Abstract
A highly efficient twostep simultaneous iterative computer method is established here for solving polynomial equations. A suitable special type of correction helps us to achieve a very high computational efficiency as compared to the existing methods so far in the literature. Analysis of simultaneous scheme proves that its convergence order is 14. Residual graphs are also provided to demonstrate the efficiency and performance of the newly constructed simultaneous computer method in comparison with the methods given in the literature. In the end, some engineering problems and some higher degree complex polynomials are solved numerically to validate its numerical performance.
1. 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. These 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 [12–18] and references therein).
The 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 fourteenthorder 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) [21–24] exist in the literature. Consider nonlinear polynomial equation of degree :with multiple real or complex exact root of respective unknown multiplicities . 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., [25–31].
2. Construction of Simultaneous Computer Methods for Multiple Roots
Considering twostep fourthorder Newton’s method [32] for finding multiple roots of nonlinear polynomial equation (1),where is the multiplicity of the root of equation (1). We would like to convert (2) into the simultaneous method for estimating all roots of (1). We use fifthorder Thukral et al. method [19] as a correction to increase the efficiency and convergence order requiring no additional evaluations of the function:
Suppose equation (1) has m distinct roots; then,
This implies
This giveswhere
For multiple roots, equation (7) can be written aswhere are now multiple roots of respective unknow multiplicities .
Replacing by in (8), we havewhere
Using (9) in (2), we havewhere and .
Thus, we have constructed a new simultaneous method (11) abbreviated as MMN14M for calculating all multiple roots of polynomial equation (1). The simultaneous method (11) requires two evaluations of the function and two evaluations of the first derivative. For multiplicity unity, i.e., , we use method (11) for determining all the distinct roots of equation (1) and abbreviate it as MMN14D.
2.1. Convergence Analysis
In this section, we discuss the convergence analysis of the twostep simultaneous method (11) which is given in the form of the following theorem.
Theorem 1. Let be the roots of equation (1) with multiplicity . If are the initial approximations of the roots, respectively, and sufficiently close to actual roots, the order of convergence of method (11) equals fourteen.
Proof. Letbe the errors in , , and approximations, respectively. Consider the first step of (11):whereThen, obviously, for distinct roots,Thus, for multiple roots, we have, from (11),where [19] and .
Thus,If it is assumed that all errors are of the same order as, say , then, from (17), we haveFrom the second equation of (11),This impliesThis impliesSince, from (18), , thus,which shows convergence order of simultaneous iterative scheme (11) is fourteen. Hence, the theorem is proved. The 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.
3. Computational Aspect
Here, we compare the computational efficiency and convergence behaviour of our new fourteenthorder method MMN14M (11) with the Midrog Petkovic method [20] of order 10 and the Gargantini–Farmer–Loizou method [21–24] 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 bywhere is the computational cost and 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 . The weights used for division, multiplication, and addition plus subtraction are , , and , respectively. For a given polynomial of degree m, the number of division, multiplication, addition, and subtraction per iteration for all roots is denoted by , , and . The cost of computation can be calculated as
Thus, (23) becomes
Apply (25) and data given in Table 1, we find the percentage ratio [20] given bywhere X and (11) are the Petkovic method (abbreviated as PJM10), GFLM15M, and our new method MMN14M, respectively. These 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.
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4. Numerical Results
Here, some numerical examples are considered in order to demonstrate the performance of our family of twostep fourteenthorder 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 Maple18 with 64 digits’ floating point arithmetic. We take as a tolerance and use the following stopping criteria for estimating the roots:where 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, represents the number of iterations and CPU represents execution time in seconds. All the numerical calculations are performed using maple18 on the computer (Processor Intel(R) Core(TM) i33110m [email protected]) with 64bit operating system. Figures 2–11 show the residue falls of the methods MMN14D, MMN14M, PJM10, GFLM15D, and GFLM15M for Examples 1–9. The 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).

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Example 1 (car stability). Application in mechanical engineering.
The design of a car suspension system requires to be balanced for getting good comfort and stability for all driving conditions and speeds. The following equations must be satisfied for stability of a design of a car which has good comfort on rough roads:LetThen, we get the following polynomial equation:having exact rootsThe initial estimations of (28) are taken asFigure 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 using simultaneous methods PJM10, MMN14D, and GFLM15D, respectively.
Figure 12 shows that has two positive roots and one negative root. However, negative root is redundant.
Thus, for car stability, the required positive roots are and . Using these values in andwe havewhich yields the following velocity for car stability:where meter is the distance between peeks and and are the horizontal speeds of the car at times t1 and t3.
Example 2. Application in civil engineering.
Figure 13(a) shows a uniform beam subject to a linearly increasing distributed load.
The equation for the elastic curve (Figure 13(b)) isWe have to find the point of maximum deflection, i.e., the value of , where :LetThen, substituting this value in (38), we get the value of maximum deflection. Use the following values in computation: , and .
The exact roots of (38) areThe initial estimations of (38) have been taken asWe 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.
Thus, for maximum deflection, we put in (38) and get the deflection equal to .
Figure 3 shows the residual graph for approximating roots of nonlinear function using simultaneous methods PJM10, MMN14D, and GFLM15D, respectively.
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Example 3 (thermodynamics). Mechanical engineering Application.
In general, mechanical engineering as well as most other scientists use thermodynamics extensively in their research work. The following polynomial is used to relate the zeropressure specific heat of dry air, , to temperature:We have to determine the temperature that corresponds to specific heat of .
Putting in the above equation, we have the following polynomial:with exact rootsThe initial estimations of (42) have been taken asWe 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 using simultaneous methods PJM10, MMN14D, and GFLM15D, respectively.
Example 4. Multiple complex roots [33].
Considerwith multiple exact roots,of the multiplicity , and , respectively. The initial estimations have been taken asFor distinct roots, we takeWe 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 and 6 and residual graph in Figures 5(a) and 5(b).
Figures 5(a) and 5(b) show the residual graph for approximating roots of nonlinear function using simultaneous methods PJM10, MMN14D, MMN14M, GFLM15D, and GFLM15M, respectively.
Example 5. Multiple complex roots.
Considerwith multiple exact roots,of the multiplicity , and , respectively. The initial estimations have been taken asFor 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 and residual graph in Figures 6(a) and 6(b).
Figures 6(a) and 6(b) show residual graph for approximating roots of nonlinear function using simultaneous methods PJM10, MMN14D, MMN14M, GFLM15D, and GFLM15M, respectively.
Example 6. Real roots with high multiplicity.
Considerwith multiple exact roots,of the multiplicity , and , respectively. The initial estimations have been taken asFor 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 and residual graph in Figures 7(a) and 7(b).
Figures 7(a) and 7(b) show the residual graph for approximating roots of nonlinear function using simultaneous methods PJM10, MMN14D, MMN14M, GFLM15D, and GFLM15M, respectively.
Example 7. Fluid permeability in biogels [34].
Specific hydraulic permeability relates the pressure gradient to fluid velocity in porous medium (agarose gel or extracellular fiber matrix) results the following nonlinear polynomial equations:orwhere is specific hydraulic permeability, radius of the fiber, and is the porosity [35]. Using and , we haveThe exact roots of (59) areWe choose the following initial estimates for simultaneous determination of all roots of (59) areFigure 8 shows residual graph for approximating roots of nonlinear function using simultaneous methods PJM10, MMN14D, and GFLM15D, respectively.
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.
Example 8. Considerwith multiple exact roots,of the multiplicity , and , respectively. The initial estimations have been taken asFor distinct roots,Figures 9(a) and 9(b) show the residual graph for approximating roots of nonlinear functions using simultaneous methods PJM10, MMN14D, MMN14M, GFLM15D, and GFLM15M, respectively.
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).
Example 9 (see [36]). The solution of Legendre polynomial.
Legendre polynomial is the solution of the Legendre differential equation:For Legendre polynomials, the recursive relation are Legendre polynomials are plotted in Figures 14(a) and 14(b). It is notable how the roots are clustered in the ends of the domain , as shown in Figure 14(a). Here, we find the roots of legendre polynomial of degree 10, i.e., :We choose the following random initial approximations of (67):Figure 14(a) shows plot of Legendre polynomial from to , while Figure 14(b) shows Legendre polynomial of degree 10 only, i.e., .
Figure 10 shows the residual graph for approximating roots of nonlinear function using simultaneous methods PJM10, MMN14D, and GFLM15D, respectively.
Numerical results for linear combination of Legendre polynomials.
Although the roots of Legendre polynomials are real and lie in a specific interval, the roots of their linear combination need not be real may be complex. Some results on these linear combinations are given below:wherePutting values of in (69), we haveWe choose the following random initial approximations of (71):Figure 11 shows the residual graph for approximating roots of nonlinear function using simultaneous methods PJM10, MMN14D, and GFLM15D, respectively.
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.
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5. Conclusion
We have developed here twostep 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
The statements made and views expressed are solely the responsibility of the author.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Authors’ Contributions
All authors contributed equally in the preparation of this manuscript.
Acknowledgments
The fourth author (YUG) would like to acknowledge that this publication was made possible by a grant from Carnegie Corporation of New York.