Computer Methodologies for the Comparison of Some Efficient Derivative Free Simultaneous Iterative Methods for Finding Roots of Non-Linear Equations

In this article, we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear equations. Convergence analysis proved that the order of convergence of the family of derivative free simultaneous iterative method is nine. Our main aim is to check out the most regularly used simultaneous iterative methods for finding all roots of non-linear equations by studying their dynamical planes, numerical experiments and CPU time-methodology. Dynamical planes of iterative methods are drawn by using MATLAB for the comparison of global convergence properties of simultaneous iterative methods. Convergence behavior of the higher order simultaneous iterative methods are also illustrated by residual graph obtained from some numerical test examples. Numerical test examples, dynamical behavior and computational efficiency are provided to present the performance and dominant efficiency of the newly constructed derivative free family of simultaneous iterative method over existing higher order simultaneous methods in literature.


Introduction
One of the ancient problems in mathematics is the estimations of roots of non-linear equation the other hand, there are lot of numerical iterative methods devoted to approximate all roots of Eq. (1) simultaneously (see, e.g., [1,[5][6][7][8] and the references therein). The SIM are popular as compared to single root finding methods due to their wider range of convergence, reliability and their applications for parallel computing as well. Further details on SIM, their convergence analysis, efficiency and parallel implementations can be seen in [9,[10][11][12][13] and references cited there in. The main objective of this article is to construct SIM which have more efficient and higher convergence order for approximating all distinct roots of Eq. (1). For the analysis and comparison of convergence behavior of simultaneous iterative methods, we use the techniques of dynamical plane with CAS MATLAB (R2011b).

Constructions of Simultaneous Method
Here, we construct a ninth order derivative free simultaneous method which is more efficient than the similar methods existing in literature.

Construction of Simultaneous Methods for Distinct Roots
Consider eighth order derivative free Kung-Traub's [4] family of iterative method (abbreviated as KF):

Convergence Analysis
Here, we discuss the convergence of iterative method SIM1: Theorem: Let f 1 ; f 2 ; …; f n be n simple roots of Eq. (1). If g 2 ; …; g ð0Þ n be the sufficiently close initial approximations to actual roots, then the order of convergence of SIM1 is nine. Proof: Let e i ¼ g i À f i ; e 0 i ¼ r i À f i ; be the errors in g i and r i approximations respectively. For simplification, we omit iteration index t. From SIM1, we have: where Now, if f i is a simple root, then for a small enough e, g i À v j is bounded away from zero, and so where z j À f j ¼ Õ ðe 8 Þ, see [4]: S i À 1 ¼Õðe 8 Þ: Thus, Eq. (4) gives: Hence, the theorem is proved.

Dynamical Studies of KF, SIM1 and SPJ1
In this section, we discuss the dynamical study of KF, SIM1 and [14] method (abbreviated as SPJ1). We have discussed the dynamical behavior of simultaneous methods to show global convergence as dynamical planes of single root finding methods may have divergence regions which do not exist in simultaneous methods. Let us recall some basic concepts of this study. For more details on the dynamical behavior of the iterative methods one can consult [2] and [15]. Taking a rational map < f : C ! C, where C is a complex plane, the orbit g 0 2 C defines a set such as, orbðgÞ An attracting point g Ã 2 C defines basins of attraction <ðg Ã Þ as the set of starting points whose orbit tends to g Ã . To generate basins of attraction, we take grid 2000 Â 2000 of square ½À2:5 Â 2:5 2 2 C. To each root of Eq. (1), we assign a color to which the corresponding orbit of the iterative methods starts and converges to a fixed point. Take color map as Jet. We take f ðg i Þ j j< 10 À5 and maximum numbers of iterations are chosen as 5 due to wider convergence region of simultaneous methods. Dark black points are assigned, if the orbit of the iterative methods does not converge to root after 5 iterations. We obtained basins of attractions for the following three test function f 1 ðgÞ ¼ g 4 þ g 2 þ g À 1 and f 2 ðgÞ ¼ g 6 þ g À 1 and f 3 ðgÞ ¼ sin gÀ1

Computational Aspects
Here, we discuss the computational efficiency and convergence behavior of the [14] method (abbreviated as SPJ1) and the new method SIM1. As presented in [14], the efficiency index ð " EÞ is used to estimate the efficiency of iterative method as: where G in [14], denotes the cost of computation and r, the order of convergence.  Using Eq. (11) and data in Tab. 1, we find the percentage ratio ðSIM 1; SPJ 1Þ [14] as: Figs. 11-12, graphically illustrates these percentage ratios. Figs. 11-12, clearly show that the newly constructed simultaneous method SIM1 is more efficient as compared to Petkovic method (SPJ1).

Numerical Results
Here, some numerical test examples are considered in order to show the performance of simultaneous ninth order derivative free method SIM1. We compare our method with [14] method (SPJ1) of convergence order ten for distinct roots. All the numerical calculations are done by using Maple 18 with 64 digits floating point arithmetic. We take 2 ¼ 10 À30 as tolerance and use as a stopping criteria.
Tests examples from [16][17][18] are provided in Tabs. 2-3. In all Tables, CO denotes the order of convergence, a, parameter valued in SIM1, n; the number of iterations and CPU , execution time in seconds. Figs. 13-16, show that residue fall of the methods SIM 1 and SPJ 1 for the numerical test examples 1 À 2, shows that method SIM1 is more efficient as compared to SPJ 1. We observe that numerical results of the method SIM 1 are comparable with SPJ 1 method on same number of iteration.     We also calculate the CPU execution time, as all the calculations are done using Maple 18 on (Processor Intel(R) Core(TM) i3-3110m CPU@2.4 GHz with 64-bit Operating System). We observe from Tables that CPU time of the methods SIM1 is comparable or better than method SPJ1, showing the efficiency of our family of derivative free methods SIM1 as compared to them. Algorithm for simultaneous iterative method Step 1: Given g Step 2: Set g Step 3: For a given 2 > 0; if g Step 4: Set t ¼ t þ 1 and go to Step 1.

Example 1 [18]:
Consider f 4 ðgÞ ¼ e gðgÀ1ÞðgÀ2ÞðgÀ3Þ À 1 (14) with exact roots: The initial estimates have been taken as: with exact roots are f 1 ¼ À5; f 2 ¼ À2; f 3 ¼ 2: The initial estimates have been taken as: The acidity of a saturated solution of magnesium hydroxide in hydrochloric acid HCl is given by for the hydronium ion concentration we obtained the following non-linear equation with exact roots are 2:4 , À3:0 AE 2:3i up to one decimal places. The initial estimates have been taken as:

Conclusions
We have developed here derivate free family of simultaneous methods of order nine for determining all the roots of non-linear equations. It must be pointed out that so far there exists derivative free method of order four only in the literature. We have made here comparison with method SPJ1 of order 10 involving derivative. The dynamical behavior/basins of attractions of our family of simultaneous methods SIM1 is also discussed here to show the global convergence. An example of single root finding derivative free method of order 8 of King-Traub is discussed to show that the single root finding methods may have divergence region. The computational efficiency of our method SIM1 is very large as compare to the method SPJ1 as given in Tabs. 2-4, which is also obvious from Figs. 11-12. We have made the numerical comparison with SPJ1 method. From Tabs. 2-4 and Figs. 1, 4, 7, 13-18, we observe that our numerical results are comparable or better in term of absolute error, number of iterations and CPU time and for log of residual graphs and lapsed time of dynamical planes.