JMATHJournal of Mathematics2314-47852314-4629Hindawi10.1155/2021/66435146643514Research ArticleInverse Numerical Iterative Technique for Finding all Roots of Nonlinear Equations with Engineering Applicationshttps://orcid.org/0000-0002-2980-5801ShamsMudassir1https://orcid.org/0000-0002-6474-045XRafiqNaila2AhmadBabar3MirNazir Ahmad1MustafaGhulam1Department of Mathematics and StatisticsRiphah International University I-14Islamabad 44000Pakistanriphah.edu.pk2Department of MathematicsNUMLIslamabadPakistannuml.edu.pk3Department of MathematicsComsats University IslamabadIslamabad 44000Pakistancomsats.edu.pk20214120212021241020202411202021220204120212021Copyright © 2021 Mudassir Shams et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce here a new two-step derivate-free inverse simultaneous iterative method for estimating all roots of nonlinear equation. It is proved that convergence order of the newly constructed method is four. Lower bound of the convergence order is determined using Mathematica and verified with theoretical local convergence order of the method introduced. Some nonlinear models which are taken from physical and engineering sciences as numerical test examples to demonstrate the performance and efficiency of the newly constructed modified inverse simultaneous methods as compared to classical methods existing in literature are presented. Dynamical planes and residual graphs are drawn using MATLAB to elaborate efficiency, robustness, and authentication in its domain.

1. Introduction

A wide range of problems in physical and engineering sciences can be formulated as a nonlinear equation:(1)fr=0.

The most ancient and popular iterative technique for approximating single roots of (1) is Newton’s method  which has local quadratic convergence:(2)sk=rkfrkfrk,k=0,1,.

Nedzibove et al., in , presented the inverse method of the same order corresponding to method (2):(3)sk=rk2frkrkfrk+frk.

In the last few years, lot of work has been carried out on numerical iterative methods which approximate single root at a time of (1). There is another class of derivative-free iterative methods which approximates all roots of (1) simultaneously. The simultaneous iterative methods for approximating all roots of (1) are very popular due to their global convergence and parallel implementation on computer (see, e.g., Weierstrass , Kanno , Proinov , Petkovi´c , Mir , Nourein , Aberth , and reference cited there in ).

Among derivative-free simultaneous methods, Weierstrass–Dochive  method (abbreviated as WDK) is the most attractive method given by(4)sik=rikwrik,where(5)wrik=frikj=1jinrikrjk,i,j=1,2,3,,n,is Weierstrass’ Correction. Method (4) has local quadratic convergence.

Nedzibove  introduced a new modification to (4), that is, an inverse method to WDK abbreviated as IWDK, i.e.,(6)uik=rik2j=1jinrikrjkrikj=1jinrikrjk+frik.

The main aim of this paper is to construct a two-step inverse method of convergence order four.

2. Construction of Family of Simultaneous Method for Distinct Roots

We modify the Weierstrass method (4) as follows:(7)zik=uikfuikj=1jinuikujk,where uik=rikfrik/j=1jinrikrjk and denote it by WDK2. Let us now convert method (7) into inverse iterative method as follows:(8)zik=uik2j=1jinuikujkuikj=1jinuikujk+fuik,where uik=rik2j=1jinrikrjk/rikj=1jinrikrjk+frik.

Thus, method (8) is a two-step inverse method abbreviated as IWM2.

2.1. Convergence Analysis

We prove here that convergence order of the IWM2 method is four.

Let DCn be an open convex subset, Γ:DCn and m times differentiable operator Γ1r,,ΓnrT be continuous, and the sequence rkkN be defined by rk+1=Γrk:(9)rk=r1k,,rnkrik+1=Γirk,i1,...,n,kN,where norm in Cn is defined as r=maxr1,,rn.

Theorem 1.

Let X and Y be normed spaces. Take an open convex subset D of X for a u times Frēchet differential operator Γ, i.e., Γ:DY Then, for any x, y D(10)ΓyΓxj=1q1j!Γjxyxyxjtimesyxqq!supζx,yΓqζ.

Using Theorem 1, we have the following.

Theorem 2.

Let βD if

Γβ=β

Γβ=Γβ=Γβ==Γuβ=0

Then, there exists s>0 such that, for any r0D,r0β<s, the sequence rk+1=ΓrkkN converges to β.

Proof.

Let s0>0 be such that(11)v0=rC:rβs0D.

C0=maxΓur0zv0, and there exists 0<ss0 such that(12)C0sqq!<sC0q!1/q1<s,where v=rCn:rβs. Using hypothesis (2), rv; then, (ii) and Theorem 1 implies(13)Γrβ=ΓrΓβj=1q11j!Γjβrβrβjtimes1q!rβqsupζβ,rΓqζqC0sqq!<s.

Thus, Γrv. Using the above relation for r=rk, we have(14)rk+1β=ΓrkβC0u!rkβq.

Using (14), recursively, we have(15)rkβC0u!rkβqC0u!C0u!rkβqqC0q!1+q++qkr0βqkC0q!1/q1sqk0 for k0.

Thus, from inequality (14), rkNk is at least q. Now, consider IWM2 as a vector function, i.e., Γr=Γ1r,,Γnr, where(16)Γizi=ui2ui+fui/j=1jinuiuj,where ui=ri2ri+fri/j=1jinrirj.

For a fixed point β=β1,,βn, it is not difficult to prove Γiζ/ri=2Γiζ/rirj=3Γiζ/2rirj=0 and higher order partial derivative is not equal to zero. Thus, IWM2 has at least fourth-order convergence.

Theorem 3.

Let ζ1,,ζn be simple roots of (1) and for sufficiently close initial distinct estimations r10,,rn0 of the roots, respectively; IWM2 has then convergence order 4.

Proof.

Consider εi=rikζi, εi=uikζi, and εi=zikζi be the errors in rik, uik, and zik, respectively. For simplicity, we omit iteration index k. From first step of IWM2, we have(17)uiζi=riζirifri/j=1jinrirjri+fri/j=1jinrirj.

Thus, we obtain(18)εi=εi1jij=1nriζj/rirj1+fri/j=1jinrirj=εi1jij=1nriζj/rirj+fri/j=1jinrirj1+fri/j=1jinrirj.

Using the expression jij=1nriζj/rirj1=kinεk/rirkjik1riζk/rirj  in (18), we have(19)εi=εiεi/rijij=1nriζj/rirjkinεk/rirkjik1riζk/rirj1+εk/rijij=1nriζj/rirj.

If we assume all errors are of the same order, i.e., εi=εk=Oε; then, we have(20)εi=ε21/rijij=1nriζj/rirjkin1/rirkjik1riζk/rirj1+εk/rijij=1nriζj/rirj=Oε2.

From second-step of IWM2, we have(21)ziζi=uiζiuifui/j=1jinuiujri+fui/j=1jinuiuj.

Thus, we obtain(22)εi=εi1jij=1nuiζj/uiuj1+fui/j=1jinuiuj=εi1jij=1nuiζj/uiuj+fui/j=1jinuiuj1+fui/j=1jinuiuj.

As from the above argument jij=1nuiζj/uiuj1=kinεk/uiukjik1uiζk/uiuj using in (22), we have(23)εi=εiεi/uijij=1nuiζj/uiujkinεk/uiukjik1uiζk/uiuj1+εk/uijij=1nuiζj/uiuj.

If we assume all errors are of the same order, i.e., εi=εk=Oε; then,(24)εi=ε21/uijij=1nuiζj/uiujkin1/uiskjik1uiζk/uiuj1+εk/uijij=1nuiζj/uiuj=Oε2=Oε22=Oε4.

Hence, the theorem is proved.

2.1.1. Using CAS for Verification of Convergence Order

Consider(25)fr=rθrϕrφ,and the first component of Γ1r iterative schemes to find zeros of (25), rk+1=Γrk, simultaneously. In order to verify Theorem 2 conditions, we have to express the differential of an operator Γr in terms of their partial derivate of its component as Γir:(26)Γ1rr1Γ1rr2Γ1rr3,2Γ1rr122Γ1rr1r22Γ1rr222Γ1rr2r3,3Γ1rr133Γ1rr12r23Γ1rr1r223Γ1rr233Γ1rr22r3,,and so on.

The lower bound of the convergence obtained until the first nonzero element of the row is found. The Mathematica code is given for each of the consider methods as follows.

Weierstrass–Dochive Method (WDK):(27)Γ1r1,r2,r3rfrj=1jinrirj,i,j=1,,n,In 1DΓ1r1,r2,r3,r1/.r1θ,r2ϕ,r3φ,Out 10,In 2DΓ1r1,r2,r3,r2/.r1θ,r2ϕ,r3φ,Out 20,In 2DΓ1r1,r2,r3,r2/.r1θ,r2ϕ,r3φ,Out 20,In 3Simplify DΓ1r1,r2,r3,r1,r2/.r1θ,r2ϕ,r3φ,Out 31θ+ϕ.

Modified Inverse Weierstrass Method:(28)Γ1r1,r2,r3r2j=1jinrirjrj=1jinrirj+fr,In 1:=DΓ1r1,r2,r3,r2r1θ,r2ϕ,r3φ,Out 1:=0,In 2:=DΓ1r1,r2,r3,r3r1θ,r2ϕ,r3φ,Out 2:=0,In 3:=Simplify DΓ1r1,r2,r3,r1,r1r1θ,r2ϕ,r3φ,Out 3:=2θθϕθφθϕφ.

WDK2 Method:(29)Γ1r1,r2,r3ufuj=1jinuiuj,where u=rfr/j=1jinrirj,(30)In 1DΓ1r1,r2,r3,r1/.r1θ,r2ϕ,r3φ,Out 10,,In 13Simplify DΓ1r1,r2,r3,r1,r3,r1,r2r1θ,r2ϕ,r3φ,Out 1312θ2.

IWM2 Method:(31)Γ1r1,r2,r3u2j=1jinuiujuj=1jinuiuj+fu,where u=rj=1jinrirj/rj=1jinrirj+fr,(32)In 1DΓ1r1,r2,r3,r1r1θ,r2ϕ,r3φ,Out 10,,In 14Simplify DΓ1r1,r2,r3,r1,r1,r1,r1,r1θ,r2ϕ,r3φ,Out 1424θ3.

(1) Basins of Attraction. To provoke the basins of attraction of iterative schemes WDK, IWDK, WDK2, and IWM2 for the root of nonlinear equation, we execute the real and imaginary parts of the starting approximation as two axes over a mesh of 250×250 in complex plane. Using rk+1rk<103 as a stopping criteria and maximum number of iterations as 25. We allow different colors to mark to which root the iterative scheme converges and black in other case. Color brightness in basins shows less number of iterations. For the generation of basins, we consider the following four nonlinear functions, i.e., f1r=logr+er+1 and f2r=sinr1/2cosr3/2+1.

The elapsed time from Table 1 and brightness in color in Figure 1(d)2(d) shows the dominance behavior of IWM2 over WDK, IWDK, and WDK2, respectively.

Elapsed time in seconds.

MethodWDKIWDKWDK2IWM2
f1r0.129370.1422070.3231900.107267
f2r0.1609210.238890.4319360.153851

(a), (b), (c), and (d) show basins of attraction for nonlinear function f1r=r3+r40 of the iterative methods WDK, IWDK, WDK2, and IWM2 respectively.

The elapsed time from Table 1 and brightness in color in Figure 2(d) show the dominance behavior of IWM2 over WDK, IWDK, and WDK2, respectively.

(a), (b), (c), and (d) show basins of attraction for nonlinear function f2r=sinr1/2cosr3/2+1 of the iterative methods WDK, IWDK, WDK2, and IWM2, respectively.

Computational time in seconds of WDK2 and IWM2 for nonlinear function f3rf6r, respectively.

3. Numerical Results

Some nonlinear models from engineering and physical sciences are considered to illustrate the performance and efficiency of WDK2 and IWM2 using CAS Maple 18 with 64 digits floating point arithmetic for all computer calculations. We approximate the roots of (1) rather than the exact roots which depend on computer precision , and the following stopping criteria are used to terminate the computer program:(33)ei=rik+1rik2<,where ei represents the absolute error. We take =1030. In Tables 25, CO represents convergence order of iterative schemes WDK2 and IWM2, respectively.

Simultaneous finding of all roots.

Methode16e26e36e46
WDK20.00.06.8e − 666.8e − 66
IWM20.00.01.2e − 892.4e − 86

Simultaneous finding of all roots.

Methode13e23e33
WDK28036.08036.020.2
IWM24.9e − 974.9e − 971.7e − 110

Simultaneous finding of all roots.

Methode13e23e33e43
WDK20.20.40.50.7
IWM24.8e − 379.4e − 360.0010.004

Simultaneous finding of all roots.

Methodse13e23e33
WDK29.39.37.5
IWM23.9e − 737.0e − 731.3e − 102
3.1. Applications in Engineering

In this section, we discuss some applications in engineering.

Example 1 (see [<xref ref-type="bibr" rid="B24">24</xref>]).

Fractional Conversion.

As expression described in [25, 26],(34)f3r=r47.79075r3+14.7445r2+2.511r1.674,is the fractional conversion of nitrogen, hydrogen feed at 250 atm. and 227 k.

The exact roots of (34) are(35)ζ1=3.9485+0.3161i,ζ2=3.94850.3161i,ζ3=0.3841,ζ4=0.2778.

The initial calculated values of (34) have been taken as follows:(36)r10=3.5+0.3i,r20=3.50.3i,r30=0.3+0.01i,r40=1.8+0.01i.

Table 2 clearly shows the dominance behavior of IWM2 over WDK2 iterative method in terms of CPU time in seconds and absolute error on same number of iterations k for nonlinear function. f3r.

Example 2 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

Van der Waal’s Fluid Model.

A Van der Waals fluid is the one which satisfies the equation of state:(37)p=Rθvbav2,where R, a, and b are positive constants, P is the pressure, θ is the absolute temperature, and v volume. We obtain a nonlinear equation(38)P+3v3v1=8T,by setting P=27b2p/a, T=27Rbθ/8a, and r=v/3b Taking P=6 and T = 2 in (37), we have(39)18r3+13r2+9r3=0or(40)f4r=18r3+13r2+9r3.

The exact roots of (40) are(41)ζ1=0.4767630.702381i,ζ2=0.476763+0.702381i,ζ3=0.2313104.

The initial calculated values of (40) have been taken as follows:(42)r10=0.40.7i,r20=0.40.7i,r30=0.2.

Table 3 clearly shows the dominance behavior of IWM2 over the WDK2 iterative method in terms of CPU time in seconds and absolute error on the same number of iterations k for nonlinear function f4r.

Example 3 (see [<xref ref-type="bibr" rid="B27">27</xref>]).

Continuous Stirred Tank Reactor (CSTR).

An isothermal stirred tank reactor (CSTR) is considered here. Items A and R are fed to the reactor at rates of Q and q-Q, respectively. Complex reaction developed in the reactor is given as follows:(43)A+RB,B+RC,C+RD,C+RE.

For a simple feedback control system, this problem was first tested by Douglas (see ). During his searching, he designed the following equation of transfer function of the reactor:(44)Hc2.98r+2.25r+1.45r+2.852r+4.35=1.

Hc being the gain of the proportional controller. This transfer function yields the following nonlinear equation by taking Hc=0:(45)f5r=r4+11.50t3+47.49r2+83.06325r+51.23266875=0.

The transfer function has the four negative real roots, i.e., r1=1.45,r2=2.85,r3=2.85, and r4=4.45

The initial calculated values of (45) have been taken as follows:(46)r10=1.0,r20=1.1,r30=2.2,r40=3.9.

Table 4 clearly shows the dominance behavior of IWM2 over the WDK2 iterative method in terms of CPU time in seconds and absolute error on same number of iterations k for nonlinear function f5r.

Example 4.

(see ). Predator-Prey Model.

Consider the Predator-Prey model in which the predation rate is denoted by(47)Pr=kr3a3+r3,a,k>0,where r is the number of aphids as preys  and lady bugs as a predator. Obeying the Mathusian Model, the growth rate of aphids is defined as Gr=r1r,r1>0. To find the solution of the problem, we take the aphid density for which Pr=Gr implies(48)r1r3kr2+r1a3=0.

Taking k = 30 (aphids eaten rate), a = 20 (number of aphids), and r1=21/3 (rate per hour) in (48), we obtain(49)f6r=0.7937005260r330r2+6349.604208.

The exact roots of (49) are(50)ζ1=25.198,ζ2=25.198,ζ3=12.84.

The initial estimates for f6r has been taken as follows:(51)r10=1.8+8.7i,r20=1.88.7i,r30=0.1+0.1i.

Table 5 clearly shows the dominance behavior of IWM2 over WDK2 iterative method in terms of CPU time in seconds and absolute error on the same number of iterations k for nonlinear function f6r.

4. Conclusion

In this work, new two-step derivative-free inverse iterative methods of convergence order 4 for the simultaneous approximations of all roots of a nonlinear equation (1) are introduced and discussed. Dynamical planes and basins of attraction are presented to show the global convergence behavior of inverse simultaneous iterative methods and two-step classical Weierstrass method. Brightness in color in the dynamical planes of IWM2 shows less number of iteration steps as compared to classical simultaneous methods WDK2 for finding all roots of (1). The results of numerical test examples from Tables 25, CPU time from Figure 3, and residual error from Figures 47, corroborate with theoretical analysis and illustrate the effectiveness and rapid convergence of our proposed derivative-free inverse simultaneous iterative method as compared to the WDK2.

Error graph of WDK2 and IWM2 for f3r, respectively.

Error graph of WDK2 and IWM2 for f4r, respectively.

Error graph of WDK2 and IWM2 for f5r, respectively.

Error graph of WDK2 and IWM2 for f6r, respectively.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this article.

Authors’ Contributions

All authors’ contributed equally in the preparation of this manuscript.

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