# On Dynamics of Iterative Techniques for Nonlinear Equation with Applications in Engineering

Authors: Mudassir Shams; Nazir Ahmad Mir; Naila Rafiq; A. Othman Almatroud; Saima Akram
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this article, we construct an optimal family of iterative methods for finding the single root and then extend this family for determining all the distinct as well as multiple roots of single-variable nonlinear equations simultaneously. Convergence analysis is presented for both the cases to show that the optimal order of convergence is 4 in the case of single root finding methods and 6 for simultaneous determination of all distinct as well as multiple roots of a nonlinear equation. The computational cost, basins of attraction, efficiency, log of residual, and numerical test examples show that the newly constructed methods are more efficient as compared to the existing methods in the literature.


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## Body

## 1. Introduction

To solve a nonlinear equation,(1)fs=0is the oldest problem of science, in general, and mathematics, in particular. The nonlinear equations have diverse applications in many areas of science and engineering. In general, to find the roots of (1), we look towards iterative schemes, which can be further classified as to approximate the single root and all roots of (1). In this article, we are going to work on both types of iterative methods. Many iterative methods of different convergence orders already exist in the literature (see [1–11]) to approximate the roots of (1). Ostrowski [12] defined the efficiency index I to classify these iterative methods in terms of their convergence order k and number of function evaluations or its derivatives per iteration, say v, i.e.,(2)I=k1/v.An iterative method is said to be optimal according to the Kung–Traub conjecture [13], if(3)H=2v−1.holds, where H is the optimal order of convergence. The aforementioned methods are used to approximate one root at a time. But, mathematicians are also interested in finding all the roots of (1) simultaneously. This is due to the fact that simultaneous iterative methods are very popular due to their wider region of convergence, are more stable as compared to single root finding methods, and can be implemented for parallel computing as well. More detail on single as well as simultaneous determination of all roots can be found in [1, 10–28] and the reference cited therein.The most famous of the single root finding methods is the classical Newton–Raphson method:(4)si+1=si−fsif′si,i=1,2,….Method (4) is optimal with an efficiency of 1.41 by the Traub conjecture. If we use Weierstrass’ correction [26]:(5)fsif′si=ws=fsi∏j=1ns−sj,in (4), then, we get classical Weierstrass–Dochev methods to approximate all roots of nonlinear equation (1) as(6)si+1=si−fsi∏j=1j≠insi−sj,i,j=1,2,3,…,n.Method (6) has a convergence order 2. Later, Aberth [29] presented the 3rd order simultaneous method, given as(7)si+1=si−11/Nsi−∑j=1j≠in1/si−sj,where Nsi=f′si/fsi.First of all, we construct a family of optimal fourth-order methods using the procedure of weight function and then convert it into simultaneous iterative methods for finding all distinct as well as multiple roots of nonlinear equation (1).


## 2. Construction of Methods and Convergence Analysis

King [9] presented the following optimal fourth-order method (abbreviated as E1):(8)yi=si−fsif′si,zi=yi−fsi+βfyifsi+β−2fyifyif′si.Cordero et al. [7] gave the fourth-order optimal method as follows (abbreviated as E2):(9)yi=si−fsif′si,zi=si−fsi+fyif′si−fyi22fsi+fyifsi2f′si.Chun [4] in 2008 gave the fourth-order optimal method as follows (abbreviated as E3):(10)yi=si−fsif′si,zi=si−fsif′si1+fyifsi+2fyifsi2.Maheshwari [11] gave the fourth-order optimal method as follows (abbreviated as E4):(11)yi=si−fsif′si,zi=si−1f′sifsi2fyi−fsi−fyi2fsi.Chun [3] gave the fourth-order optimal method as follows (abbreviated as E5):(12)yi=si−fsif′si,zi=si−fsif′si−fyif′sifsifsi−fyi2.Kou et al. [10] gave the fourth-order optimal method as follows (abbreviated as E6):(13)yi=si−fsif′si,zi=si−fsi2+fyi2f′sifsi−fyi.Behzad [5] give the fourth-order optimal method as follows (abbreviated as E7):(14)yi=si−fsif′si,zi=si−fsif′si−fsi+2fyifsi+fyi2fyif′si.Chun [2] in 2006 gave the fourth-order optimal method as follows (abbreviated as E8):(15)yi=si−fsif′si,zi=yi−fyif′sifsi+2fyifsi.Ostrowski [12] give the fourth-order optimal method as follows (abbreviated as E9):(16)yi=si−fsif′si,zi=yi−si−yifyifsi−2fyi.Here, we propose the following two families of iterative methods:(17)yi=si−fsif′si,zi=yi−fyif′siβfyi2+Ψu3,where u=fyi/fsi and β is a real number.For the iteration schemes (17), we have the following convergence theorem by using CAS Maple 18 and error relation of the iterative schemes defined in (17) is found.Theorem 1.
Letζ∈I be a simple root of a sufficiently differential able function f:I⊆R⟶R in an open interval I. If s0 is sufficiently close to ζ and Ψu is a real-valued function satisfying Ψ0=1,Ψ′0=2/3, and Ψ″0<∞, then the convergence order of the family of iterative method (17) is four and the error equation is given by(18)ei+1=113c23−c2c3−32c23Ψ″0ei4+Oei5,where cm=fmζ/m!f′ζ,m≥2.Proof.
Letζ be a simple root of f and si=ζ+ei. By Taylor’s series expansion of fsi around s=ζ, taking fζ=0, we get(19)fsi=f′ζei+c2ei2+c3ei3+c4ei4+Oei5,and(20)f′si=f′ζ1+2c2ei+3c3ei2+4c4ei3+Oei4.
Dividing (19) by (20), we have(21)fsif′si=e−c2ei2+2c2+2c3ei3+Oei4,so(22)yi=ζ+c2ei2+−2c22+2c3ei3+⋯,fyi=c2ei2+−4c22+2c3ei3+⋯.
Now,(23)fyif′si=c2ei2+−4c22+2c3ei2+⋯,fyifsi=u=c2ei+−3c22+2c3ei2+⋯.
ExpandingΨu about the origin, we have(24)Ψu=Ψ0+Ψ′0u+12Ψ″0u2+⋯=Ψ0+Ψ′0c2ei+−3Ψ′0c22+2Ψ′0c3+12Ψ′0c22ei2+⋯,βfyi2+Ψu3=Ψ30+3Ψ′0Ψ20c2ei+⋯,(25)ei+1=−Ψ30c2+c22ei2+−3Ψ0Ψ′0c22+4Ψ30c22−2Ψ30c3−2c22+2c3ei3+⋯.
By puttingΨ0=1andΨ′0=2/3 in (25), we have(26)ei+1=113c23−c2c3−32c23Ψ″0ei4+Oei5.
Hence, it proves the theorem.


### 2.1. The Concrete Fourth-Order Methods

We now construct some concrete forms of the family of methods described by (17). Let us take the function Ψu satisfying the conditions of Theorem 1.Therefore, we get the following new three iterative methods with arbitrary constantβ∈ℜ and by choosing different weight functions given in Table 1:Table 1
Weight functions for methods.
S. no.Ψu,u=fyi/fsi,Ψ0=1,Ψ′0=2/31Ψu=1+2/3u2Ψu=1+2/3u+αu23Ψu=1/1−2/3uNote. u=fyi/fsi and α∈ℜ.Concrete method 1 (abbreviated as Q1):(27)yi=si−fsif′si,zi=yi−fyif′siβfyi2+1+23u3.Concrete method 2 (abbreviated as Q2):(28)yi=si−fsif′si,zi=yi−fyif′siβfyi2+1+23u+αu23.Concrete method 3 (abbreviated as Q3):(29)yi=si−fsif′si,zi=yi−fyif′siβfyi2+11−2/3u3,where u=fyi/fsi and β∈ℜ.


### 2.2. Complex Dynamical Study of Families of Iterative Methods

Here, we discuss the dynamical study of iterative methods (Q1–Q3 and E1–E9). We investigate the regions, from where we take the initial estimates to achieve the roots of the nonlinear equation. We actually numerically approximate the domain of attractions of the roots as a qualitative measure, that is, how the iterative methods depend on the choice of initial estimates? To answer this question on the dynamical behavior of the iterative methods, we investigate the dynamics of the methods (Q1–Q3) and compare it with (E1–E9). Let us recall some basic concepts of this study in the background contexture of complex dynamics. For more details on the dynamical behavior of the iterative methods, one can consult [6, 30, 31]. Taking a rational function ℜf:C⟶C, where C denotes the complex plane, the orbit s0∈C is defined as a set such as orbs=s0,ℜfs0,ℜf2s0,…,ℜfms0,…. The convergence of orbs⟶s∗ is understood in the sense, if limx⟶∞Rks=s∗ exists. A point s0∈C is known as periodic with a minimal period m if Rms0=s0 holds, where m is the smallest positive integer. A periodic point for m=1 is known as fixed, attracting if Rk′s<1, repelling if Rk′s>1, and neutral otherwise. An attracting point s0∈C defines the basin of attraction as the set of starting points whose orbit tends to s∗. The closure of the set of its repelling periodic points of a rational map is known as the Julia set denoted by JR, and its complement is the Fatou set denoted by FR. The iterative methods when applied to find the roots of (1) provide the rational map. But, we are interested in the basins of attraction of the roots of nonlinear function (1). Fatou set FQ contains the basins of attraction of different roots is a well-known fact. In general, the Julia set (a fractal and rational map) behaves as unstable in this region. For the dynamical and graphical point of view, we take 2000×2000 grids of square −2.5,2.52∈C2 . To each root of (1), we assign a color to which the corresponding orbit of the iterative method starts and converges to a fixed point. Take color map as Jet. We use si+1−si<10−3 as a stopping criteria, and the maximum number of iterations is taken as 20. We mark a dark blue point, if the orbit of the iterative method does not converge to the root after 20 iterations which means it has a distance greater than to any other root. Different color is used for different roots. Iterative methods have different basins of attraction distinguished by their colors. In basins, brightness in color represents the number of iterations to achieve the root of (1). Note that the darkest blue regions denote the lack of convergence to any root of (1). Finally, in Tables 2–5, we present the elapsed time of basins of attraction corresponding to iterative maps (Q1–Q3 and E1–E9) using the tic-toc command in code using MATLAB (R2011b). Figures 1–4 show the basins of attraction of iterative methods (Q1–Q3 and E1–E9) for nonlinear functions f1s=s3+s−1, f2s=e−s−1, f3s=s3+2.87s2−10.28/4.62−s, and f4s=sinhs+2/2sinhs−3/3, respectively. By observing the basins of attraction, we can easily judge the stability of iterative methods (Q1–Q3 and E1–E9). Elapsed time, divergent regions, and brightness in color presents that Q1–Q3 is better than E1–E9.Table 2
Elapsed time for basins of attraction.
f1s=s3+s−1Q1Q2Q3E1E2E3E4E5E6E7E8E917.401312.180111.832113.74511.861211.660323.204212.587815.145612.622617.936012.9451Table 3
Elapsed time for basins of attraction.
f2s=e−s−1Q1Q2Q3E1E2E3E4E5E6E7E8E912.554810.44879.484310.51349.024611.468622.609912.350712.547111.233158.80948.6127Table 4
Elapsed time for basins of attraction.
f3s=s3+2.87s2−10.28/4.62−sQ1Q2Q3E1E2E3E4E5E6E7E8E920.205814.082612.746514.616713.442914.460222.101415.695519.323515.068920.477115.9503Table 5
Elapsed time for basins of attraction.
f4s=sinhs+2/2sinhs−3/3Q1Q2Q3E1E2E3E4E5E6E7E8E935.662139.645340.063933.843539.189538.964567.923346.362156.981033.991442.239128.6589Figure 1
Basins of attraction of iterative methods (Q1–Q3 and E1–E9) for nonlinear function f1s.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Figure 2
Basins of attraction of iterative methods (Q1–Q3 and E1–E9) for nonlinear function f2s.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Figure 3
Basins of attraction of iterative methods (Q1–Q3 and E1–E9) for nonlinear function f3s.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Figure 4
Basins of attraction of iterative methods (Q1–Q3 and E1–E9) for nonlinear function f4s.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Figures1(a)–1(l) present the basins of attraction of methods (Q1–Q3 and E1–E9) for a nonlinear function f1s=s3+s−1. In Figures 1(a)–1(l), brightness of the color in basins shows the less number of iterations for convergence of iterative methods. Table 2 shows the elapsed time of Q1–Q3 and E1–E9.Figures2(a)–2(l) present the basins of attraction of methods (Q1–Q3 and E1–E9) for a nonlinear function f2s=e−s−1. In Figures 2(a)–2(l), brightness of the color in basins shows the less number of iterations for convergence of iterative methods. Table 3 shows the elapsed time of Q1–Q3 and E1–E9.Figures3(a)–3(l) present the basins of attraction of methods (Q1–Q3 and E1–E9) for a nonlinear function f3s=s3+2.87s2−10.28/4.62−s. In Figures 3(a)–3(l), brightness of the color in basins shows the less number of iterations for convergence of iterative methods. Table 4 shows the elapsed time of Q1–Q3 and E1–E9.Figures4(a)–4(l) present the basins of attraction of methods (Q1–Q3 and E1–E9) for a nonlinear function f4s=sinhs+2/2sinhs−3/3. In Figures 4(a)–4(l), brightness of the color in basins shows the less number of iterations for convergence of iterative methods. Table 5 shows the elapsed time of Q1–Q3 and E1–E9.


## 2.1. The Concrete Fourth-Order Methods

We now construct some concrete forms of the family of methods described by (17). Let us take the function Ψu satisfying the conditions of Theorem 1.Therefore, we get the following new three iterative methods with arbitrary constantβ∈ℜ and by choosing different weight functions given in Table 1:Table 1
Weight functions for methods.
S. no.Ψu,u=fyi/fsi,Ψ0=1,Ψ′0=2/31Ψu=1+2/3u2Ψu=1+2/3u+αu23Ψu=1/1−2/3uNote. u=fyi/fsi and α∈ℜ.Concrete method 1 (abbreviated as Q1):(27)yi=si−fsif′si,zi=yi−fyif′siβfyi2+1+23u3.Concrete method 2 (abbreviated as Q2):(28)yi=si−fsif′si,zi=yi−fyif′siβfyi2+1+23u+αu23.Concrete method 3 (abbreviated as Q3):(29)yi=si−fsif′si,zi=yi−fyif′siβfyi2+11−2/3u3,where u=fyi/fsi and β∈ℜ.


## 2.2. Complex Dynamical Study of Families of Iterative Methods

Here, we discuss the dynamical study of iterative methods (Q1–Q3 and E1–E9). We investigate the regions, from where we take the initial estimates to achieve the roots of the nonlinear equation. We actually numerically approximate the domain of attractions of the roots as a qualitative measure, that is, how the iterative methods depend on the choice of initial estimates? To answer this question on the dynamical behavior of the iterative methods, we investigate the dynamics of the methods (Q1–Q3) and compare it with (E1–E9). Let us recall some basic concepts of this study in the background contexture of complex dynamics. For more details on the dynamical behavior of the iterative methods, one can consult [6, 30, 31]. Taking a rational function ℜf:C⟶C, where C denotes the complex plane, the orbit s0∈C is defined as a set such as orbs=s0,ℜfs0,ℜf2s0,…,ℜfms0,…. The convergence of orbs⟶s∗ is understood in the sense, if limx⟶∞Rks=s∗ exists. A point s0∈C is known as periodic with a minimal period m if Rms0=s0 holds, where m is the smallest positive integer. A periodic point for m=1 is known as fixed, attracting if Rk′s<1, repelling if Rk′s>1, and neutral otherwise. An attracting point s0∈C defines the basin of attraction as the set of starting points whose orbit tends to s∗. The closure of the set of its repelling periodic points of a rational map is known as the Julia set denoted by JR, and its complement is the Fatou set denoted by FR. The iterative methods when applied to find the roots of (1) provide the rational map. But, we are interested in the basins of attraction of the roots of nonlinear function (1). Fatou set FQ contains the basins of attraction of different roots is a well-known fact. In general, the Julia set (a fractal and rational map) behaves as unstable in this region. For the dynamical and graphical point of view, we take 2000×2000 grids of square −2.5,2.52∈C2 . To each root of (1), we assign a color to which the corresponding orbit of the iterative method starts and converges to a fixed point. Take color map as Jet. We use si+1−si<10−3 as a stopping criteria, and the maximum number of iterations is taken as 20. We mark a dark blue point, if the orbit of the iterative method does not converge to the root after 20 iterations which means it has a distance greater than to any other root. Different color is used for different roots. Iterative methods have different basins of attraction distinguished by their colors. In basins, brightness in color represents the number of iterations to achieve the root of (1). Note that the darkest blue regions denote the lack of convergence to any root of (1). Finally, in Tables 2–5, we present the elapsed time of basins of attraction corresponding to iterative maps (Q1–Q3 and E1–E9) using the tic-toc command in code using MATLAB (R2011b). Figures 1–4 show the basins of attraction of iterative methods (Q1–Q3 and E1–E9) for nonlinear functions f1s=s3+s−1, f2s=e−s−1, f3s=s3+2.87s2−10.28/4.62−s, and f4s=sinhs+2/2sinhs−3/3, respectively. By observing the basins of attraction, we can easily judge the stability of iterative methods (Q1–Q3 and E1–E9). Elapsed time, divergent regions, and brightness in color presents that Q1–Q3 is better than E1–E9.Table 2
Elapsed time for basins of attraction.
f1s=s3+s−1Q1Q2Q3E1E2E3E4E5E6E7E8E917.401312.180111.832113.74511.861211.660323.204212.587815.145612.622617.936012.9451Table 3
Elapsed time for basins of attraction.
f2s=e−s−1Q1Q2Q3E1E2E3E4E5E6E7E8E912.554810.44879.484310.51349.024611.468622.609912.350712.547111.233158.80948.6127Table 4
Elapsed time for basins of attraction.
f3s=s3+2.87s2−10.28/4.62−sQ1Q2Q3E1E2E3E4E5E6E7E8E920.205814.082612.746514.616713.442914.460222.101415.695519.323515.068920.477115.9503Table 5
Elapsed time for basins of attraction.
f4s=sinhs+2/2sinhs−3/3Q1Q2Q3E1E2E3E4E5E6E7E8E935.662139.645340.063933.843539.189538.964567.923346.362156.981033.991442.239128.6589Figure 1
Basins of attraction of iterative methods (Q1–Q3 and E1–E9) for nonlinear function f1s.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Figure 2
Basins of attraction of iterative methods (Q1–Q3 and E1–E9) for nonlinear function f2s.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Figure 3
Basins of attraction of iterative methods (Q1–Q3 and E1–E9) for nonlinear function f3s.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Figure 4
Basins of attraction of iterative methods (Q1–Q3 and E1–E9) for nonlinear function f4s.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Figures1(a)–1(l) present the basins of attraction of methods (Q1–Q3 and E1–E9) for a nonlinear function f1s=s3+s−1. In Figures 1(a)–1(l), brightness of the color in basins shows the less number of iterations for convergence of iterative methods. Table 2 shows the elapsed time of Q1–Q3 and E1–E9.Figures2(a)–2(l) present the basins of attraction of methods (Q1–Q3 and E1–E9) for a nonlinear function f2s=e−s−1. In Figures 2(a)–2(l), brightness of the color in basins shows the less number of iterations for convergence of iterative methods. Table 3 shows the elapsed time of Q1–Q3 and E1–E9.Figures3(a)–3(l) present the basins of attraction of methods (Q1–Q3 and E1–E9) for a nonlinear function f3s=s3+2.87s2−10.28/4.62−s. In Figures 3(a)–3(l), brightness of the color in basins shows the less number of iterations for convergence of iterative methods. Table 4 shows the elapsed time of Q1–Q3 and E1–E9.Figures4(a)–4(l) present the basins of attraction of methods (Q1–Q3 and E1–E9) for a nonlinear function f4s=sinhs+2/2sinhs−3/3. In Figures 4(a)–4(l), brightness of the color in basins shows the less number of iterations for convergence of iterative methods. Table 5 shows the elapsed time of Q1–Q3 and E1–E9.


## 3. Generalization to Simultaneous Methods

Suppose nonlinear equation (1) has n roots. Then, fs and f′s can be approximated as(30)fs=∏j=1ns−sj,f′s=∑k=1n∏j≠kj=1ns−sj.This implies(31)f′sfs=∑j=1n1s−sj=11/s−si−∑j=1j≠in1/s−sj.This gives the Aberth–Ehrlich method (7):(32)si+1=si−11/Nsi−∑j=1j≠in1/si−sj,whereNsi=fsif′si.Now from (31), an approximation of fsi/f′si is formed by replacing sj with ztj as follows:(33)fsif′si=11/Nsi−∑j=1j≠in1/si−ztj,t=1,2,3.Using (33) in (4), we have the following method for finding all the distinct roots:(34)si+1=si−11/Nsi−∑j=1j≠in1/si−ztj.In case of multiple roots, we have the following method:(35)si+1=si−11/Nsi−∑j=1j≠inσj/si−ztj,t=1,2,3,where σi is the multiplicity of the root si and(36)z1j=yj−fyjf′sjβfyj2+1+23u∗3,z2j=yj−fyjf′sjβfyj2+1+23u∗+αu∗23,z3j=yj−fyjf′sjβfyj2+11−2/3u∗3,in which yj=sj−fsj/f′sj and u∗=fyj/fsj. Using correction ztj, we get the following new family of simultaneous iterative methods for extracting multiple roots of nonlinear equation (1):(37)si+1=si−σi1/Nsi−∑j=1j≠inσj/si−ztj,t=1,2,3.Thus, we have constructed new three simultaneous iterative methods (37) abbreviated as SM1–SM3.


### 3.1. Convergence Analysis

In this section, we discuss the convergence analysis of a family of simultaneous methods (SM1–SM3) which is given in form of the following theorem. Obviously, convergence for the method (34) will follow from the convergence of the method (SM1–SM3) from Theorem 2 when the multiplicities of the roots are one.Theorem 2.
Letζ1,ζ2,…,ζn be the n number of simple roots of nonlinear equation (1). If s10, s20, s30, …, sn0 be the initial approximations of the roots, respectively, and sufficiently close to actual roots, then the order of convergence of the method (SM1–SM3) equals six.Proof.
Letεi=si−ζi and εi′=si+1−ζi be the errors in si and si+1 approximations, respectively. Then, obviously for distinct roots(38)1Nsi=f′sifsi=∑j=1n1si−ζj=1si−ζi+∑j≠ij=1n1si−ζj.
Thus, for multiple roots, we have from (37) that(39)si+1−ζi=si−ζi−σiσi/si−ζi+∑j≠ij=1nσj/si−ζi−∑j≠ij=1nσj/si−ztj,εi′=εi−σiσi/εi+∑j≠ij=1n−σjztj−ζj/si−ζjsi−ztj=εi−σiεiσi+εi∑j≠ij=1n−σjztj−ζj/si−ζjsi−ztj=εi−σiεiσi+εi∑j≠ij=1nEiεj4,where ztj−ζj=εj4 from (18) and Ei=−σj/si−ζjsi−ztj. Thus,(40)εi′=εi2∑j≠ij=1nEiεj4σi+εi∑j≠ij=1nEiεj4.
If it is assumed that absolute values of all errorsεjj=1,2,3,… are of the same order, say εj=Oε, then from (40), we have(41)εi′=Oεi6.
Hence, the theorem is proved.


## 3.1. Convergence Analysis

In this section, we discuss the convergence analysis of a family of simultaneous methods (SM1–SM3) which is given in form of the following theorem. Obviously, convergence for the method (34) will follow from the convergence of the method (SM1–SM3) from Theorem 2 when the multiplicities of the roots are one.Theorem 2.
Letζ1,ζ2,…,ζn be the n number of simple roots of nonlinear equation (1). If s10, s20, s30, …, sn0 be the initial approximations of the roots, respectively, and sufficiently close to actual roots, then the order of convergence of the method (SM1–SM3) equals six.Proof.
Letεi=si−ζi and εi′=si+1−ζi be the errors in si and si+1 approximations, respectively. Then, obviously for distinct roots(38)1Nsi=f′sifsi=∑j=1n1si−ζj=1si−ζi+∑j≠ij=1n1si−ζj.
Thus, for multiple roots, we have from (37) that(39)si+1−ζi=si−ζi−σiσi/si−ζi+∑j≠ij=1nσj/si−ζi−∑j≠ij=1nσj/si−ztj,εi′=εi−σiσi/εi+∑j≠ij=1n−σjztj−ζj/si−ζjsi−ztj=εi−σiεiσi+εi∑j≠ij=1n−σjztj−ζj/si−ζjsi−ztj=εi−σiεiσi+εi∑j≠ij=1nEiεj4,where ztj−ζj=εj4 from (18) and Ei=−σj/si−ζjsi−ztj. Thus,(40)εi′=εi2∑j≠ij=1nEiεj4σi+εi∑j≠ij=1nEiεj4.
If it is assumed that absolute values of all errorsεjj=1,2,3,… are of the same order, say εj=Oε, then from (40), we have(41)εi′=Oεi6.
Hence, the theorem is proved.


## 4. Computational Aspect

Here, we compare the computational efficiencies of the Petkovic et al. [21] method and the new methods (SM1–SM3). As presented in [21], the efficiency of an iterative method can be estimated using the efficiency index given by(42)ELm=loguD,where D is the computational cost and u is the order of convergence of the iterative method. Arithmetic operations per iteration with certain weight depending on the execution time of operation are used to evaluate the computational cost D . The weights used for division, multiplication, and addition plus subtraction are wd,wm,andwAS, respectively. For a given polynomial of degree m and n roots, the number of division, multiplication, and addition and subtraction per iteration for all roots are denoted by Dm,Mm, and ASm. The cost of computation can be calculated as(43)D=Dm=wasASm+wmMm+wdDm.Thus, (42) becomes(44)ELm=loguwasASm+wmMm+wdDm.Consider that the number of operations of a complex polynomial with real and complex roots reduces to the operation of real arithmetic, which is given in Table6 as a polynomial degree m taking the dominant term of order m2. Applying (44) and the data given in Table 6, we calculate the percentage ratio ρSM1–SM3,PJ6 [21] given by(45)ρSM1–SM3,PJ6=ELSM1–SM3ELPJ6−1×100,ρPJ6,SM1–SM3=ELPJ6ELSM1–SM3−1×100,where PJ6 is the Petkovic method [21] of order 4. Figures 5(a)–5(d) graphically illustrate these percentage ratios. Figures 5(a)–5(c) show the computational efficiency of methods (SM1–SM3) w.r.t the method PJ6, and Figure 5(d) shows the computational efficiency of the method PJ6 w.r.t (SM1–SM3). It is evident from Figures 5(a)–5(d) that the newly constructed simultaneous methods (SM1–SM3) are more efficient as compared to PJ6 [21].Table 6
The number of basic operations.
MethodsCOASmMmDmSM166m2+Om6m2+Om2m2+OmSM267m2+Om5m2+Om2m2+OmSM366m2+Om5m2+Om2m2+OmPJ668m2+Om6m2+Om2m2+OmFigure 5
Computational efficiency of iterative methods SM1–SM3 w.r.t PJ6.
(a)(b)(c)(d)


## 5. Numerical Results

Here, some numerical examples are considered in order to demonstrate the performance of our family of one-step fourth-order single root finding methods (Q1–Q3) and sixth-order simultaneous methods (SM1–SM3), respectively. We compare our family of optimal fourth-order single root finding methods (Q1–Q3) with E1–E9 methods. The family of simultaneous methods (SM1–SM3) of order six is compared with thePJ6 method [21] of the same order. All the computations are performed using CAS Maple 18 with 9000 (64 digits floating point arithmetic in case of simultaneous methods) significant digits. For single root finding methods, the stopping criteria is as follows:(46)ifsi<∈,iisik−α<∈,whereas(47)ei=fsi2,for simultaneous methods. We take ∈=10−600 for the single root finding method and ∈=10−30 for simultaneous determination of all roots of nonlinear equation (1).Numerical tests examples from [32–34] are provided in Tables 7–13. In Tables 7, 9, 11, and 12, we present the numerical results for simultaneous determination of all roots, while Tables 8, 10, and 13 represent for single root finding methods. In all tables, CO represents the convergence order; n, the number of iterations; ρ, the computational order of convergence; and CPU, the computational time in seconds. Table 14 shows the values of arbitrary parameters α and β used in iterative methods Q1–Q3 for test Examples 1–3.Table 7
Simultaneously finding of all roots.
f5s=s3+2.87s2−10.28/4.62−sζ1=2.0021,ζ2=−3.3304,ζ3=−1.5417(0)S1,2 = 2.5, (0) S3 = −7.4641, (0) S4 = −0.5359MethodCOCPUne1e2e3PJ660.03252.0e − 401.1e − 381.5e − 39SM1–SM360.01550.03.2e − 270.0Table 8
Comparison of the optimal 4th order single root finding methods forf5s.
Methodsi6−αfi6siCPUρf5s=s3+2.87s2−10.28/4.62−S,s0=2.5,CO=4Q12.0e − 581.0e − 760.0624.0Q23.8e − 632.0e − 810.0314.0Q32.1e − 601.1e − 780.0314.0E11.6e − 678.5e − 860.0624.0E24.8e − 662.4e − 840.0314.0E31.8e − 679.5e − 860.0314.0E43.6e − 661.8e − 840.0634.0E53.2e − 651.6e − 830.5314.0E63.1e − 671.9e − 8500784.0E71.3e − 647.1e − 830.0464.0E83.2e − 651.6e − 830.0474.0E96.9e − 693.5e − 870.0624.0Table 9
Simultaneously finding of all roots off6s.
f6s=s4−7.7907s3+2.511s−1.674ζ1=3.9485+0.3161i,ζ2=3.9485−0.3161i,ζ3=−0.3841,ζ4=0.2778S10=3.5+0.3i,S20=3.5−0.3i,S30=0.3+0.01i,S40=1.8+0.01iMethodCOCUPne1e2e3e4PJ660.03144.6e − 627.0e − 623.7e − 930.0SM1–SM360.01540.00.01.0e − 1371.0e − 133Table 10
Comparison of the optimal 4th order single root finding methods forf6s.
Methodsi6−αfi6siCPUρf6S=s4−7.79075s3+14.744s2+2.5111s−1.674,s0=0.27,CO=4Q19.5e − 19473.0e − 77830.0794.0Q21.3e − 21202.9e − 84770.0744.0Q35.9e − 20023.2e − 80040.0734.0E14.1e − 20166.5e − 80610.0784.0E21.4e − 19341.6e − 77340.0624.0E32.8e − 20161.5e − 80610.0474.0E52.1e − 19349.1e − 77340.0634.0E58.6e − 19062.6e − 76190.0624.0E67.4e − 19708.4e − 78760.0784.0E71.0e − 18816.7e − 75230.0794.0E88.6e − 19062.6e − 76190.0474.0E92.9e − 20831.1e − 83290.0784.0Table 11
Simultaneously finding of all roots forf7s.
f7−1s−sins−1/2sins−2/2sins−2.5/2ζ1−1,ζ2−2,ζ3−2.5S10=−0.2,S20=1.7,S30=3MethodCOCPUne1e2e3PJ660.04730.010.010.2e − 5SM1–SM360.03138.8e − 35.1e − 54.3e − 4Table 12
Simultaneously finding of all roots off7−1s.
f7−2s=sin3s−1/2sin3s−2/2sin3s−2.5/2ζ1−1,ζ2−2,ζ3−2.5,each with multiplicity3S10=−0.2,S20=1.7,S30=3MethodCOCPUne1e2e3PJ660.04735.5e − 162.7e − 82.2e − 17SM1–SM360.03131.8e − 211.7e − 133.9e − 18Table 13
Comparison of the optimal 4th order single root finding methodsf7−1s.
Methodsi6−αfi6siCPUρf7−1s=sins−1/2sins−2/2sins−2.5/2,s0=1.68956,CO=4Q16.7e − 5197.8e − 20740.4224.0Q24.2e − 5575.1e − 22270.7344.0Q38.8e − 5331.7e − 21291.3284.0E11.2e − 5367.5e − 21451.3284.0E297e − 5163.5e − 20611.3444.0E31.0e − 5362.7e − 21451.3444.0E41.6e − 5753.1e − 20501.3754.0E58.0e − 5071.9e − 20250.9064.0E62.2e − 5258.4e − 21001.3754.0E78.2e − 4992.4e − 19930.0.8914.0E88.0e − 5071.9e − 20251.13284.0E92.5e − 5508.3e − 22001.3294.0Table 14
Parameter values used in iterative methods Q1, Q2, and Q3 for Examples1–3.
MethodExample1Example2Example3βαβαβαQ110.005—10.005—10.005—Q25.0051.0015.0051.0015.0051.001Q3−5.005—15.0005—−5.005—We also calculate the CPU execution time, as all the calculations are done using Maple 18 (Processor Intel(R) Core(TM) i3-3110m CPU@2.4 GHz with 64-bit operating system). We observe from tables that CPU time of the methods SM1–SM3 is comparable or better than that of thePJ6 method [21], showing the efficiency of our methods (SM1–SM3) as compared to them.


## 6. Applications in Engineering

In this section, we consider two examples from engineering.Example 1.
(beam designing model, see [34]).
An engineer considers that a problem of embedments of a sheet-pile wall results a nonlinear function given as(48)f5s=s3+2.87s2−10.284.62−s.
The exact roots of (48) are ζ1=2.0021, ζ2=−3.3304, and ζ3=−1.5417 as shown in Figure 6.
The initial estimates forf5s are taken as(49)s10=2.5,s20=−7.4641,s30=−0.5359.Figure 6
Exact roots of nonlinear equationf5s.Example 2.
(fractional conversion, see [32]).
As the expression described in [14, 35],(50)f6s=s4−7.79075s3+14.7445s2+2.511s−1.674is the fractional conversion of nitrogen, with hydrogen feed at 250 atm. and 227 k. The exact roots of (50) are ζ1=3.9485+0.3161i,ζ2=3.9485−0.3161i,ζ3=−0.3841,andζ4=0.2778. Real roots of (50) are shown in Figure 7. The initial estimates for f6s are taken as(51)s10=3.5+0.3i,s20=3.5−0.3i,s30=−0.3+0.01i,s40=1.8+0.01i.Figure 7
Exact roots of nonlinear equationf6s.Example 3.
(see [33], for simultaneous determination of distinct and multiple roots).
Here, we consider another standard test function for the demonstration of convergence behavior of newly constructed methods.
Consider(52)f7−1s=sins−12sins−22sins−2.52,with exact roots ζ1=1,ζ2=2,andζ3=2.5 as shown in Figure 8. The initial guessed values have been taken as(53)s10=−0.2,s20=1.7,s30=3.
For distinct roots, we use method (34), and for multiple roots, method (37):(54)f7−2s=sin3s−12sin3s−22sin3s−2.52.
Figures9(e)–9(g) show the residual fall of iterative methods (Q1–Q3 and E1–E9), Figures 9(a), 9(b), and 9(d) present the residual fall of simultaneous iterative methods (SM1–SM3 and PJ6) for nonlinear functions f5s, f6s, and f7−1s when the multiplicity of roots is simple, and Figure 9(c) presents the residual fall for multiple roots for a nonlinear function f7−2s, respectively.
Table13 shows the values of parameters αandβ used in iterative methods Q1–Q3 for test Examples 1–3.Figure 8
Exact real roots of nonlinear equationf7s.Figure 9
(a) Residual fall of simultaneous methods SM1−SM3; PJ6 for nonlinear functionsf5s for simple roots. (b) Residual fall of simultaneous methods SM1−SM3; PJ6 for nonlinear functions f6s for simple roots. (c) Residual fall of simultaneous methods SM1−SM3; PJ6 for nonlinear functions f7s for multiple roots. (d) Residual fall of simultaneous methods SM1−SM3; PJ6 for nonlinear functions f7s for simple roots. (e) Residual fall of iterative methods Q1−Q3; E1−E9 for nonlinear equation f5s. (f) Residual fall of iterative methods Q1−Q3; E1−E9 for nonlinear equation f6s. (g) Residual fall of iterative methods Q1−Q3; E1−E9 for nonlinear equation f7s.
(a)(b)(c)(d)(e)(f)(g)


## 7. Conclusion

We have developed here three families of single-step single root finding methods of optimal convergence order four and three families of simultaneous methods of order six, respectively. From Tables7–13 and Figures 1–5 and 8, we observe that our methods (Q1–Q3 and SM1–SM3) are superior in terms of efficiency, stability, CPU time, and residual error as compared to the methods E1–E9 and PJ6 method, respectively.



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