most important difficulty of iterative methods in practical problems.Besides, in practice one wants to find all the real solutions of a nonlinear function at the same time.
To remedy such shortcomings, only a few approaches such as interval methods (see, e.g., [19] or [20]) have been    given to the literature up to now.For instance, Yun in [21] applied a numerical integration based technique toward this goal.His technique is robust, but it might be time consuming in case of having so many zeros in an interval.
Herein we provide a predictor-corrector algorithm based on Mathematica 8 software consisting of two parts.In the first part, we apply the procedure given by Wagon in [10] for extracting robust initial approximations for all the simple real zeros.And second is to apply any of the presented optimal methods in this paper as a corrector to increase the accuracy of the solution in a short piece of time, when high precision computing is needed.To illustrate the procedure, we start by the fact that we attempt to find all the axis crossings; that is to say, we will not attempt to capture simple zeros at which there is no axis crossing.
Toward this end and as Wagon did in [10], we can use the options of plotting as a function and to find the crossings.Thus, one picks out the points from the normal graphics form of the plot .Note that the setting of MeshFunctions must be a pure function and cannot be just .
To further illustrate, consider the following example in which the whole of the procedure is to extract all the simple real zeros of a nonlinear equation f We used PerformanceGoal-> "Quality" and WorkingPrecision-> 128 in order to obtain high reliable initial approximations for the crossing of the function to the -axis.Note that some of such considerations for some problems could also be changed adaptively for providing better feedbacks.In what follows, we extract the initial guesses, sort them, and obtain the (estimate) number of zeros in the considered interval (see Algorithm 2).
The plot of the function  is given in Figure 5, which clearly reveals the difficulty in obtaining all the real solutions.But the hybrid algorithm descried in this section could be applied for all the real simple solutions.For this test, we have 40 zeros which must be found in high precision accuracy.The initial list of approximations would be {−0.995857,−0.989387, −0.972279, −0.953544, −0.92376, −0.905373,In the above piece of code, each member of the list of initial approximations obtained from the predictor step of our algorithm will be corrected until its residual, that is, |(  )| ≤ 10 −1000 , while we work with 2000 number of fixed point arithmetics.Note that the other iterative solvers could be coded in a similar way.
An interesting point is that one may ask that higher optimal order is equal to higher computational load and thus the whole procedures might be inefficient in terms of computational time.We clearly respond that there is no such thing for optimal schemes and especially when we mix them with a convergence guaranteed algorithm; that is to say, the computation of functional evaluations is much more expensive than operational cost when working in high Without the theoretical number of real zeros in the desired interval, it would be also difficult to choose the number of initial guesses.It must be noted that this approach must only be taken into account for finding the real simple zeros of nonlinear equations.In fact, if the nonlinear function has zeros with multiplicity, then the fast approach of Wagon must be replaced by a verified method, as the ones discussed in [22] to extract all the simple and multiple real solutions of a nonlinear function as the seeds for the simple and multiple zero-finders developed in this paper.

Concluding Remarks
This paper has contributed two optimal eighth-and sixteenth-order methods for solving nonlinear equations using generic functions (weight function) in the computer programming package Mathematica 8.It was observed that any derived method from the new optimal methods possesses 1.682 and 1.741 as the optimal computational efficiency indices.
From Tables 1 and 2 and the tested examples, we can conclude that all implemented methods for nonlinear functions converge fast in a shorter piece of time in contrast to the lower order schemes when a list of powerful initial guesses is available for all the simple real zeros in the interval.
The application of one of the new schemes, that is, (18), was given by producing beautiful fractal pictures useful in computer graphics as Kalantari mentioned.We note that keeping tighter conditions will produce fractals with much more quality and reliability.For further application, refer to [4,23,24].
We also used the programming package Mathematica 8 in our calculations and gave the necessary cautions and pieces of codes for the users to implement them in their own problems as easily as possible.We have designed a hybrid algorithm to capture all the simple real solutions of nonlinear equations as rapidly as possible using a similar technique introduced by Stan Wagon.The algorithm worked efficiently for very hard test problems.And thus, the proposed algorithm could be easily used in practical problems.(A.1)

Figure 5 :
Figure 5: The plot of the function () along with its finitely many simple real zeros on the interval [−1, 1].
1, 2, . ... Note that we defined the truncated Maclaurin series (in the above way purposely) as if to construct the most general iterative methods without memory in the programming package Mathematica.

Table 2 :
Comparison of different optimal methods for finding all the zeros of g[x ].We have 47 zeros in this interval which shows the difficulty of finding all the simple real zeros.Hopefully the hybrid algorithm given above works well in this case for finding all the simple real zeros in the interval.Again, the new optimal sixteenth-order method beats the other schemes.