A Highly Efficient Computer Method for Solving Polynomial Equations Appearing in Engineering Problems

Department of Mathematics, National University of Modern Languages, Islamabad, Pakistan Department of Mathematics and Statistics, Riphah International University I-14, Islamabad 44 000, Pakistan Quantum Leap Africa (QLA), AIMS Rwanda Centre, Remera Sector KN 3, Kigali, Rwanda Institut de Mathématiques et de Sciences Physiques (IMSP/UAC), Laboratoire de Topologie Fondamentale, Computationnelle et Leurs Applications (Lab-ToFoCApp), BP 613 Porto-Novo, Benin African Center for Advanced Studies, P.O. Box 4477, Yaounde, Cameroon


Introduction
Determining the roots of polynomial equations is among the oldest problems in mathematics, whereas the polynomial equations have a wide range of applications in science and engineering. For example, aerospace engineers may use polynomials to determine acceleration of a rocket or jet or even stability of an aeroplane and mechanical engineers use polynomials to design engines and machines. Simultaneous methods are very popular as compared to the methods for individual finding of the roots. ese methods have a wider region of convergence, are more stable, and can be implemented for parallel computing. More details on simultaneous methods, their convergence properties, computational efficiency, and parallel implementation may be found in the works of Cosnard et al. [1], Kanno et al. [2], Proinov et al. [3], Sendov et al. [4] Ikhile [5], Mir at al. [6], Wahab et al. [7], Cholakov [8], Proinov and Ivanov [9], Iliev [10], and Kyncheva [11]. Nowadays, mathematicians are working on iterative methods for finding all the zeros of polynomial simultaneously (see [12][13][14][15][16][17][18] and references therein). e main objective of this paper is to develop simultaneous method which not only has a higher convergence order but also is more efficient as compared to existing methods. A very high computational efficiency for the newly constructed scheme for finding distinct as well as multiple roots is achieved by using a suitable corrections [19] which enable us to achieve fourteenth-order convergence with minimal number of functional evaluations in each step. So far among the higher order simultaneous methods, only the Midrog Petkovic method [20] of order ten and the Gargantini-Farmer-Loizou method of 2N + 1 convergence order (where N is positive integer) [21][22][23][24] exist in the literature. Consider nonlinear polynomial equation of degree m: with multiple real or complex exact root ζ 1 , . . . , ζ n of respective unknown multiplicities σ 1 , . . . , σ n (σ 1 + · · · + σ n � m). Generally, the multiplicity of roots is not given in advance. However, research studies are working on numerical methods which approximate the unknown multiplicity of roots, see, e.g., [25][26][27][28][29][30][31].

Construction of Simultaneous Computer Methods for Multiple Roots
Considering two-step fourth-order Newton's method [32] for finding multiple roots of nonlinear polynomial equation (1), where σ i is the multiplicity of the root ζ i of equation (1). We would like to convert (2) into the simultaneous method for estimating all roots of (1). We use fifth-order ukral et al. method [19] as a correction to increase the efficiency and convergence order requiring no additional evaluations of the function: f ′ y (k) .
Suppose equation (1) has m distinct roots; then, is implies For multiple roots, equation (7) can be written as where ζ 1 , . . . , ζ n are now multiple roots of respective unknow multiplicities σ 1 , . . . , σ n (σ 1 + · · · + σ n � m). Replacing t j by z j in (8), we have where 2 Mathematical Problems in Engineering Using (9) in (2), we have . us, we have constructed a new simultaneous method (11) abbreviated as MMN14M for calculating all multiple roots of polynomial equation (1). e simultaneous method (11) requires two evaluations of the function and two evaluations of the first derivative. For multiplicity unity, i.e., σ i � 1, i � 1, . . . , n, we use method (11) for determining all the distinct roots of equation (1) and abbreviate it as MMN14D.

Convergence Analysis.
In this section, we discuss the convergence analysis of the two-step simultaneous method (11) which is given in the form of the following theorem. Theorem 1. Let ζ 1 , . . . , ζ n be the roots of equation (1) with multiplicity σ 1 , . . . , σ n (σ 1 + · · · + σ n � m). If t (0) 1 , . . . , t (0) n are the initial approximations of the roots, respectively, and sufficiently close to actual roots, the order of convergence of method (11) equals fourteen.
be the errors in t (k) i , y (k) i , and u (k) i approximations, respectively. Consider the first step of (11): where en, obviously, for distinct roots, us, for multiple roots, we have, from (11), Mathematical Problems in Engineering If it is assumed that all errors ∈ j (j � 1, 2, 3, . . .) are of the same order as, say |∈ j | � O|∈|, then, from (17), we have From the second equation of (11), is implies is implies Since, from (18), ∈ i ′ � O(∈) 7 , thus, 4 Mathematical Problems in Engineering which shows convergence order of simultaneous iterative scheme (11) is fourteen. Hence, the theorem is proved. e above results are equally valid for complex polynomial by performing real arithmetic. Numerical Examples 4 and 5 for complex polynomials are provided to verify its validity. □

Computational Aspect
Here, we compare the computational efficiency and convergence behaviour of our new fourteenth-order method MMN14M (11) with the Midrog Petkovic method [20] of order 10 and the Gargantini-Farmer-Loizou method [21][22][23][24] of order 15 (abbreviated as GFLM15M for multiple and GFLM15D for distinct roots). As presented in [20], the efficiency of an iterative method can be estimated using the efficiency index given by where d is the computational cost and r is the order of convergence of the iterative method. We use arithmetic operation per iteration with certain weight depending on the execution time of operation to evaluate the computational cost d. e weights used for division, multiplication, and addition plus subtraction are w as , w m , and w d , respectively. For a given polynomial of degree m, the number of division, multiplication, addition, and subtraction per iteration for all roots is denoted by AS m , M m , and D m . e cost of computation can be calculated as us, (23) becomes Apply (25) and data given in Table 1, we find the percentage ratio ρ( (11), (X)) [20] given by where X and (11) are the Petkovic method (abbreviated as PJM10), GFLM15M, and our new method MMN14M, respectively. ese ratios are graphically displayed in Figure 1(a)-1(d). It is evident from Figure 1(a)-1(d) that the new method (11) is more efficient as compared to the PJM10 and GFLM15M methods.

Numerical Results
Here, some numerical examples are considered in order to demonstrate the performance of our family of two-step fourteenth-order simultaneous methods, namely, MMN14D (for multiplicity unity) and MMN14M (for multiple roots) (11). We compare our family of methods with J. Džunic, M. S. Petkovic, and L. D. Petkovic [20] method of order ten for distinct roots (abbreviated as the PJM10 method) and with the Gargantini-Farmer-Loizou method (GFLM15D and GFLM15M) of order 15, respectively. All the computations are performed using Maple-18 with 64 digits' floating point arithmetic. We take ∈ � 10 − 30 as a tolerance and use the following stopping criteria for estimating the roots: where e (k) i represents the absolute error of function values. Numerical tests' examples from [6,17,20,33] are taken and compared on the same number of iterations and provided in Tables 2-15. In all the tables, n represents the number of iterations and CPU represents execution time in seconds. All the numerical calculations are performed using maple-18 on the computer (Processor Intel(R) Core(TM) i3-3110m CPU@2.4GHz) with 64-bit operating system. Figures 2-11 show the residue falls of the methods MMN14D, MMN14M, PJM10, GFLM15D, and GFLM15M for Examples 1-9. e residual falls show that the methods MMN14D and MMN14M are more efficient as compared to PJM10, GFLM15D, and GFLM15M methods. We observe that numerical results of the methods MMN14M and MMN14D are better than PJM10, GFLM15D, and GFLM15M methods in terms of absolute errors and CPU time (Algorithm 1).
Example 1 (car stability). Application in mechanical engineering. e design of a car suspension system requires to be balanced for getting good comfort and stability for all driving conditions and speeds. e following equations must be satisfied for stability of a design of a car which has good comfort on rough roads: Let en, we get the following polynomial equation: Mathematical Problems in Engineering 5 Step 1: for given initial estimates t (0) i (i � 1, 2, . . . , n), tolerance ∈ > 0, and iterations p, set k � 0.
Step 2: calculate ) and σ j is the multiplicity of actual multiple roots ζ j .
having exact roots e initial estimations of (28) are taken as Figure 12 shows that (28) has two positive roots which are determined in 3 iterations by PJM10, GFLM15D, and MMN14D methods, and the comparison is shown in Table 2 We observe that MMN14D has better performance in terms of CPU time and absolute errors as compared to PJM10 and GFLM15D, respectively. Residual errors of MMN14D are also very less as compared to PJM10 and GFLM15D as shown by residual graph for this polynomial in Figure 2. Figure 2 shows residual graph for approximating roots of nonlinear function f 1 (t) using simultaneous methods PJM10, MMN14D, and GFLM15D, respectively. Figure 12 shows that f 1 (t) has two positive roots and one negative root. However, negative root is redundant.
Example 2. Application in civil engineering. Figure 13(a) shows a uniform beam subject to a linearly increasing distributed load. e equation for the elastic curve (Figure 13(b)) is We have to find the point of maximum deflection, i.e., the value of t, where f ′ (t) � 0: Let en, substituting this value in (38), we get the value of maximum deflection. Use the following values in   (39) e initial estimations of (38) have been taken as

Log of Residual
We observe that the method MMN14D is superior in terms of numerical results, CPU time, and error as compared to PJM10 and GFLM15D as shown in Table 3 and residual graph by Figure 3.

(41)
We have to determine the temperature that corresponds to specific heat of 1.2(kJ/kgK).
Putting C ρ � 1.2 in the above equation, we have the following polynomial: We observe that our method, namely, MMN14D, has better performance in terms of numerical results, CPU time, and residual errors as compared to PJM10 and GFLM15D as shown in Table 4 and residual graph in Figure 4. shows   residual graph for approximating roots of nonlinear function f 3 (t) using simultaneous methods PJM10, MMN14D, and GFLM15D, respectively. Example 4. Multiple complex roots [33]. Consider with multiple exact roots, of the multiplicity σ 1 � 14, σ 2 � 12, σ 3 � 14, and σ 4 � 10, respectively. e initial estimations have been taken as For distinct roots, we take We observe that our methods, namely, MMN14D and MMN14M, have better performance in terms of numerical results, CPU time, and residual errors as compared to PJM10, GFLM1515D, and GFLM15M as shown in Table 5  Consider with multiple exact roots, of the multiplicity σ 1 � 100, σ 2 � 200, σ 3 � 300, and σ 4 � 400, respectively. e initial estimations have been taken as For distinct roots, We observe that our methods, namely, MMN14D and MMN14M, have better performance in terms of numerical results, CPU time, and residual errors as compared to PJM10, GFLM15D, and GFLM15M as shown in Tables 7 and  8 with multiple exact roots, of the multiplicity σ 1 � 40, σ 2 � 30, σ 3 � 20, and σ 4 � 10, respectively. e initial estimations have been taken as For distinct roots, We observe that our method, namely, MMN14D and MMN14M, have better performance in terms of numerical results, CPU time, and residual errors as compared to PJM10, GFLM15D, and GFLM15M as shown in Tables 9 and  10  Example 7. Fluid permeability in biogels [34].
Specific hydraulic permeability relates the pressure gradient to fluid velocity in porous medium (agarose gel or or where k is specific hydraulic permeability, R e radius of the fiber, and t is the porosity [35]. Using k � 0.4655 and R e � 100 * 10 − 9 , we have f 7 (t) � − 100 * 10 − 9 t 3 + 9.3100 * t 2 − 18.6200 * t + 9.3100.
We observe that our method, namely, MMN14D, has better performance in terms of numerical results, CPU time, and residual errors as compared to PJM10 and GFLM15D as shown in Table 11 and residual graph in Figure 8.
We observe that our methods, namely, MMN14D and MMN14M, have better performance in terms of numerical results, CPU time, and residual errors as compared to PJM10, GFLM15D, and GFLM15M as shown in Tables 12  and 13 and residual graph in Figures 9(a) and 9(b).
Numerical results for linear combination of Legendre polynomials.
We observe that our methods, namely, MMN14D and MMN14M, have better performance in terms of numerical results, CPU time, and residual error as compared to PJM10, GFLM15D, and GFLM15M as shown in Tables 2-15 and residual graph in Figures 2-11, respectively.

Conclusion
We have developed here two-step simultaneous computer methods of order fourteen for solving nonlinear polynomial equations, one for determining all the distinct roots, namely, MMN14D, and the other for determining multiple roots of nonlinear polynomial equations, namely, MMN14M. From comparison of numerical results, as depicted in Tables 1-17, computational efficiency (Figures 1(a) and 1(d)) and graphical representations of residual errors are shown in Figures 2-11; we observe that our methods (11) of 14th order are superior in terms of efficiency, CPU time, and residual errors as compared to the Petkovic method PJM10 and the Gargantini-Farmer-Loizou method GFLM15D and GFLM15M. Using the similar ways, we can introduce more higher order and efficient methods.

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