One of the ancient problems in mathematics is the estimations of roots of non-linear equation

(1) f(η)=0.

There are number of applications of non-linear equation in science and engineering. Newton’s method is a numerical iterative scheme which finds a single root at a time. The simultaneous iterative method (SIM) such as, Weirstrass [1] method is used to find all the distinct roots of Eq. (1). The iterative methods for finding single root of non-linear polynomial equation have been studied by [2–4] and many others. On the other hand, there are lot of numerical iterative methods devoted to approximate all roots of Eq. (1) simultaneously (see, e.g., [1,5–8] 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 [9,10–13] 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).

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

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 ℜf:C→C , where C is a complex plane, the orbit η0∈C defines a set such as, orb(η)={η0,ℜf(η0),ℜf2(η0),...,ℜfm(η0),...} . The convergence orb(η)→η∗ is understood in the sense if limk→∞⁡ℜ(η)=η∗ exist. A point η0∈C known as attracting, if |ℜk′(η)|<1 . An attracting point η∗∈C defines basins of attraction ℜ(η∗) as the set of starting points whose orbit tends to η∗ . To generate basins of attraction, we take grid 2000×2000 of square [−2.5×2.5]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(ηi)|<10−5 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 f1(η)=η4+η2+η−1 and f2(η)=η6+η−1 and f3(η)=sin⁡(η−12)sin⁡(η−22)sin⁡(η−2.52). The root of f1(η) are 0.2 + 1.3i, 0.2 − 1.3i, −1, 0.5, roots of f2(η) are −1.1, −0.4 + 1i, −0.4 − 1i, 0.6 + 0.7i, 0.6 − 0.7i, 0.7 and root of f3(η) are 1, 2, 2.5 correct up to 1-decimal place. Brightness in color in Figs. 1–9 means less number of iterations. Finally, in Fig. 10, we present Elapsed time of basins of attraction corresponding to iterative map KF, SIM1 and SPJ1 using tic-toc command in MATLAB (R2011b).

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 (E¯) is used to estimate the efficiency of iterative method as:

(9) E¯(m)=log⁡rG,

where G in [14], denotes the cost of computation and r, the order of convergence.

(10) G=G(m)=wasASm+wmMm+wdDm.

Thus, Eq. (9) becomes:

(11) E¯(m)=log⁡rwasASm+wmMm+wdDm.

Using Eq. (11) and data in Tab. 1, we find the percentage ratio Ω(SIM1,SPJ1) [14] as:

(12) Ω(SIM1,SPJ1)=(E¯(SIM1)E¯(SPJ1)−1)×100

(13) Ω(SPJ1,SIM1)=(E¯(SPJ1)E¯(SIM1)−1)×100

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).

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 ∈ =10−30 as tolerance and use as a stopping criteria.

e⌣i=‖ηi(t+1)−ηi(t)‖2

Tests examples from [16–18] are provided in Tabs. 2–3. In all Tables, CO denotes the order of convergence, α , 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 SIM1 and SPJ1 for the numerical test examples 1−2 , shows that method SIM1 is more efficient as compared to SPJ1 . We observe that numerical results of the method SIM1 are comparable with SPJ1 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 η1(0),η2(0),η3(0),...,ηn(0) for t = 0, such that

σi(t)=ηi(t)−f(ηi(t))Πnj≠ij=1⁡(ηi(t)−zj(t)).,(i,j=1,2,...,n),wherezj(t)=uj(t)−(f(σj(t))f′(vj(t))(σj(t)−ηj(t)+f(ηj(t))f(ηj(t))−f′(uj(t))ηj(t)−uj(t))(f(ηj(t))−f(uj(t)))(f′(vj(t))−f(uj(t))))+(f(σj(t))f(σj(t))−f(uj(t))σj(t)−uj(t)),anduj(t)=σj(t)−(f(σj(t))f′(vj(t))(f′(vj(t))−f(σj(t)))(f(ηj(t))−f(σj(t))ηj(t)−σj(t)))(f(σj(t))f′(ηj(t))),σj(t)=ηj(t)−αf(ηj(t))2f′(vj(t))−f(ηj(t)),vj(t)=ηj(t)+αf(ηj(t)).

Step 2: Set ηi(t+1)=σi(t)

Step 3: For a given 0,\>if\eta _i^{\left( {t + 1} \right)} - \eta _i^{\left( t \right)}{_2} < \in $]]>∈>0,if‖ηi(t+1)−ηi(t)‖2<∈ , then stop.

Step 4: Set t=t+1 and go to Step 1.

Example 1 [18]:

Consider

(14) f4(η)=eη(η−1)(η−2)(η−3)−1

with exact roots:

ζ1=0,ζ2=1,ζ3=2,ζ4=3.

The initial estimates have been taken as:

η1(0)=0.1,η2(0)=0.8,η3(0)=1.8,η4(0)=2.9,

Example 2 [17]:

Consider

(15) f5(η)=η3+5η2−4η−20+cos⁡(η3+5η2−4η−20)−1

with exact roots are ζ1=−5,ζ2=−2,ζ3=2.

The initial estimates have been taken as:

η1(0)=−5.1,η2(0)=−1.8,η3(0)=1.9.

Example 3 [16]:

The acidity of a saturated solution of magnesium hydroxide in hydrochloric acid HCl is given by

(16) 3.64×10−11[H3O+]=[H3O+]+3.6×10−4

for the hydronium ion concentration [H3O+]. If we set η=104[H3O+], we obtained the following non-linear equation

(17) f6(η)=η3+3.6η2−36.4

with exact roots are 2.4 , −3.0±2.3 up to one decimal places. The initial estimates have been taken as: η1(0)=2.45,η2(0)=−3.0261+2.3834i,η3(0)=−3.0261−2.3834i.

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.