# Inverse Numerical Iterative Technique for Finding all Roots of Nonlinear Equations with Engineering Applications

Authors: Mudassir Shams; Naila Rafiq; Babar Ahmad; Nazir Ahmad Mir
Journal: Journal of Mathematics
 (2021)

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## Abstract

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.


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## Body

## 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 [1] which has local quadratic convergence:(2)sk=rk−frkf′rk,k=0,1,….Nedzibove et al., in [2], presented the inverse method of the same order corresponding to method (2):(3)sk=rk2f′rkrkf′rk+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 [3], Kanno [4], Proinov [5], Petkovi´c [6], Mir [7], Nourein [8], Aberth [9], and reference cited there in [10–22]).Among derivative-free simultaneous methods, Weierstrass–Dochive [23] method (abbreviated as WDK) is the most attractive method given by(4)sik=rik−wrik,where(5)wrik=frik∏j=1j≠inrik−rjk,i,j=1,2,3,…,n,is Weierstrass’ Correction. Method (4) has local quadratic convergence.Nedzibove [2] introduced a new modification to (4), that is, an inverse method to WDK abbreviated as IWDK, i.e.,(6)uik=rik2∏j=1j≠inrik−rjkrik∏j=1j≠inrik−rjk+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=uik−fuik∏j=1j≠inuik−ujk,where uik=rik−frik/∏j=1j≠inrik−rjk and denote it by WDK2. Let us now convert method (7) into inverse iterative method as follows:(8)zik=uik2∏j=1j≠inuik−ujkuik∏j=1j≠inuik−ujk+fuik,where uik=rik2∏j=1j≠inrik−rjk/rik∏j=1j≠inrik−rjk+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.LetD∈Cn be an open convex subset, Γ:D⟶Cn and m times differentiable operator Γ1r,…,ΓnrT be continuous, and the sequence rkk∈N be defined by rk+1=Γrk:(9)rk=r1k,…,rnk⟺rik+1=Γirk,∀i∈1,...,n,k∈N,where norm in Cn is defined as r=maxr1,…,rn.Theorem 1.
LetX and Y be normed spaces. Take an open convex subset D of X for a u times Frēchet differential operator Γ, i.e., Γ:D⟶Y Then, for any x, y ∈D(10)Γy−Γx−∑j=1q−1j!Γjxy−x…y−x︸j−times≤y−xqq!supζ∈x,yΓqζ.Using Theorem1, we have the following.Theorem 2.
Letβ∈D if(i)
Γβ=β(ii)
Γβ=Γ′β=Γ″β=⋯=Γuβ=0
Then, there existss>0 such that, for any r0∈D,r0−β<s, the sequence rk+1=Γrkk∈N converges to β.Proof.
Lets0>0 be such that(11)v0=r∈C:r−β≤s0⊂D.
C0=maxΓur0z∈v0, and there exists 0<s≤s0 such that(12)C0sqq!<s⟺C0q!1/q−1<s,where v=r∈Cn:r−β≤s. Using hypothesis (2), r∈v; then, (ii) and Theorem 1 implies(13)Γr−β=Γr−Γβ−∑j=1q−11j!Γjβr−β…r−β︸j−times≤1q!r−βqsupζ∈β,rΓqζq≤C0sqq!<s.
Thus,Γr∈v. Using the above relation for r=rk, we have(14)rk+1−β=Γrk−β≤C0u!rk−βq.
Using (14), recursively, we have(15)rk−β≤C0u!rk−βq≤C0u!C0u!rk−βqq≤⋯≤C0q!1+q+⋯+qkr0−βqk≤C0q!1/q−1sqk⟶0fork⟶0.
Thus, from inequality (14), rk∈Nk is at least q. Now, consider IWM2 as a vector function, i.e., Γr=Γ1r,…,Γnr, where(16)Γizi=ui2ui+fui/∏j=1j≠inui−uj,whereui=ri2ri+fri/∏j=1j≠inri−rj.
For a fixed pointβ=β1,…,βn, it is not difficult to prove ∂Γiζ/∂ri=∂2Γiζ/∂ri∂rj=∂3Γiζ/∂2ri∂rj=0 and higher order partial derivative is not equal to zero. Thus, IWM2 has at least fourth-order convergence.Theorem 3.
Letζ1,…,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.
Considerεi=rik−ζi, εi′=uik−ζi, and εi″=zik−ζi be the errors in rik, uik, and zik, respectively. For simplicity, we omit iteration index k. From first step of IWM2, we have(17)ui−ζi=ri−ζi−rifri/∏j=1j≠inri−rjri+fri/∏j=1j≠inri−rj.
Thus, we obtain(18)εi′=εi1−∏j≠ij=1nri−ζj/ri−rj1+fri/∏j=1j≠inri−rj=εi1−∏j≠ij=1nri−ζj/ri−rj+fri/∏j=1j≠inri−rj1+fri/∏j=1j≠inri−rj.
Using the expression∏j≠ij=1nri−ζj/ri−rj−1=∑k≠inεk/ri−rk∏j≠ik−1ri−ζk/ri−rj [2] in (18), we have(19)εi′=εiεi/ri∏j≠ij=1nri−ζj/ri−rj−∑k≠inεk/ri−rk∏j≠ik−1ri−ζk/ri−rj1+εk/ri∏j≠ij=1nri−ζj/ri−rj.
If we assume all errors are of the same order, i.e.,εi=εk=Oε; then, we have(20)εi′=ε21/ri∏j≠ij=1nri−ζj/ri−rj−∑k≠in1/ri−rk∏j≠ik−1ri−ζk/ri−rj1+εk/ri∏j≠ij=1nri−ζj/ri−rj=Oε2.
From second-step of IWM2, we have(21)zi−ζi=ui−ζi−uifui/∏j=1j≠inui−ujri+fui/∏j=1j≠inui−uj.
Thus, we obtain(22)εi″=εi′1−∏j≠ij=1nui−ζj/ui−uj1+fui/∏j=1j≠inui−uj=εi′1−∏j≠ij=1nui−ζj/ui−uj+fui/∏j=1j≠inui−uj1+fui/∏j=1j≠inui−uj.
As from the above argument∏j≠ij=1nui−ζj/ui−uj−1=∑k≠inεk′/ui−uk∏j≠ik−1ui−ζk/ui−uj using in (22), we have(23)εi″=εi′εi′/ui∏j≠ij=1nui−ζj/ui−uj−∑k≠inεk′/ui−uk∏j≠ik−1ui−ζk/ui−uj1+εk′/ui∏j≠ij=1nui−ζj/ui−uj.
If we assume all errors are of the same order, i.e.,εi′=εk′=Oε′; then,(24)εi″=ε′21/ui∏j≠ij=1nui−ζj/ui−uj−∑k≠in1/ui−sk∏j≠ik−1ui−ζk/ui−uj1+εk′/ui∏j≠ij=1nui−ζj/ui−uj=Oε′2=Oε22=Oε4.
Hence, the theorem is proved.


#### 2.1.1. Using CAS for Verification of Convergence Order

Consider(25)fr=r−θr−ϕr−φ,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)∂Γ1r∂r1∂Γ1r∂r2∂Γ1r∂r3,∂2Γ1r∂r12∂2Γ1r∂r1∂r2∂2Γ1r∂r22∂2Γ1r∂r2∂r3,∂3Γ1r∂r13∂3Γ1r∂r12∂r2∂3Γ1r∂r1∂r22∂3Γ1r∂r23∂3Γ1r∂r22∂r3,⋮⋮⋮⋮⋮…,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,r3≔r−fr∏j=1j≠inri−rj,i,j=1,…,n,In1≔DΓ1r1,r2,r3,r1/.r1⟶θ,r2⟶ϕ,r3⟶φ,Out1≔0,In2≔DΓ1r1,r2,r3,r2/.r1⟶θ,r2⟶ϕ,r3⟶φ,Out2≔0,In2≔DΓ1r1,r2,r3,r2/.r1⟶θ,r2⟶ϕ,r3⟶φ,Out2≔0,In3≔SimplifyDΓ1r1,r2,r3,r1,r2/.r1⟶θ,r2⟶ϕ,r3⟶φ,Out3≔1−θ+ϕ.Modified Inverse Weierstrass Method:(28)Γ1r1,r2,r3≔r2∏j=1j≠inri−rjr∏j=1j≠inri−rj+fr,In1:=DΓ1r1,r2,r3,r2r1⟶θ,r2⟶ϕ,r3⟶φ,Out1:=0,In2:=DΓ1r1,r2,r3,r3r1⟶θ,r2⟶ϕ,r3⟶φ,Out2:=0,In3:=SimplifyDΓ1r1,r2,r3,r1,r1r1⟶θ,r2⟶ϕ,r3⟶φ,Out3:=2θθ−ϕθ−φθϕφ.WDK2 Method:(29)Γ1r1,r2,r3≔u−fu∏j=1j≠inui−uj,where u=r−fr/∏j=1j≠inri−rj,(30)In1≔DΓ1r1,r2,r3,r1/.r1⟶θ,r2⟶ϕ,r3⟶φ,Out1≔0,⋮,In13≔SimplifyDΓ1r1,r2,r3,r1,r3,r1,r2r1⟶θ,r2⟶ϕ,r3⟶φ,Out13≔−−12θ2.IWM2 Method:(31)Γ1r1,r2,r3≔u2∏j=1j≠inui−uju∏j=1j≠inui−uj+fu,where u=r∏j=1j≠inri−rj/r∏j=1j≠inri−rj+fr,(32)In1≔DΓ1r1,r2,r3,r1r1⟶θ,r2⟶ϕ,r3⟶φ,Out1≔0,⋮,In14≔SimplifyDΓ1r1,r2,r3,r1,r1,r1,r1,r1⟶θ,r2⟶ϕ,r3⟶φ,Out14≔24θ3.(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+1−rk<10−3 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=sinr−1/2cosr−3/2+1.The elapsed time from Table1 and brightness in color in Figure 1(d)–2(d) shows the dominance behavior of IWM2 over WDK, IWDK, and WDK2, respectively.Table 1
Elapsed time in seconds.
MethodWDKIWDKWDK2IWM2f1r0.129370.1422070.3231900.107267f2r0.1609210.238890.4319360.153851Figure 1
(a), (b), (c), and (d) show basins of attraction for nonlinear functionf1r=r3+r−40 of the iterative methods WDK, IWDK, WDK2, and IWM2 respectively.
(a)(b)(c)(d)The elapsed time from Table1 and brightness in color in Figure 2(d) show the dominance behavior of IWM2 over WDK, IWDK, and WDK2, respectively.Figure 2
(a), (b), (c), and (d) show basins of attraction for nonlinear functionf2r=sinr−1/2cosr−3/2+1 of the iterative methods WDK, IWDK, WDK2, and IWM2, respectively.
(a)(b)(c)(d)Figure 3
Computational time in seconds of WDK2 and IWM2 for nonlinear functionf3r−f6r, respectively.


## 2.1. Convergence Analysis

We prove here that convergence order of the IWM2 method is four.LetD∈Cn be an open convex subset, Γ:D⟶Cn and m times differentiable operator Γ1r,…,ΓnrT be continuous, and the sequence rkk∈N be defined by rk+1=Γrk:(9)rk=r1k,…,rnk⟺rik+1=Γirk,∀i∈1,...,n,k∈N,where norm in Cn is defined as r=maxr1,…,rn.Theorem 1.
LetX and Y be normed spaces. Take an open convex subset D of X for a u times Frēchet differential operator Γ, i.e., Γ:D⟶Y Then, for any x, y ∈D(10)Γy−Γx−∑j=1q−1j!Γjxy−x…y−x︸j−times≤y−xqq!supζ∈x,yΓqζ.Using Theorem1, we have the following.Theorem 2.
Letβ∈D if(i)
Γβ=β(ii)
Γβ=Γ′β=Γ″β=⋯=Γuβ=0
Then, there existss>0 such that, for any r0∈D,r0−β<s, the sequence rk+1=Γrkk∈N converges to β.Proof.
Lets0>0 be such that(11)v0=r∈C:r−β≤s0⊂D.
C0=maxΓur0z∈v0, and there exists 0<s≤s0 such that(12)C0sqq!<s⟺C0q!1/q−1<s,where v=r∈Cn:r−β≤s. Using hypothesis (2), r∈v; then, (ii) and Theorem 1 implies(13)Γr−β=Γr−Γβ−∑j=1q−11j!Γjβr−β…r−β︸j−times≤1q!r−βqsupζ∈β,rΓqζq≤C0sqq!<s.
Thus,Γr∈v. Using the above relation for r=rk, we have(14)rk+1−β=Γrk−β≤C0u!rk−βq.
Using (14), recursively, we have(15)rk−β≤C0u!rk−βq≤C0u!C0u!rk−βqq≤⋯≤C0q!1+q+⋯+qkr0−βqk≤C0q!1/q−1sqk⟶0fork⟶0.
Thus, from inequality (14), rk∈Nk is at least q. Now, consider IWM2 as a vector function, i.e., Γr=Γ1r,…,Γnr, where(16)Γizi=ui2ui+fui/∏j=1j≠inui−uj,whereui=ri2ri+fri/∏j=1j≠inri−rj.
For a fixed pointβ=β1,…,βn, it is not difficult to prove ∂Γiζ/∂ri=∂2Γiζ/∂ri∂rj=∂3Γiζ/∂2ri∂rj=0 and higher order partial derivative is not equal to zero. Thus, IWM2 has at least fourth-order convergence.Theorem 3.
Letζ1,…,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.
Considerεi=rik−ζi, εi′=uik−ζi, and εi″=zik−ζi be the errors in rik, uik, and zik, respectively. For simplicity, we omit iteration index k. From first step of IWM2, we have(17)ui−ζi=ri−ζi−rifri/∏j=1j≠inri−rjri+fri/∏j=1j≠inri−rj.
Thus, we obtain(18)εi′=εi1−∏j≠ij=1nri−ζj/ri−rj1+fri/∏j=1j≠inri−rj=εi1−∏j≠ij=1nri−ζj/ri−rj+fri/∏j=1j≠inri−rj1+fri/∏j=1j≠inri−rj.
Using the expression∏j≠ij=1nri−ζj/ri−rj−1=∑k≠inεk/ri−rk∏j≠ik−1ri−ζk/ri−rj [2] in (18), we have(19)εi′=εiεi/ri∏j≠ij=1nri−ζj/ri−rj−∑k≠inεk/ri−rk∏j≠ik−1ri−ζk/ri−rj1+εk/ri∏j≠ij=1nri−ζj/ri−rj.
If we assume all errors are of the same order, i.e.,εi=εk=Oε; then, we have(20)εi′=ε21/ri∏j≠ij=1nri−ζj/ri−rj−∑k≠in1/ri−rk∏j≠ik−1ri−ζk/ri−rj1+εk/ri∏j≠ij=1nri−ζj/ri−rj=Oε2.
From second-step of IWM2, we have(21)zi−ζi=ui−ζi−uifui/∏j=1j≠inui−ujri+fui/∏j=1j≠inui−uj.
Thus, we obtain(22)εi″=εi′1−∏j≠ij=1nui−ζj/ui−uj1+fui/∏j=1j≠inui−uj=εi′1−∏j≠ij=1nui−ζj/ui−uj+fui/∏j=1j≠inui−uj1+fui/∏j=1j≠inui−uj.
As from the above argument∏j≠ij=1nui−ζj/ui−uj−1=∑k≠inεk′/ui−uk∏j≠ik−1ui−ζk/ui−uj using in (22), we have(23)εi″=εi′εi′/ui∏j≠ij=1nui−ζj/ui−uj−∑k≠inεk′/ui−uk∏j≠ik−1ui−ζk/ui−uj1+εk′/ui∏j≠ij=1nui−ζj/ui−uj.
If we assume all errors are of the same order, i.e.,εi′=εk′=Oε′; then,(24)εi″=ε′21/ui∏j≠ij=1nui−ζj/ui−uj−∑k≠in1/ui−sk∏j≠ik−1ui−ζk/ui−uj1+εk′/ui∏j≠ij=1nui−ζj/ui−uj=Oε′2=Oε22=Oε4.
Hence, the theorem is proved.


### 2.1.1. Using CAS for Verification of Convergence Order

Consider(25)fr=r−θr−ϕr−φ,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)∂Γ1r∂r1∂Γ1r∂r2∂Γ1r∂r3,∂2Γ1r∂r12∂2Γ1r∂r1∂r2∂2Γ1r∂r22∂2Γ1r∂r2∂r3,∂3Γ1r∂r13∂3Γ1r∂r12∂r2∂3Γ1r∂r1∂r22∂3Γ1r∂r23∂3Γ1r∂r22∂r3,⋮⋮⋮⋮⋮…,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,r3≔r−fr∏j=1j≠inri−rj,i,j=1,…,n,In1≔DΓ1r1,r2,r3,r1/.r1⟶θ,r2⟶ϕ,r3⟶φ,Out1≔0,In2≔DΓ1r1,r2,r3,r2/.r1⟶θ,r2⟶ϕ,r3⟶φ,Out2≔0,In2≔DΓ1r1,r2,r3,r2/.r1⟶θ,r2⟶ϕ,r3⟶φ,Out2≔0,In3≔SimplifyDΓ1r1,r2,r3,r1,r2/.r1⟶θ,r2⟶ϕ,r3⟶φ,Out3≔1−θ+ϕ.Modified Inverse Weierstrass Method:(28)Γ1r1,r2,r3≔r2∏j=1j≠inri−rjr∏j=1j≠inri−rj+fr,In1:=DΓ1r1,r2,r3,r2r1⟶θ,r2⟶ϕ,r3⟶φ,Out1:=0,In2:=DΓ1r1,r2,r3,r3r1⟶θ,r2⟶ϕ,r3⟶φ,Out2:=0,In3:=SimplifyDΓ1r1,r2,r3,r1,r1r1⟶θ,r2⟶ϕ,r3⟶φ,Out3:=2θθ−ϕθ−φθϕφ.WDK2 Method:(29)Γ1r1,r2,r3≔u−fu∏j=1j≠inui−uj,where u=r−fr/∏j=1j≠inri−rj,(30)In1≔DΓ1r1,r2,r3,r1/.r1⟶θ,r2⟶ϕ,r3⟶φ,Out1≔0,⋮,In13≔SimplifyDΓ1r1,r2,r3,r1,r3,r1,r2r1⟶θ,r2⟶ϕ,r3⟶φ,Out13≔−−12θ2.IWM2 Method:(31)Γ1r1,r2,r3≔u2∏j=1j≠inui−uju∏j=1j≠inui−uj+fu,where u=r∏j=1j≠inri−rj/r∏j=1j≠inri−rj+fr,(32)In1≔DΓ1r1,r2,r3,r1r1⟶θ,r2⟶ϕ,r3⟶φ,Out1≔0,⋮,In14≔SimplifyDΓ1r1,r2,r3,r1,r1,r1,r1,r1⟶θ,r2⟶ϕ,r3⟶φ,Out14≔24θ3.(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+1−rk<10−3 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=sinr−1/2cosr−3/2+1.The elapsed time from Table1 and brightness in color in Figure 1(d)–2(d) shows the dominance behavior of IWM2 over WDK, IWDK, and WDK2, respectively.Table 1
Elapsed time in seconds.
MethodWDKIWDKWDK2IWM2f1r0.129370.1422070.3231900.107267f2r0.1609210.238890.4319360.153851Figure 1
(a), (b), (c), and (d) show basins of attraction for nonlinear functionf1r=r3+r−40 of the iterative methods WDK, IWDK, WDK2, and IWM2 respectively.
(a)(b)(c)(d)The elapsed time from Table1 and brightness in color in Figure 2(d) show the dominance behavior of IWM2 over WDK, IWDK, and WDK2, respectively.Figure 2
(a), (b), (c), and (d) show basins of attraction for nonlinear functionf2r=sinr−1/2cosr−3/2+1 of the iterative methods WDK, IWDK, WDK2, and IWM2, respectively.
(a)(b)(c)(d)Figure 3
Computational time in seconds of WDK2 and IWM2 for nonlinear functionf3r−f6r, respectively.


## 2.1.1. Using CAS for Verification of Convergence Order

Consider(25)fr=r−θr−ϕr−φ,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)∂Γ1r∂r1∂Γ1r∂r2∂Γ1r∂r3,∂2Γ1r∂r12∂2Γ1r∂r1∂r2∂2Γ1r∂r22∂2Γ1r∂r2∂r3,∂3Γ1r∂r13∂3Γ1r∂r12∂r2∂3Γ1r∂r1∂r22∂3Γ1r∂r23∂3Γ1r∂r22∂r3,⋮⋮⋮⋮⋮…,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,r3≔r−fr∏j=1j≠inri−rj,i,j=1,…,n,In1≔DΓ1r1,r2,r3,r1/.r1⟶θ,r2⟶ϕ,r3⟶φ,Out1≔0,In2≔DΓ1r1,r2,r3,r2/.r1⟶θ,r2⟶ϕ,r3⟶φ,Out2≔0,In2≔DΓ1r1,r2,r3,r2/.r1⟶θ,r2⟶ϕ,r3⟶φ,Out2≔0,In3≔SimplifyDΓ1r1,r2,r3,r1,r2/.r1⟶θ,r2⟶ϕ,r3⟶φ,Out3≔1−θ+ϕ.Modified Inverse Weierstrass Method:(28)Γ1r1,r2,r3≔r2∏j=1j≠inri−rjr∏j=1j≠inri−rj+fr,In1:=DΓ1r1,r2,r3,r2r1⟶θ,r2⟶ϕ,r3⟶φ,Out1:=0,In2:=DΓ1r1,r2,r3,r3r1⟶θ,r2⟶ϕ,r3⟶φ,Out2:=0,In3:=SimplifyDΓ1r1,r2,r3,r1,r1r1⟶θ,r2⟶ϕ,r3⟶φ,Out3:=2θθ−ϕθ−φθϕφ.WDK2 Method:(29)Γ1r1,r2,r3≔u−fu∏j=1j≠inui−uj,where u=r−fr/∏j=1j≠inri−rj,(30)In1≔DΓ1r1,r2,r3,r1/.r1⟶θ,r2⟶ϕ,r3⟶φ,Out1≔0,⋮,In13≔SimplifyDΓ1r1,r2,r3,r1,r3,r1,r2r1⟶θ,r2⟶ϕ,r3⟶φ,Out13≔−−12θ2.IWM2 Method:(31)Γ1r1,r2,r3≔u2∏j=1j≠inui−uju∏j=1j≠inui−uj+fu,where u=r∏j=1j≠inri−rj/r∏j=1j≠inri−rj+fr,(32)In1≔DΓ1r1,r2,r3,r1r1⟶θ,r2⟶ϕ,r3⟶φ,Out1≔0,⋮,In14≔SimplifyDΓ1r1,r2,r3,r1,r1,r1,r1,r1⟶θ,r2⟶ϕ,r3⟶φ,Out14≔24θ3.(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+1−rk<10−3 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=sinr−1/2cosr−3/2+1.The elapsed time from Table1 and brightness in color in Figure 1(d)–2(d) shows the dominance behavior of IWM2 over WDK, IWDK, and WDK2, respectively.Table 1
Elapsed time in seconds.
MethodWDKIWDKWDK2IWM2f1r0.129370.1422070.3231900.107267f2r0.1609210.238890.4319360.153851Figure 1
(a), (b), (c), and (d) show basins of attraction for nonlinear functionf1r=r3+r−40 of the iterative methods WDK, IWDK, WDK2, and IWM2 respectively.
(a)(b)(c)(d)The elapsed time from Table1 and brightness in color in Figure 2(d) show the dominance behavior of IWM2 over WDK, IWDK, and WDK2, respectively.Figure 2
(a), (b), (c), and (d) show basins of attraction for nonlinear functionf2r=sinr−1/2cosr−3/2+1 of the iterative methods WDK, IWDK, WDK2, and IWM2, respectively.
(a)(b)(c)(d)Figure 3
Computational time in seconds of WDK2 and IWM2 for nonlinear functionf3r−f6r, 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+1−rik2<∈,where ei represents the absolute error. We take ∈=10−30. In Tables 2–5, CO represents convergence order of iterative schemes WDK2 and IWM2, respectively.Table 2
Simultaneous finding of all roots.
Methode16e26e36e46WDK20.00.06.8e − 666.8e − 66IWM20.00.01.2e − 892.4e − 86Table 3
Simultaneous finding of all roots.
Methode13e23e33WDK28036.08036.020.2IWM24.9e − 974.9e − 971.7e − 110Table 4
Simultaneous finding of all roots.
Methode13e23e33e43WDK20.20.40.50.7IWM24.8e − 379.4e − 360.0010.004Table 5
Simultaneous finding of all roots.
Methodse13e23e33WDK29.39.37.5IWM23.9e − 737.0e − 731.3e − 102


### 3.1. Applications in Engineering

In this section, we discuss some applications in engineering.Example 1 (see [24]).
Fractional Conversion.
As expression described in [25, 26],(34)f3r=r4−7.79075r3+14.7445r2+2.511r−1.674,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=3.9485−0.3161i,ζ3=−0.3841,ζ4=0.2778.
The initial calculated values of (34) have been taken as follows:(36)r10=3.5+0.3i,r20=3.5−0.3i,r30=−0.3+0.01i,r40=1.8+0.01i.
Table2 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 [6]).
Van der Waal’s Fluid Model.
A Van der Waals fluid is the one which satisfies the equation of state:(37)p=Rθv−b−av2,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+3v3v−1=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+9r−3=0or(40)f4r=18r3+13r2+9r−3.
The exact roots of (40) are(41)ζ1=−0.476763−0.702381i,ζ2=−0.476763+0.702381i,ζ3=0.2313104.
The initial calculated values of (40) have been taken as follows:(42)r10=−0.4−0.7i,r20=−0.4−0.7i,r30=0.2.
Table3 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 [27]).
Continuous Stirred Tank Reactor (CSTR).
An isothermal stirred tank reactor (CSTR) is considered here. Items A andR 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+R⟶B,B+R⟶C,C+R⟶D,C+R⟶E.
For a simple feedback control system, this problem was first tested by Douglas (see [28]). 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,andr4=−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.
Table4 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 [16]). 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 [6] and lady bugs as a predator. Obeying the Mathusian Model, the growth rate of aphids is defined as Gr=r1∗r,r1∗>0. To find the solution of the problem, we take the aphid density for which Pr=Gr implies(48)r1∗r3−kr2+r1∗a3=0.
Takingk = 30 (aphids eaten rate), a = 20 (number of aphids), and r1∗=2−1/3 (rate per hour) in (48), we obtain(49)f6r=0.7937005260r3−30r2+6349.604208.
The exact roots of (49) are(50)ζ1=25.198,ζ2=25.198,ζ3=12.84.
The initial estimates forf6r has been taken as follows:(51)r10=1.8+8.7i,r20=1.8−8.7i,r30=0.1+0.1i.
Table5 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.


## 3.1. Applications in Engineering

In this section, we discuss some applications in engineering.Example 1 (see [24]).
Fractional Conversion.
As expression described in [25, 26],(34)f3r=r4−7.79075r3+14.7445r2+2.511r−1.674,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=3.9485−0.3161i,ζ3=−0.3841,ζ4=0.2778.
The initial calculated values of (34) have been taken as follows:(36)r10=3.5+0.3i,r20=3.5−0.3i,r30=−0.3+0.01i,r40=1.8+0.01i.
Table2 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 [6]).
Van der Waal’s Fluid Model.
A Van der Waals fluid is the one which satisfies the equation of state:(37)p=Rθv−b−av2,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+3v3v−1=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+9r−3=0or(40)f4r=18r3+13r2+9r−3.
The exact roots of (40) are(41)ζ1=−0.476763−0.702381i,ζ2=−0.476763+0.702381i,ζ3=0.2313104.
The initial calculated values of (40) have been taken as follows:(42)r10=−0.4−0.7i,r20=−0.4−0.7i,r30=0.2.
Table3 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 [27]).
Continuous Stirred Tank Reactor (CSTR).
An isothermal stirred tank reactor (CSTR) is considered here. Items A andR 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+R⟶B,B+R⟶C,C+R⟶D,C+R⟶E.
For a simple feedback control system, this problem was first tested by Douglas (see [28]). 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,andr4=−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.
Table4 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 [16]). 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 [6] and lady bugs as a predator. Obeying the Mathusian Model, the growth rate of aphids is defined as Gr=r1∗r,r1∗>0. To find the solution of the problem, we take the aphid density for which Pr=Gr implies(48)r1∗r3−kr2+r1∗a3=0.
Takingk = 30 (aphids eaten rate), a = 20 (number of aphids), and r1∗=2−1/3 (rate per hour) in (48), we obtain(49)f6r=0.7937005260r3−30r2+6349.604208.
The exact roots of (49) are(50)ζ1=25.198,ζ2=25.198,ζ3=12.84.
The initial estimates forf6r has been taken as follows:(51)r10=1.8+8.7i,r20=1.8−8.7i,r30=0.1+0.1i.
Table5 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 2–5, CPU time from Figure 3, and residual error from Figures 4–7, 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.Figure 4
Error graph of WDK2 and IWM2 forf3r, respectively.Figure 5
Error graph of WDK2 and IWM2 forf4r, respectively.Figure 6
Error graph of WDK2 and IWM2 forf5r, respectively.Figure 7
Error graph of WDK2 and IWM2 forf6r, respectively.



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