One of the most primal problem of science and engineering is locating the zeros of polynomial of degree k with arbitrary real coefficient.

(1) f(x)=∑ i=1kaixi,ak≠0.

Let ξ1,…,ξk denote all the simple zeros of Eq. (1). According to Abel’s impossibility theorem [1] “There is no solution in radical to general polynomial with arbitrary co-efficient of degree five or higher” we therefore look toward numerical schemes to approximate zeros of polynomial Eq. (1).

There exist a lot of numerical schemes in literature which approximate single zeros at a time (see, e.g., [2–4]). Here, we consider the following family of numerical schemes [5]:

(2) x(n+1)=y(n)-f(x(n))f′(x(n))((f(x(n)))2f(y(n))+2f(x(n))f(y(n))+β(f(y(n)))3(f(x(n)))3),

where y(n)=x(n)-f(x(n))f′(x(n)) and β∈R.

The numerical scheme Eq. (2) approximate single zero of Eq. (1) at a time.

Beside these single roots finding methods, mathematicians and engineers are interested in simultaneous numerical schemes which approximate all roots simultaneously. More detail on simultaneous methods, their global convergence and parallel implementation on computer algebra system (CAS) and stability are found in [6,7] and reference cite there in [8–14].

Therefore, the main aim of this research article is to construct a derivative free numerical scheme which approximates all real zero of Eq. (1). Using CAS-Mathematica, we find the lower bound of convergence to verify convergence order theoretically. Computational order of convergence [15] and convergence history for random initial approximations are graphed to show the efficiency and performance of numerical schemes as compared to other existing methods of same order. Log of residual graph, graphs of computational order of convergence and local computational order of convergence [16] support the global convergence behavior of our newly constructed numerical scheme for estimating all real zeros of Eq. (1).

Corresponding to numerical schemes

(3) y(n)=x(n)-f(x(n))f′(x(n)),

for approximating all zeros of Eq. (1), the numerical method is

(4) xi(n+1)=xi(n)-f(xi(n))∏ j≠ij=1 k(xi(n)-xj(n)).(i=1,…,k)

This numerical method is well known Weierstrass method [17] for approximating all zeros of polynomial Eq. (1) having local quadratic convergence. Using in analogical way as above, we convert Eq. (2) into simultaneous method as:

(5) xi(n+1)=yi(n)-f(xi(n))∏ j≠ij=1 k(xi(n)-xj(n))((f(xi(n)))2f(yi(n))+2f(xi(n))f(yi(n))+β(f(yi(n)))3(f(xi(n)))3),

where yi(n)=xi(n)-f(xi(n))∏ j≠ij=1 k(xi(n)-xj(n)) and β∈R. Thus, we have constructed here, a new derivative free family of numerical schemes (abbreviated as MD).

Nedzhibov et al. [18] in 2005, present the following cubic convergence derivative free family of simultaneous numerical schemes (abbreviated as ND) as:

(6) xi(n+1)=xi(n)-f(xi(n))∏ j≠ij=1 k(xi(n)-xj(n))(1+f(yi(n))f(xi(n))-2βf(yi(n))).

Here, some numerical examples from [20,21] are considered with and without random initial approximation to estimate all real zeros of polynomial equation of higher degree. All the computations are performed using Mat Lab R@2011 with 64 digits floating point arithmetic. We take ∈=10-30 as tolerance and use the following stopping criteria ei=|xi(n+1)-xi(n)|<∈, where e_i represents the absolute error. We compare our iterative schemes MD with ND of the same convergence order. In all numerical calculations, we take β=2.

Example 1: Consider a stirred tank reactor (SCRT) in which two items A and R are feed in reactor at Q and q-Q rate respectively. A complex reaction

a+r→b;b+r→c;c+r→r;c+r→e, develops the following function

(16) hc2.98(x+2.25)(x+1.45)(x+2.85)2(x+4.35)=1.

We obtained an algebraic polynomial equation of degree 4 by taking ℏc=0 in Eq. (16) f1(x)=x4+11.5x3+47.49x2+83.06325x+51.32366875=0 with exact roots: ζ1=-1.45,ζ2=-2.85,ζ3=-2.85,ζ4=-4.45

Convergence history and computational order of convergence graph (Figs. 1, 4, 7 and 10) of numerical schemes ND, MD are obtained by taking the following random initial guessed valued in our computer program i.e., X1=[0.032601;0.5612;0.88187;0.66918], where X1=[xi(0),i=1,…,4]. Using random initial guessed value X1, the iterative scheme MD converges to exact zeros after 100 iteration by consuming 16.848368 s CPU time while ND converges after 155 iterations and consumes 37.249147 s.

Convergence rates increase by taking the following initial guessed value: x1(0)=-1.0,x2(0)=-1.1,x3(0)=-2.2,x4(0)=-3.9.

The numerical results are presented in Tabs. 1–6. In all Tabs. 1–6, CO presents convergence order, CPU, presents CPU time and local computational order of convergence (LCOC) by σ.

Example 2: Consider

(17) f2(x)=x5-3x4-23x3+51x2+94x-120

with exact roots: ζ1=1,ζ2=-2,ζ3=3,ζ4=-4,ζ5=5.

For convergence history and computational order of convergence graph (Figs. 2, 5, 8 and 11) of numerical schemes ND, MD, we used the following initial guessed valued in our computer program i.e., X2=[0.81472;0.90579;0.12699;0.91338;0.63236] where X2=[xi(0),i=1,…,5]. Using random initial guessed values X2, the iterative scheme MD converges to exact zeros after 92 iteration by consuming 27.056098 s CPU time while ND converges after 98 iterations and consumes 31.444833 s.

Convergence rates increase by taking the following initial guessed value: x1(0)=0.9,x2(0)=-1.9,x3(0)=2.9,x4(0)=-3.9,x5(0)=4.9

Example 3: Consider

(18) f3(x)=x7-27x6+282x5-1410x4+3249x3-1923x2-3532x+3360

with exact roots: ζ1=-1,ζ2=1,ζ3=3,ζ4=4,ζ5=7,ζ6=5,ζ7=8.

For convergence history and computational order of convergence graph (Figs. 3, 6, 9 and 12) of numerical schemes ND, MD are obtained by taking the following initial guessed valued in our computer program i.e., X3=[0.77029;0.35022;0.66201;0.41616;0.84193;0.83193;0.83292;0.25644], where X3=[xi(0),i=1,…,7]. Using random initial guessed value X3, the iterative scheme MD converges to exact zero after 19 iterations by consuming 13.7714 s CPU time while ND converges after 23 iterations and consumes 18.120062 s.

Convergence rates increase by taking the following initial guessed value: x1(0)=-0.9,x2(0)=0.9,x3(0)=2.5,x4(0)=3.9,x5(0)=6.9,x6(0)=4.5,x7(0)=7.9.

Here, we have developed a family of derivative free method for approximating all real zeros of polynomial. Lower bound of convergence of iterative methods MD and ND are calculated using CAS-Mathematica. Using Mat Lab, we graph convergence history and computational order of convergence. From Tabs. 1–6 and Figs. 1–15, we observe that our method MD is much better in terms of convergence history, computational order of convergence, numerical results, log of residual and local computational order of convergence as compared to ND method.