Inverse Family of Numerical Methods for Approximating All Simple and Roots with Multiplicity of Nonlinear Polynomial Equations with Engineering Applications

Department of Mathematics and Statistics, Riphah International University I-14, Islamabad 44 000, Pakistan Department of Mathematics, National University of Modern Languages, Islamabad, Pakistan Deperament of Mathematics, Yildiz Technical University, Faculty of Arts and Science, Esenler 34 210, Istanbul, Turkey Science and Math Program, Asian University for Women, Chattogram 4000, Bangladesh School of Mechanical and Mechatronic Engineering University of Technology Sydney (UTS), Sydney, Australia


Introduction
Considering nonlinear polynomial equation of degree n, f(r) � r n + a n−1 r n− 1 + · · · + a 0 � Π n j�1 r − ζ j � r − ζ i * Π n j≠i j�1 r − ζ j , (1) with arbitrary real or complex coefficient a n−1 , . . . , a 0 . Let ζ 1 . . . ζ n denote all the simple or complex roots of (1) with multiplicity σ 1 . . . σ n . Newton's method [11] is one of the most basic and ancient methods that is used to estimate single roots of (1) at a time as below: f ′ r (t) , (t � 0, 1, . . .). (2) Iterative method (2) has local quadratic convergence. Nedzhibov et al. [13] presented corresponding inverse numerical technique of the same convergence order as Here, we propose the following family of the optimal second-order convergence method for finding simple roots of (1) as where α ∈ R. Method (4) is optimal and the convergence order of (4) is 2 if ζ is a simple root of (1) and ∈ � r − ζ. e error equation of (4) is obtained using Maple-18: or Corresponding inverse methods of (4) is constructed as r (t) f ′ r (t) + f r (t) 1/ 1 − αf r (t) /1 + f r (t) .
Besides simple root finding methods [3-5, 13, 15, 16, 18-20] in literature, there exists another class of numerical methods which estimate all real and complex roots of (1) at a time, known as simultaneous methods. Simultaneous numerical iterative schemes are very prevalent due to their global convergence properties and its parallel execution on computers [1, 6, 8-10, 12, 14]. e most prominent method among simultaneous derivative-free iterative technique is the Weierstrass-Dochive [17] method (abbreviated as MWM1), which is defined as where is Weierstrass' correction. Method (10) has local quadratic convergence. For finding all multiple roots of (1), we use the following correction [17]: where σ is the multiplicity of the roots. (7), we get a new family of inverse modified Weierstrass method (abbreviated as MWM2):

Construction of the Inverse Simultaneous Method
Inverse simultaneous iterative method (13) can also be written as us, we construct a new derivative-free family of inverse iterative simultaneous scheme (13), abbreviated as MWM2, for estimating all distinct roots of (1). To estimate all multiple roots of (1), we use correction (12) instead of (11) in (7).
i − ζ i be the errors in r i and s i respectively. For the simplicity of the calculation, we omit the iteration index. en, (15) or Using (15), we have Mathematical Problems in Engineering us, we obtain Using the expression If all the errors are assumed of the same order, i.e., Hence, it is proved.

Lower Bound of Convergence of MWM1 and MWM2
Computer algebra system, Mathematica, has been used to find the lower bound of convergence of MWM1 and MWM2. Consider where ϱ 1 , ϱ 2 , and ϱ 3 are exact zeros of (23). e first component of ℵ 1 (r) (where r � [r 1, r 2, r 3 ]) of numerical iterative methods is for finding zeros of (23), r (t+1) � ℵ(r (t) ), simultaneously. We have to express the derivatives of ℵ(r), i.e., the partial derivatives of ℵ(r) with respect to r are as follows: and so on. We obtain the lower bound of convergence order till the first nonzero element of row is found.

Numerical Results
Some engineering problems are considered to demonstrate the performance and effectiveness of the simultaneous method, MWM2 and MWM1. For computer calculations, we use CAS-Maple-18, and the following stopping criteria for termination of computer are programmed: where e (t) i signifies the absolute error. In Tables 1-5, C-Time represents computational time in second.

Engineering Applications.
Some engineering applications are deliberated in this section in order to show the feasibility of the present work.
Example 1. (see [2]). Considering a physical problem of beam positioning results in the following nonlinear polynomial equation: e exact root of (30), ζ 1,2 , is 2 with multiplicity 2 and the remaining other two roots are simple, i.e., . We take the following initial estimates: � −0.5354.
(31) Table 1 clearly demonstrates the superiority of MWM2 over MWM1 in terms of predicted absolute error and CPU time for guesstimating all real roots of (30) on the same number of iterations n � 3.

Example 2.
(see [16]). In this engineering application, we consider a reactor of stirred tank. Items H 1 and H 2 are fed to the reactor at rates of ß and q-ß, respectively. Composite reaction improves in the apparatus as below: Douglas et al. [7] first examined this complex control system and obtained the following nonlinear polynomial equation: where T c is the gain of the proportional controller. By taking T c � 0, we have f(r) � r 4 + 11.50r 3 + 47.49r 2 + 83.06325r + 51.23266875 � 0.   [4]). Consider the function e problem describes the fractional alteration of nitrogen-hydrogen (NH) feed into ammonia at 250 atm pressure and 500 o C temperature. Since the (37) is of order four, it has four roots:       Table 3 evidently shows the supremacy behavior of MWM2 over MWM1 in terms of estimated absolute error and in CPU time on the same number of iterations n � 8 for guesstimating all real and complex roots of (37). Minuscule alteration of nitrogen-hydrogen (NH) feed into ammonia lies between (0,1); therefore, our desire root is ζ 1 up to 1900 decimal places: Remaining other approximating roots are ζ 2 � −1. Example 4. (see [8]).Consider with multiple exact roots: e initial estimations have been taken as For distinct roots, Table 4 evidently shows the supremacy behavior of MWM2 over MWM1 in terms of estimated absolute error and in CPU time on the same number of iterations n � 5 for guesstimating all real and complex roots of (41).
Example 5. (see [5]). e sourness of a soaked solution of magnesium-hydroxide (MgOH) in hydroelectric acid (HCl) is given by with exact roots of (46), ζ 1 � 2.4 and ζ 2,3 � −3.0 ± 2.3i up to one decimal places. e initial estimates have been taken as (47) Table 5 evidently illustrates the supremacy behavior of MWM2 over MWM1 in terms of estimated absolute error and in CPU time on the same number of iterations n � 8 for guesstimating all real and complex roots of (46). Example 6. (see [21]). In general, mechanical engineering, as well as the majority of other scientists, uses thermodynamics extensively in their research work. e following polynomial is used to relate the zero-pressure specific heat of dry air, C ρ , to temperature: C ρ � 1.9520 × 10 − 14 r 4 − 9.5838 × 10 − 11 r 3 + 9.7215   (50) Table 6 clearly illustrates the supremacy behavior of MWM1 over MWM2 in estimated absolute error and in CPU time on the same number of iterations n � 4 for guesstimating all real and complex roots of (49).

Conclusion
A new derivative-free family of inverse numerical methods of convergence order 2 for simultaneous estimations of all distinct and multiple roots of (1) was introduced and discussed in this paper. Tables 1-5 and Figure 1   e results of numerical test cases from Tables 1-5, CPU time, and residual error graph from Figure 3 demonstrated the effectiveness and rapid convergence of our proposed iterative method MWM2 as compared to MWM1.

Data Availability
No data were used to support this study. Disclosure e statements made and views expressed are solely the responsibility of the authors.