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

Authors: Mudassir Shams; Naila Rafiq; Nasreen Kausar; Shams Forruque Ahmed; Nazir Ahmad Mir; Suvash Chandra Saha
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

A new inverse family of the iterative method is interrogated in the present article for simultaneously estimating all distinct and multiple roots of nonlinear polynomial equations. Convergence analysis proves that the order of convergence of the newly constructed family of methods is two. The computer algebra systems CAS-Mathematica is used to determine the lower bound of convergence order, which justifies the local convergence of the newly developed method. Some nonlinear models from physics, chemistry, and engineering sciences are considered to demonstrate the performance and efficiency of the newly constructed family of inverse simultaneous methods in comparison to classical methods in the literature. The computational time in seconds and residual error graph of the inverse simultaneous methods are also presented to elaborate their convergence behavior.


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## Body

## 1. Introduction

Considering nonlinear polynomial equation of degreen,(1)fr=rn+an−1rn−1+⋯+a0=Πnj=1r−ζj=r−ζi∗Πnj≠ij=1r−ζj,with arbitrary real or complex coefficient an−1,…,a0. Let ζ1…ζn denote all the simple or complex roots of (1) with multiplicity σ1…σn. 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:(2)st=rt−frtf′rt,t=0,1,….Iterative method (2) has local quadratic convergence. Nedzhibov et al. [13] presented corresponding inverse numerical technique of the same convergence order as(3)st=rt2f′rtrtf′rt+frt.Here, we propose the following family of the optimal second-order convergence method for finding simple roots of (1) as(4)st=rt−frtf′rt11−αfrt/1+frt,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. The error equation of (4) is obtained using Maple-18:(5)st−ζrt−ζ2⟶−α+C2;Ckr=fkxk!f′x,or(6)st−ζ=O∈2.Corresponding inverse methods of (4) is constructed as(7)st=rt2∗f′rtrtf′rt+frt1/1−αfrt/1+frt.Ifζ is an exact root of (1), fζ=0, and the following is obtained:(8)gr=rt2∗f′rtrtf′rt+frt1/1−αfrt/1+frt,gζ=ζfixed point,(9)g′ζ=0.Inverse iterative schemes (7) are second-order convergence as it is easy to prove g′′ζ≠0.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].The most prominent method among simultaneous derivative-free iterative technique is the Weierstrass–Dochive [17] method (abbreviated as MWM1), which is defined as(10)sit=rit−writ,where(11)fritf′rit=writ=fritΠnj≠ij=1rit−rjt,i,j=1,2,3,…,n,is Weierstrass’ correction. Method (10) has local quadratic convergence. For finding all multiple roots of (1), we use the following correction [17]:(12)w∗rit=fritΠnj≠ij=1rit−rjtσj,i,j=1,2,3,…,n,where σ is the multiplicity of the roots.


## 2. Construction of the Inverse Simultaneous Method

Using Weierstrass correctionwrit=frit/∏j≠ij=1nrit−rjt in (7), we get a new family of inverse modified Weierstrass method (abbreviated as MWM2):(13)sit=rit21+1−αfritrit1+1−αfrit+writ1+frit.Inverse simultaneous iterative method (13) can also be written as(14)sit=rit−ritwrit1+fritrit1+1−αfrit+writ1+frit.Thus, 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).


### 2.1. Convergence Framework

In this section, we demonstrate convergence theorem of inverse iterative scheme MWM2.Theorem 1.
Letζ1,…,ζn be single zero of (1) and for necessarily close primary distinct guess r10,…,rn0 of the zero, respectively; then, MWM2 has local 2nd-order convergence.Proof.
Letεi=rit−ζiandεi′=sit−ζi be the errors in ri and si respectively. For the simplicity of the calculation, we omit the iteration index. Then,(15)si=ri21+1−αfriri1+1−αfri+wri1+fri,or(16)si=ri−riwri1+friri1+1−αfri+wri1+fri.
Usingfri/∏j≠ij−1nri−rj=wri in (15), we have(17)si=ri−wri1+fri1+1−αfri+wri/ri1+fri.
Thus, we obtain(18)si−ζi=ri−ζi−wri1+fri1+1−αfri+wri/ri1+fri,(19)εi′=εi1−∏j≠ij=1nri−ζj/ri−rj1+εi∏j≠ij=1nri−ζj1+1−αεi∏j≠ij=1nri−ζj+∏j≠ij=1nri−ζj/ri−rj1+εi∏j≠ij=1nri−ζj,(20)=ϵi1+1−αϵi∏j≠ij=1nri−ζj1+1−αϵi∏j≠ij=1nri−ζj+∏j≠ij=1nri−ζj/ri−rj1+ϵi∏j≠ij=1nri−ζj.
Using the expression∏j≠ij=1nri−ζj/ri−rj−1=∑k≠inϵk/ri−rkɛ∏j≠ik−1ri−ζk/ri−rj [9] in (20), we have(21)εi′=εi1+1−αεi∏j≠ij=1nri−ζj1+1−αεi∏j≠ij=1nri−ζj+∑k≠inεk/ri−rk∏j≠ik−1ri−ζk/ri−rj+11+εi∏j≠ij=1nri−ζj.
If all the errors are assumed of the same order, i.e.,ϵi=ϵk=Oϵ, then(22)εi′=ε1+1−αεiɛ∏j≠ij=1nri−ζj1+1−αεi∏j≠ij=1nri−ζj+∑k≠inεk/ri−rk∏j≠ik−1ri−ζk/ri−rj+11+εi∏j≠ij=1nri−ζj,=Oε2.
Hence, it is proved.


## 2.1. Convergence Framework

In this section, we demonstrate convergence theorem of inverse iterative scheme MWM2.Theorem 1.
Letζ1,…,ζn be single zero of (1) and for necessarily close primary distinct guess r10,…,rn0 of the zero, respectively; then, MWM2 has local 2nd-order convergence.Proof.
Letεi=rit−ζiandεi′=sit−ζi be the errors in ri and si respectively. For the simplicity of the calculation, we omit the iteration index. Then,(15)si=ri21+1−αfriri1+1−αfri+wri1+fri,or(16)si=ri−riwri1+friri1+1−αfri+wri1+fri.
Usingfri/∏j≠ij−1nri−rj=wri in (15), we have(17)si=ri−wri1+fri1+1−αfri+wri/ri1+fri.
Thus, we obtain(18)si−ζi=ri−ζi−wri1+fri1+1−αfri+wri/ri1+fri,(19)εi′=εi1−∏j≠ij=1nri−ζj/ri−rj1+εi∏j≠ij=1nri−ζj1+1−αεi∏j≠ij=1nri−ζj+∏j≠ij=1nri−ζj/ri−rj1+εi∏j≠ij=1nri−ζj,(20)=ϵi1+1−αϵi∏j≠ij=1nri−ζj1+1−αϵi∏j≠ij=1nri−ζj+∏j≠ij=1nri−ζj/ri−rj1+ϵi∏j≠ij=1nri−ζj.
Using the expression∏j≠ij=1nri−ζj/ri−rj−1=∑k≠inϵk/ri−rkɛ∏j≠ik−1ri−ζk/ri−rj [9] in (20), we have(21)εi′=εi1+1−αεi∏j≠ij=1nri−ζj1+1−αεi∏j≠ij=1nri−ζj+∑k≠inεk/ri−rk∏j≠ik−1ri−ζk/ri−rj+11+εi∏j≠ij=1nri−ζj.
If all the errors are assumed of the same order, i.e.,ϵi=ϵk=Oϵ, then(22)εi′=ε1+1−αεiɛ∏j≠ij=1nri−ζj1+1−αεi∏j≠ij=1nri−ζj+∑k≠inεk/ri−rk∏j≠ik−1ri−ζk/ri−rj+11+εi∏j≠ij=1nri−ζj,=Oε2.
Hence, it is proved.


## 3. 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(23)fr=r−ϱ1r−ϱ2r−ϱ3,where ϱ1, ϱ2, and ϱ3 are exact zeros of (23). The first component of ℵ1r (where r=r1,r2,r3) of numerical iterative methods is for finding zeros of (23), rt+1=ℵrt, simultaneously. We have to express the derivatives of ℵr, i.e., the partial derivatives of ℵr with respect to r are as follows:(24)∂ℵ1r∂r1∂ℵ1r∂r2∂ℵ1r∂r3,∂12ℵr∂r12∂12ℵr∂r1∂r2∂12ℵr∂r22∂2Γ1r∂r2∂r3,∂13ℵr∂r13∂13ℵr∂r12∂r2∂13ℵr∂r1∂r22∂13ℵr∂r23∂13ℵr∂r22∂r3,and so on.We obtain the lower bound of convergence order till the first nonzero element of row is found. The Mathematica notebook codes are used for the following MWM1 and MWM2:Weierstrass–Dochive method, MWM1:(25)ℵ1r1,r2,r3≕r1−fr1r1−r2∗r1−r3,(26)In1≕Dℵ1r1,r2,r3,r1r1⟶ϱ1,r2⟶ϱ2,r3⟶ϱ3,Out1≕0,In2≕Dℵ1r1,r2,r3,r2r1⟶ϱ1,r2⟶ϱ2,r3⟶ϱ3,Out2≕0,In3≕Dℵ1r1,r2,r3,r3r1⟶ϱ1,r2⟶ϱ2,r3⟶ϱ3,Out3≕0,In4≔SimplifyDℵ1r1,r2,r3,r1,r2r1⟶ϱ1,r2⟶ϱ2,r3⟶ϱ3,Out4≔0,In6≕SimplifyDℵ1r1,r2,r3,r1,r2r1⟶ϱ1,r2⟶ϱ2,r3⟶ϱ3,Out6≕1ϱ1−ϱ2.Modified inverse family of iterative schemes, MWM2:(27)ℵ1r1,r2,r3≕r121+1−αfr1r11+1−αfr1+r1−r2r1−r31+fr1,(28)In1≕Dℵ1r1,r2,r3,r2/r1⟶ϱ1,r2⟶ϱ2,r3⟶ϱ3,Out1≕0,In2≕Dℵ1r1,r2,r3,r3/r1⟶ϱ1,r2⟶ϱ2,r3⟶ϱ3,Out2≕0,In3≔Dℵ1r1,r2,r3,r3/.r1⟶ϱ1,r2⟶ϱ2,r3⟶ϱ3,Out3=0,In4≔SimplifyDℵ1r1,r2,r3,r1,r1.r1⟶ϱ1,r2⟶ϱ2,r3⟶ϱ3,Out4=4+8∗α∗ϱ1ϱ33.


## 4. 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:(29)eit=rit+1−rit<∈=10−30,where eit signifies the absolute error. In Tables 1–5, C-Time represents computational time in second.Table 1
Residual error for finding all distinct roots.
MethodC-Timee13e23e33e43σ2MWM11.103DivDiv1.3e − 270.02.101MWM20.0480.00.00.00.02.307Residual error for finding all multiple rootsMWM20.3120.00.00.00.02.735Table 2
Residual error for finding all distinct roots.
MethodC-Timee17e27e37e47σ6MWM12.1190.070.020.10.12.125MWM20.1153.5e − 3242.0e − 3190.0050.0052.968Table 3
Residual error for finding all distinct roots.
MethodC-Timee18e28e38e48σ7MWM11.7391.1e − 9Div3.6e − 13Div1.341MWM20.0322.0e − 3174.0e − 2891.6e − 1394.0e − 2893.167Table 4
Residual error for finding all distinct roots.
MethodC-Timee15e25e35e45σ4MWM10.1254.7e − 261.2e − 289.1e − 280.2e − 262.314MWM20.0461.3e − 273.1e − 300.00.02.753Residual error for finding all multiple roots at n = 1MWM20.1030.00.00.00.0Table 5
Residual error for finding all distinct roots.
MethodC-Timee18e28e38σ7MWM10.0670.06.4e − 276.4e − 272.101MWM20.0430.00.00.02.231


### 4.1. 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:(30)fr=r4+4r3−24r2+16r+1,=r−22r2+8r+4.
The exact root of (30), ζ1,2, is 2 with multiplicity 2 and the remaining other two roots are simple, i.e., ζ3=−4−23 and ζ3=−4+23. We take the following initial estimates:(31)r10=1.17,r20=1.17,r30=−7.4641,r40=−0.5354.
Table1 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 H1 and H2 are fed to the reactor at rates of ß and q-ß, respectively. Composite reaction improves in the apparatus as below:(32)H1+H2⟶H3;H3+H2⟶H4;H4+H2⟶H5;H4+H2⟶H6.
Douglas et al. [7] first examined this complex control system and obtained the following nonlinear polynomial equation:(33)−2.98∗r+2.25r+1.45∗r+2.852∗r+4.35=1Tc,where Tc is the gain of the proportional controller. By taking Tc=0, we have(34)fr=r4+11.50r3+47.49r2+83.06325r+51.23266875=0.
The exact distinct roots of (34) are calculated as ζ1=−1.45,ζ2=−2.85,ζ3=−2.85,andζ4=−4.45, and we take the following initial guessed values:(35)r10=−1.0,r20=−1.1,r30=−1.8,r30=−3.9.
Table2 evidently illustrates the supremacy behavior of MWM2 over MWM1 in terms of the estimated absolute error and in CPU time on the same number of iterations n = 7 for guesstimating all real roots of (34).Example 3.
(see [4]). Consider the function(36)84−r2r26−3r22−r−0.186=0,(37)fr=8r4−62.326r3+117.956r2+20.088r−13.392.
The problem describes the fractional alteration of nitrogen-hydrogen (NH) feed into ammonia at 250 atm pressure and500o C temperature. Since the (37) is of order four, it has four roots:(38)ζ1=0.2777,ζ2=3.9485+0.3161i,ζ3=−0.3840,ζ4=3.9485−0.3161i.
The initial approximated value for (27) is taken as(39)r10=0.4,r20=3.7+0.5i,r30=−0.4,r40=3.7−0.5i.
Table3 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 ζ1 up to 1900 decimal places:(40)ζ1=1.12956568412579833521973452e−1912.
Remaining other approximating roots areζ2  = −1.283 404 526e − 1457-8.93 219 631 521e − 1457i, ζ3  = −8.21 745 235 223 462e − 1091 + 0i, and ζ4= −1.2 834 045 268 801e − 1457 + 8.99 321 963152e − 1457i.Example 4.
(see [8]).Consider(41)fr=r−0.3−0.6i100r−0.1−0.7i200r−0.7−0.5i300r−0.3−0.4i400,with multiple exact roots:(42)ζ1=0.3+0.6i,ζ2=0.1+0.7i,ζ3=0.7+0.5i,ζ4=0.3+0.4i,
The initial estimations have been taken as(43)r10=0.301+0.601i,r20=0.100+0.702i,r30=0.702+0.489i,r40=0.289−0.400i,
For distinct roots,(44)f∗r=r−0.3−0.6ir−0.1−0.7ir−0.7−0.5ir−0.3−0.4i,
Table4 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]). The sourness of a soaked solution of magnesium-hydroxide (MgOH) in hydroelectric acid (HCl) is given by(45)3.64×10−11H3O+=H3O++3.6×10−4,for the cranium ion concentration H3O+. If we set r=104H3O+, we obtain the following polynomial:(46)fr=r3+3.6r2−36.4,with exact roots of (46), ζ1=2.4andζ2,3=−3.0±2.3i up to one decimal places. The initial estimates have been taken as(47)r10=2.45,r20=−3.0261+2.3834i,r30=−3.0261−2.3834i.
Table5 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. The following polynomial is used to relate the zero-pressure specific heat of dry air, Cρ, to temperature:(48)Cρ=1.9520×10−14r4−9.5838×10−11r3+9.7215×10−8r2+1.671×10−4r+0.99403.
The temperature that corresponds to specific heat of1.2kJ/kgK needs to be determined. Putting Cρ=1.2 in (48), we have(49)fr=1.9520×10−14r4−9.5838×10−11r3+9.7215×10−8r2+1.671×10−4r+0.99403.with exact roots ζ1=1126.009751, ζ2=2536.837119+910.5010371i, and ζ3=−1289.950382,2536.837119−910.5010371i. The initial estimations of (49) have been taken as(50)r10=1126,r20=2536+910i,r30=−1289,r40=2536−910i.
Table6 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).Table 6
Residual error for finding all distinct roots.
MethodC-Timee14e24e34e44σ3MWM10.2034.1e − 253.6e − 307.1e − 215.6e − 232.131MWM20.1155.0e − 384.8e − 378.4e − 327.9e − 322.707


## 4.1. 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:(30)fr=r4+4r3−24r2+16r+1,=r−22r2+8r+4.
The exact root of (30), ζ1,2, is 2 with multiplicity 2 and the remaining other two roots are simple, i.e., ζ3=−4−23 and ζ3=−4+23. We take the following initial estimates:(31)r10=1.17,r20=1.17,r30=−7.4641,r40=−0.5354.
Table1 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 H1 and H2 are fed to the reactor at rates of ß and q-ß, respectively. Composite reaction improves in the apparatus as below:(32)H1+H2⟶H3;H3+H2⟶H4;H4+H2⟶H5;H4+H2⟶H6.
Douglas et al. [7] first examined this complex control system and obtained the following nonlinear polynomial equation:(33)−2.98∗r+2.25r+1.45∗r+2.852∗r+4.35=1Tc,where Tc is the gain of the proportional controller. By taking Tc=0, we have(34)fr=r4+11.50r3+47.49r2+83.06325r+51.23266875=0.
The exact distinct roots of (34) are calculated as ζ1=−1.45,ζ2=−2.85,ζ3=−2.85,andζ4=−4.45, and we take the following initial guessed values:(35)r10=−1.0,r20=−1.1,r30=−1.8,r30=−3.9.
Table2 evidently illustrates the supremacy behavior of MWM2 over MWM1 in terms of the estimated absolute error and in CPU time on the same number of iterations n = 7 for guesstimating all real roots of (34).Example 3.
(see [4]). Consider the function(36)84−r2r26−3r22−r−0.186=0,(37)fr=8r4−62.326r3+117.956r2+20.088r−13.392.
The problem describes the fractional alteration of nitrogen-hydrogen (NH) feed into ammonia at 250 atm pressure and500o C temperature. Since the (37) is of order four, it has four roots:(38)ζ1=0.2777,ζ2=3.9485+0.3161i,ζ3=−0.3840,ζ4=3.9485−0.3161i.
The initial approximated value for (27) is taken as(39)r10=0.4,r20=3.7+0.5i,r30=−0.4,r40=3.7−0.5i.
Table3 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 ζ1 up to 1900 decimal places:(40)ζ1=1.12956568412579833521973452e−1912.
Remaining other approximating roots areζ2  = −1.283 404 526e − 1457-8.93 219 631 521e − 1457i, ζ3  = −8.21 745 235 223 462e − 1091 + 0i, and ζ4= −1.2 834 045 268 801e − 1457 + 8.99 321 963152e − 1457i.Example 4.
(see [8]).Consider(41)fr=r−0.3−0.6i100r−0.1−0.7i200r−0.7−0.5i300r−0.3−0.4i400,with multiple exact roots:(42)ζ1=0.3+0.6i,ζ2=0.1+0.7i,ζ3=0.7+0.5i,ζ4=0.3+0.4i,
The initial estimations have been taken as(43)r10=0.301+0.601i,r20=0.100+0.702i,r30=0.702+0.489i,r40=0.289−0.400i,
For distinct roots,(44)f∗r=r−0.3−0.6ir−0.1−0.7ir−0.7−0.5ir−0.3−0.4i,
Table4 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]). The sourness of a soaked solution of magnesium-hydroxide (MgOH) in hydroelectric acid (HCl) is given by(45)3.64×10−11H3O+=H3O++3.6×10−4,for the cranium ion concentration H3O+. If we set r=104H3O+, we obtain the following polynomial:(46)fr=r3+3.6r2−36.4,with exact roots of (46), ζ1=2.4andζ2,3=−3.0±2.3i up to one decimal places. The initial estimates have been taken as(47)r10=2.45,r20=−3.0261+2.3834i,r30=−3.0261−2.3834i.
Table5 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. The following polynomial is used to relate the zero-pressure specific heat of dry air, Cρ, to temperature:(48)Cρ=1.9520×10−14r4−9.5838×10−11r3+9.7215×10−8r2+1.671×10−4r+0.99403.
The temperature that corresponds to specific heat of1.2kJ/kgK needs to be determined. Putting Cρ=1.2 in (48), we have(49)fr=1.9520×10−14r4−9.5838×10−11r3+9.7215×10−8r2+1.671×10−4r+0.99403.with exact roots ζ1=1126.009751, ζ2=2536.837119+910.5010371i, and ζ3=−1289.950382,2536.837119−910.5010371i. The initial estimations of (49) have been taken as(50)r10=1126,r20=2536+910i,r30=−1289,r40=2536−910i.
Table6 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).Table 6
Residual error for finding all distinct roots.
MethodC-Timee14e24e34e44σ3MWM10.2034.1e − 253.6e − 307.1e − 215.6e − 232.131MWM20.1155.0e − 384.8e − 378.4e − 327.9e − 322.707


## 5. 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 clearly show that computational order of convergence of the proposed and existing methods are agreed with the theoretical results. Simulation time, from Figure 2, clearly indicates the supremacy of our newly proposed method MWM2 over existing Weierstrass method MWM1. The 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.Figure 1
Computational order of convergence [22] of simultaneous methods, MWM1 and MWM2.Figure 2
Simulation time in second of simultaneous methods, MWM1 and MWM2, for approximating all roots of polynomial equations used in Examples1–6.Figure 3
(a-f) Error graph of MWM1 and MWM2 for nonlinear polynomial equations (30), (34), (37), (41), (46) and (49), respectively. (a) Residual error graph for Example 1. (b) Residual error graph for Example 2. (c) Residual error graph for Example 3. (d) Residual error graph for Example 4. (e) Residual error graph for Example 5. (f) Residual error graph for Example 6.
(a)(b)(c)(d)(e)(f)



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