On highly efficient derivative-free family of numerical methods for solving polynomial equation simultaneously

A highly efficient new three-step derivative-free family of numerical iterative schemes for estimating all roots of polynomial equations is presented. Convergence analysis proved that the proposed simultaneous iterative method possesses 12th-order convergence locally. Numerical examples and computational cost are given to demonstrate the capability of the method presented.


Construction of simultaneous method
Consider well-known three-step Newton methods [26] of convergence order eight as follows: where and replacing where * where . Thus, we have constructed a new simultaneous iterative method (12), which is abbreviated as NIM12.

Convergence aspect
In this section, we prove that method NIM12 has local convergence order 12.
Theorem 1 Let ζ 1 , . . . , ζ n be the n simple roots of (1). If r (0) 1 , . . . , r (0) n are the initial estimates of the roots respectively and sufficiently close to actual roots, then NIM12 has a convergence order 12.
iζ i be the errors in r i , s i , u i , and v i , respectively. From (12), the first step of NIM12, we have where [2]. For a simple root ζ and small enough , Thus, (13) gives From the second step of NIM12, we have where , For a simple root ζ and small enough , s j | is bounded away from zero, so Since from (15), i = O ( ) 3 . Thus, From the third step of NIM12, we have where With the same argument used in (16), we have Therefore, Since from (17) i =O( ) 6 , we obtain Hence, (21) proves 12th order convergence.

Computational aspect
In this section, we compare the computational efficiencies of methods NIM10 and NIM12. As presented in [11], we can formulate the efficiency indices as follows: where The cost of computation is represented by Q [11] and convergence order by r given as Using the expression of Q in (24), we have The number of operations of real arithmetic of a complex polynomial with real and complex roots reduces to operations of real arithmetic as given in Table 1. Figure 1(a)-(b) shows the percentage ratios of NIM10 and NIM12. It is evident from Fig. 1(a)-(b) that NIM12 is much better than NIM10. Figure 1(a)-(b) shows the computational efficiency of simultaneous method NIM12 and NIM10 with respect to each other. Figure 1(a)-(b) clearly shows the dominance efficiency of our newly constructed method NIM12 over NIM10.

Numerical results
For numerical calculations, we use the following stopping criteria to terminate the computer programme using Maple 18 with 125-digit floating point arithmetic: where e (t) i represents the absolute error. In all the tables, CPU means computational time in seconds. In all numerical calculations, we take α = 12/130.

Application in engineering
In this section, we also discuss some applications from engineering.
with exact roots The initial guessed values have been taken as follows: (0) r 8 = -2.2 + 0.7i. Table 2 evidently illustrates the supremacy behavior of NIM12 over NIM10 in estimated absolute error and in CPU time on the same number of iterations n = 3 for guesstimating all roots of the nonlinear polynomial equation used in Example 1. Example 2 ( [28] Fractional conversion) The expression described in [29,30] f 2 (r) = r 4 -7.79075r 3 + 14.7445r 2 + 2.511r -1.674 (27) is the fractional conversion of nitrogen, hydrogen feed at 250 atm. and 227k. The exact roots of (27) are: The initial calculated values of (27) have been taken as follows: (0) r 4 = 1.8 + 0.01i. Table 3 evidently illustrates the supremacy behavior of NIM12 over NIM10 in estimated absolute error and in CPU time on the same number of iterations n = 4 for guesstimating all roots of the nonlinear polynomial equation used in Example 2.

Example 3 ([27]
Continuous stirred tank reactor (CSTR)) An isothermal CSTR is considered here. Items E 1 and E 2 are fed to the reactor at rates of R and q-R, respectively. Complex chain reactions are developed in the reactor given as follows: This problem was first tested by Douglas (see [31]), and the following equation of transfer function of the rector was found: The transfer function has four negative real roots, i.e., r 1 = -1.45, r 2 = -2.85, r 3 = -2.85, r 4 = -4.45.
The initial calculated values of (29) have been taken as follows: (0) r 1 = -1.0, r 4 = -3.9. Table 4 evidently illustrates the supremacy behavior of NIM12 over NIM10 in estimated absolute error and in CPU time on the same number of iterations n = 4 for guesstimating all roots of the nonlinear polynomial equation used in Example 3.

Conclusion
We have developed here a family of three-step simultaneous methods of order 12 which is the highest order derivative-free simultaneous iterative method among existing methods in the literature. From Tables 1-4 and Fig. 1(a), (b), we observe that our family of derivative-free simultaneous methods NIM12 is admirable in terms of efficiency, CPU time, and residual errors as compared to the NIM10 method.