# A New Class of Halley’s Method with Third-Order Convergence for Solving Nonlinear Equations

Authors: Mohammed Barrada; Mariya Ouaissa; Yassine Rhazali; Mariyam Ouaissa
Journal: Journal of Applied Mathematics
 (2020)

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## Abstract

In this paper, we present a new family of methods for finding simple roots of nonlinear equations. The convergence analysis shows that the order of convergence of all these methods is three. The originality of this family lies in the fact that these sequences are defined by an explicit expression which depends on a parameterp where p is a nonnegative integer. A first study on the global convergence of these methods is performed. The power of this family is illustrated analytically by justifying that, under certain conditions, the method convergence’s speed increases with the parameter p. This family’s efficiency is tested on a number of numerical examples. It is observed that our new methods take less number of iterations than many other third-order methods. In comparison with the methods of the sixth and eighth order, the new ones behave similarly in the examples considered.


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## Body

## 1. Introduction

Many problems in science and engineering [1–3] can be expressed in the form of the following nonlinear scalar equations:
(1)fx=0,where fx is a real analytic function. To approximate the solution α, supposed simple, of Equation (1), we can use a fixed-point iteration method in which we find a function F, called an iteration function (I.F.) for f, and from a starting value x0 [4–6], we define a sequence
(2)xn+1=Fxnforn=0,1,2⋯.A pointα is called a fixed point of F if Fα=α. By respecting some conditions, we can guarantee the convergence of the sequence xn towards α.One of the most famous and widely methods to solve Equation (1) is the second-order Newton method given by: [3, 7–9]:
(3)xn+1=xn−fxnf′xn,n=0,1,2,⋯.In order to ameliorate the order of convergence of Newton’s method, several third-order techniques have been elaborated [1–29]. For example, Halley [1, 4, 7, 10–15], super-Halley [12, 16–20], Chebyshev [1, 6, 7, 12, 13, 29], Euler [1, 3, 5, 11, 13], Chun [16], Sharma [17, 22], Amat [12], Traub [7], Barrada et al. [23, 24], Chun and Neta [25], Jaiswal [26], Liu and Li [27], and Singh and Jaiswal [28] are interesting methods.Furthermore, considerable efforts have been made to construct higher order methods. Jaiswal [26], Kim and Geum [30], and Thukral [31], proposed some forth-order method families. Fang et al. [32] constructed some fifth-order convergent iterative methods. Thukral [31], Kou et al. [33], and Chun and Ham [34] presented three sixth-order methods. Soleymani et al. [35, 36], Bi et al. [37], Lotfi et al. [38], and Cordero et al. [39] proposed some families of the eighth-order convergence methods. Soleymani et al. [40], Lotfi et al. [38], and Artidiello et al. [41] developed some new methods of the sixteenth order.The purpose of this paper is to construct, from Halley’s method and Taylor’s polynomial, a new family of methods for finding simple roots of nonlinear equations with cubical convergence. We will show that the weight functions of his methods have particular expressions which depend on a parameterp, where p is a nonnegative integer, and that, if certain conditions are verified, the speed’s convergence of these sequences improves by increasing p. Moreover, from this study, we will study the global convergence of these methods. Finally, the efficacy of some methods of the proposed family will be tested on a number of numerical examples. A comparison with many third-order methods will be realized.


## 2. Development of New Family of Halley’s Method

One of the best-known third-order methods is Halley’s method, given by(4)xn+1=xn−fxnf′xnW0Lnn=0,1,2,3,⋯,where
(5)W0Ln=22−Ln,Ln=Lfxn=fxnf"xnf′xn2.Using second-order Taylor’s polynomial off at xn, we obtain
(6)yx=fxn+f′xnx−xn+f′′xn2x−xn2,where xn is an approximate value of α. The graph of y intersects the x-axis at some point xn+1,0, which is the solution of the equation:
(7)0=fxn+f′xnxn+1−xn+f′′xn2xn+1−xn2.Factoringxn+1−xn from the last two terms, we obtain
(8)0=fxn+xn+1−xnf′xn+f′′xn2xn+1−xn,so
(9)xn+1=xn−fxnf′xn+f′′xn/2xn+1−xn.This schema is implicit because it does not directly findxn+1 as a function of xn. It can be changed to make it explicit by replacing xn+1−xn, remaining in THE right-hand side of Equation (9), by Halley’s correction given in Equation (4), we get
(10)xn+1=xn−fxnf′xnW1Ln,where W1Ln=2−Ln/21−Ln. It is simply known as super-Halley’s method.By repeating the same scenario many times and by replacingxn+1−xn, each time, with the last correction found, we derive the following iterative process which represents a general family of Halley’s method Hp for finding simple roots:
(11)xn+1p=xnp−WpLnfxnpf′xnp,where
(12)WpLn=SpLnSp+1Lnn=0,1,2,⋯,(13)S0Ln=1andS1Ln=1−Ln2,Sp+2Ln−Sp+1Ln=−Ln2SpLn,where p is a parameter, which is a nonnegative integer.We can show thatSp can be explicitly expressed as a function of Ln on the interval I=−∞,1/2 as follows:
(14)SpLn=12p+11+Ln−11−2LnRp+1−Ln−11−2LnTpforLn∊Iwhere
(15)R=1−1−2Ln,T=1+1−2Ln.Finally, the general family of Halley’s methodHp is generated by
(16)xn+1p=xnp−WpLnfxnpf′xnp,WpLn=SpLnSp+1Lnn=0,1,2,⋯,SpLn=12p+1∑k=0p+1/2Cp+22k+11−2LnkforLn∊−∞,12This schema is simple and interesting because it regenerates both well-known and new methods. For example,Forp=0, the formula (11) corresponds to the classical Halley method H0Forp=1, the formula (11) corresponds to the famous super-Halley method H1,Forp=2⋯5,, we obtain the methods H2H3,H4, and H5 given, respectively, by the following sequences:
(17)xn+1=xn−4fxnf′xn1−Ln4−6Ln+Ln2(18)xn+1=xn−fxnf′xn4−6Ln+Ln24−8Ln+3Ln2(19)xn+1=xn−fxnf′xn8−16Ln+6Ln28−20Ln+12Ln2−Ln3(20)xn+1=xn−fxn4f′xn−Ln3+12Ln2−20Ln+8−Ln3+5Ln2−6Ln+2


## 3. Analysis of Convergence

### 3.1. Order of Convergence

Theorem 1.
Letp be a parameter where p is a non-negative integer. We suppose that the function fx has at least two continuous derivatives in the neighborhood of a zero, α. Further, assuming that f′α≠0 and x0 is sufficiently close to α, the methods defined by Equation (16) converge cubically and satisfy the error equation:
(21)en+1=c22−c3en3+Οen4forp=0,en+1=−c3en3+Οen4forp≠0,where en=xn−α is the error at nth iteration and ci=fiα/i!f′α,i=2,3⋯Proof.
Letα be a simple root of fx and en=xn−α be the error in approximating α by xn. We use the Taylor expansions [22] about α:(22)fxn=f′αen+c2en2+c3en3+c4en4+Οen5,(23)f′xn=f′α1+2c2en+3c3en2+4c4en3+Οen4,(24)f′′xn=f′α2c2+6c3en+12c4en2+Οen3,where(25)ci=fiαi!f′α,fori=2,3⋯Using (23), we obtain
(26)f′xn2=f′α21+4c2en+22c22+3c3en2+43c2c3+2c4en3+Οen4,(27)Ln=2c2en−6c22−c3en2+44c23−7c2c3+3c4en3+Οen4,(28)fxnf′xn=en−c2en2+2c22−c3en3+Οen4.Using Taylor’s series expansion [22] of WpLn about Lα leads to
(29)WpLn=WpLα+Ln−LαWp′Lα+12Ln−Lα2Wp′′Lα+ΟLn−Lα3.Knowing thatLα=0, so
(30)WpLn=Wp0+LnWp′0+12Ln2Wp′′0+ΟLn3.We have(31)Forallp∈IΝWp0=1andWp′0=12,Forallp∈IΝ∗Wp′′0=1andW0′′0=12.Thus, formula (30) becomes
(32)WpLn=1+12Ln+12Ln2Wp′′0+ΟLn3.Using Equation (27), we get
(33)WpLn=1+c2en+2Wp′′0−3c22+3c3en2+Οen3.Substituting Equation (28) and Equation (33) in formula (11), we obtain the error’s equation:
(34)en+1=21−Wp′′0c22−c3en3+Οen4.Finally, using Equation (31), we obtain
(35)en+1=c22−c3en3+Οen4forp=0,en+1=−c3en3+Οen4forp≠0,which completes the proof of the theorem.


### 3.2. Global Convergence of the Halley Families’ Methods

We will make a first study of the global convergence of six selected methods from the proposed family (Hp), in the case where they converge towards the root in a monotone fashion [6, 11, 13–15, 17, 19, 20].Lemma 2.
Let us write the I.F. off, from the sequence H0 to H5:(36)F0x=x−fxf′x22−Lfx,F1x=x−fx2f′x2−Lfx1−Lfx,F2x=x−4fxf′x1−LfxLf2x−6Lfx+4,F3x=x−fxf′xLf2x−6Lfx+43Lf2x−8Lfx+4,F4x=x−2fxf′x3Lf2x−8Lfx+4−Lf3x+12Lf2x−20Lfx+8,F5x=x−fx4f′x−Lf3x+12Lf2x−20Lfx+8−Lf3x+5Lf2x−6Lfx+2.Then, the derivatives of the I.F. off are given by
(37)F0′x=Lf2x3−2Lf′x2−Lfx2,(38)F1′x=Lf2xLfx−Lf′x21−Lfx2,(39)F2′x=Lf2x.5Lf2x−4Lf′xLf2x−2Lfx+2Lf2x−6Lfx+42,(40)F3′x=Lf2x.3Lf3x−2Lf′x5Lf2x−8Lfx+43Lf2x−8Lfx+42,(41)F4′x=Lf2x7Lf4x−2Lf′x3Lf4x−16Lf3x+48Lf2x−48Lfx+16−Lf3x+12Lf2x−20Lfx+82.


#### 3.2.1. Monotonic Convergence of the SequencesHp

Letp be a parameter where p is an integer between 0 and 5. We consider the functions gp of a real variable defined on interval I=−∞,1/2 by:
(42)g0x=32,g1x=x,g2x=5x24x2−2x+2,g3x=3x325x2−8x+4,g4x=7x423x4−16x3+48x2−48x+16,g5x=x57x4−28x3+46x2−32x+8.where β∈]α,x0[and p∈0,1,⋯,5. Knowing that Lf′x≤gpLfx and that the derivatives Fp′ of I.F. are given by (39), we deduce that Fp′x≥0 in α,b. So, x1≥α. By induction, we obtain that xn≥α for all n∈ℕ.Theorem 3.
Letp∈0,1,⋯,5,f∈Cma,b,m≥4, f′x≠0, f′′x≠0,Lfx<1/2 and Lf′x≤gpLfx, on an interval a,b containing the root α of f. Then, the sequences Hp given by Equation (16), are decreasing (resp., increasing) and converge to α from any point x0∈a,b checking fx0f′x0>0 (resp. fx0f′x0<0).Proof.
Let us consider the case wherex0f′x0>0, then x0>α. The application of the mean value theorem gives: x1−α=Fpx0−Fpα=Fp′βx0−αFurthermore, from Equation (11) we have
(43)x1−x0=−WpL0fx0f′x0,where
(44)WpL0=SpL0Sp+1L0,.SpL0=12p+1∑k=0p+1/2Cp+22k+11−2L0k.AsL0∊−∞,1/2, then SpL0>0, so WpL0>0 for all p∈0,1,⋯,5.In addition, we havefx0/f′x0>0, we deduce that x1<x0. Now, it is easy to prove by induction that xn+1≤xn for all n∈ℕ.Thereby, the sequences (16) are decreasing and converge towards a limit r∈a,b, where r≥α. So, by calculating the limit of Equation (16), we obtain
(45)r=r−frf′rWpLfr.We have Lfr<1/2 and SpLfr>0, for every real Lfr∊−∞,1/2.So,WpLfr≠0 and consequently fr=0. As α is the unique root of f in a,b, therefore r=α. This completes the proof of Theorem 3.Corollary 4.
Letp∈0,1,⋯,5,f∈Cma,b,m≥4, f′x≠0,f′′x≠0,Lf′x≤0 and 0≤Lfx<1/2 on an interval a,b containing the root α of f. The sequences Hp given by Equations (4), (10), (17), (18), (19), and (20) are decreasing (resp., increasing) and converge to α, from any point x0∈a,b checking fx0f′x0>0 (rep. fx0f′x0<0).Proof.
Taking into account thatgpt≥0 for 0≤t<1/2 for every integer p0≤p≤5, it follows that the condition Lf′x≤gpLfx of Theorem 3 is well satisfied since Lf′x≤0. By applying Theorem 3, we obtain the thesis (for example, see Section 5.1).


## 3.1. Order of Convergence

Theorem 1.
Letp be a parameter where p is a non-negative integer. We suppose that the function fx has at least two continuous derivatives in the neighborhood of a zero, α. Further, assuming that f′α≠0 and x0 is sufficiently close to α, the methods defined by Equation (16) converge cubically and satisfy the error equation:
(21)en+1=c22−c3en3+Οen4forp=0,en+1=−c3en3+Οen4forp≠0,where en=xn−α is the error at nth iteration and ci=fiα/i!f′α,i=2,3⋯Proof.
Letα be a simple root of fx and en=xn−α be the error in approximating α by xn. We use the Taylor expansions [22] about α:(22)fxn=f′αen+c2en2+c3en3+c4en4+Οen5,(23)f′xn=f′α1+2c2en+3c3en2+4c4en3+Οen4,(24)f′′xn=f′α2c2+6c3en+12c4en2+Οen3,where(25)ci=fiαi!f′α,fori=2,3⋯Using (23), we obtain
(26)f′xn2=f′α21+4c2en+22c22+3c3en2+43c2c3+2c4en3+Οen4,(27)Ln=2c2en−6c22−c3en2+44c23−7c2c3+3c4en3+Οen4,(28)fxnf′xn=en−c2en2+2c22−c3en3+Οen4.Using Taylor’s series expansion [22] of WpLn about Lα leads to
(29)WpLn=WpLα+Ln−LαWp′Lα+12Ln−Lα2Wp′′Lα+ΟLn−Lα3.Knowing thatLα=0, so
(30)WpLn=Wp0+LnWp′0+12Ln2Wp′′0+ΟLn3.We have(31)Forallp∈IΝWp0=1andWp′0=12,Forallp∈IΝ∗Wp′′0=1andW0′′0=12.Thus, formula (30) becomes
(32)WpLn=1+12Ln+12Ln2Wp′′0+ΟLn3.Using Equation (27), we get
(33)WpLn=1+c2en+2Wp′′0−3c22+3c3en2+Οen3.Substituting Equation (28) and Equation (33) in formula (11), we obtain the error’s equation:
(34)en+1=21−Wp′′0c22−c3en3+Οen4.Finally, using Equation (31), we obtain
(35)en+1=c22−c3en3+Οen4forp=0,en+1=−c3en3+Οen4forp≠0,which completes the proof of the theorem.


## 3.2. Global Convergence of the Halley Families’ Methods

We will make a first study of the global convergence of six selected methods from the proposed family (Hp), in the case where they converge towards the root in a monotone fashion [6, 11, 13–15, 17, 19, 20].Lemma 2.
Let us write the I.F. off, from the sequence H0 to H5:(36)F0x=x−fxf′x22−Lfx,F1x=x−fx2f′x2−Lfx1−Lfx,F2x=x−4fxf′x1−LfxLf2x−6Lfx+4,F3x=x−fxf′xLf2x−6Lfx+43Lf2x−8Lfx+4,F4x=x−2fxf′x3Lf2x−8Lfx+4−Lf3x+12Lf2x−20Lfx+8,F5x=x−fx4f′x−Lf3x+12Lf2x−20Lfx+8−Lf3x+5Lf2x−6Lfx+2.Then, the derivatives of the I.F. off are given by
(37)F0′x=Lf2x3−2Lf′x2−Lfx2,(38)F1′x=Lf2xLfx−Lf′x21−Lfx2,(39)F2′x=Lf2x.5Lf2x−4Lf′xLf2x−2Lfx+2Lf2x−6Lfx+42,(40)F3′x=Lf2x.3Lf3x−2Lf′x5Lf2x−8Lfx+43Lf2x−8Lfx+42,(41)F4′x=Lf2x7Lf4x−2Lf′x3Lf4x−16Lf3x+48Lf2x−48Lfx+16−Lf3x+12Lf2x−20Lfx+82.


### 3.2.1. Monotonic Convergence of the SequencesHp

Letp be a parameter where p is an integer between 0 and 5. We consider the functions gp of a real variable defined on interval I=−∞,1/2 by:
(42)g0x=32,g1x=x,g2x=5x24x2−2x+2,g3x=3x325x2−8x+4,g4x=7x423x4−16x3+48x2−48x+16,g5x=x57x4−28x3+46x2−32x+8.where β∈]α,x0[and p∈0,1,⋯,5. Knowing that Lf′x≤gpLfx and that the derivatives Fp′ of I.F. are given by (39), we deduce that Fp′x≥0 in α,b. So, x1≥α. By induction, we obtain that xn≥α for all n∈ℕ.Theorem 3.
Letp∈0,1,⋯,5,f∈Cma,b,m≥4, f′x≠0, f′′x≠0,Lfx<1/2 and Lf′x≤gpLfx, on an interval a,b containing the root α of f. Then, the sequences Hp given by Equation (16), are decreasing (resp., increasing) and converge to α from any point x0∈a,b checking fx0f′x0>0 (resp. fx0f′x0<0).Proof.
Let us consider the case wherex0f′x0>0, then x0>α. The application of the mean value theorem gives: x1−α=Fpx0−Fpα=Fp′βx0−αFurthermore, from Equation (11) we have
(43)x1−x0=−WpL0fx0f′x0,where
(44)WpL0=SpL0Sp+1L0,.SpL0=12p+1∑k=0p+1/2Cp+22k+11−2L0k.AsL0∊−∞,1/2, then SpL0>0, so WpL0>0 for all p∈0,1,⋯,5.In addition, we havefx0/f′x0>0, we deduce that x1<x0. Now, it is easy to prove by induction that xn+1≤xn for all n∈ℕ.Thereby, the sequences (16) are decreasing and converge towards a limit r∈a,b, where r≥α. So, by calculating the limit of Equation (16), we obtain
(45)r=r−frf′rWpLfr.We have Lfr<1/2 and SpLfr>0, for every real Lfr∊−∞,1/2.So,WpLfr≠0 and consequently fr=0. As α is the unique root of f in a,b, therefore r=α. This completes the proof of Theorem 3.Corollary 4.
Letp∈0,1,⋯,5,f∈Cma,b,m≥4, f′x≠0,f′′x≠0,Lf′x≤0 and 0≤Lfx<1/2 on an interval a,b containing the root α of f. The sequences Hp given by Equations (4), (10), (17), (18), (19), and (20) are decreasing (resp., increasing) and converge to α, from any point x0∈a,b checking fx0f′x0>0 (rep. fx0f′x0<0).Proof.
Taking into account thatgpt≥0 for 0≤t<1/2 for every integer p0≤p≤5, it follows that the condition Lf′x≤gpLfx of Theorem 3 is well satisfied since Lf′x≤0. By applying Theorem 3, we obtain the thesis (for example, see Section 5.1).


## 3.2.1. Monotonic Convergence of the SequencesHp

Letp be a parameter where p is an integer between 0 and 5. We consider the functions gp of a real variable defined on interval I=−∞,1/2 by:
(42)g0x=32,g1x=x,g2x=5x24x2−2x+2,g3x=3x325x2−8x+4,g4x=7x423x4−16x3+48x2−48x+16,g5x=x57x4−28x3+46x2−32x+8.where β∈]α,x0[and p∈0,1,⋯,5. Knowing that Lf′x≤gpLfx and that the derivatives Fp′ of I.F. are given by (39), we deduce that Fp′x≥0 in α,b. So, x1≥α. By induction, we obtain that xn≥α for all n∈ℕ.Theorem 3.
Letp∈0,1,⋯,5,f∈Cma,b,m≥4, f′x≠0, f′′x≠0,Lfx<1/2 and Lf′x≤gpLfx, on an interval a,b containing the root α of f. Then, the sequences Hp given by Equation (16), are decreasing (resp., increasing) and converge to α from any point x0∈a,b checking fx0f′x0>0 (resp. fx0f′x0<0).Proof.
Let us consider the case wherex0f′x0>0, then x0>α. The application of the mean value theorem gives: x1−α=Fpx0−Fpα=Fp′βx0−αFurthermore, from Equation (11) we have
(43)x1−x0=−WpL0fx0f′x0,where
(44)WpL0=SpL0Sp+1L0,.SpL0=12p+1∑k=0p+1/2Cp+22k+11−2L0k.AsL0∊−∞,1/2, then SpL0>0, so WpL0>0 for all p∈0,1,⋯,5.In addition, we havefx0/f′x0>0, we deduce that x1<x0. Now, it is easy to prove by induction that xn+1≤xn for all n∈ℕ.Thereby, the sequences (16) are decreasing and converge towards a limit r∈a,b, where r≥α. So, by calculating the limit of Equation (16), we obtain
(45)r=r−frf′rWpLfr.We have Lfr<1/2 and SpLfr>0, for every real Lfr∊−∞,1/2.So,WpLfr≠0 and consequently fr=0. As α is the unique root of f in a,b, therefore r=α. This completes the proof of Theorem 3.Corollary 4.
Letp∈0,1,⋯,5,f∈Cma,b,m≥4, f′x≠0,f′′x≠0,Lf′x≤0 and 0≤Lfx<1/2 on an interval a,b containing the root α of f. The sequences Hp given by Equations (4), (10), (17), (18), (19), and (20) are decreasing (resp., increasing) and converge to α, from any point x0∈a,b checking fx0f′x0>0 (rep. fx0f′x0<0).Proof.
Taking into account thatgpt≥0 for 0≤t<1/2 for every integer p0≤p≤5, it follows that the condition Lf′x≤gpLfx of Theorem 3 is well satisfied since Lf′x≤0. By applying Theorem 3, we obtain the thesis (for example, see Section 5.1).


## 4. Principal Advantage of the New Family

As the familyHp is governed by formula (16), depending on the parameter p, where p is a nonnegative integer, it would be interesting to look for which p values, and under which conditions, the convergence is faster.Lemma 5.
Letp∈IΝ∗. Let un and vn be defined, respectively, by the sequences xnp+1 and xnp,given by formula (16).Then, we have(46)un+1−vn+1=−fxnf′xnLn/2p+2Sp+1LnSp+2Ln.Proof.
We have the following:(47)un+1−vn+1=xn+1p+1−xn+1p=Fp+1xn−Fpxn=fxnf′xnSp+2LnSpLn−Sp+1Ln2Sp+1LnSp+2Ln.We know that(48)Sp+2Ln=Sp+1Ln−Ln2SpLn,Sp+2LnSpLn−Sp+1Ln2=Sp+1LnSpLn−Sp+1Ln−LnSpLn22=Ln2Sp+1LnSp−1Ln−SpLn2.We deduce that(49)Sp+2LnSpLn−Sp+1Ln2=Ln2pS2LnS0Ln−S1Ln2.As(50)S0Ln=1,S1Ln=1−Ln2,S2Ln=1−Ln,Then,Sp+2LnSpLn−Sp+1Ln2=−Ln/2p+2, which completes the proof of the lemma.Theorem 6.
Letp∈IΝ∗. Let un and vn be defined, respectively, by the sequences xnp+1 and xnp given by Equation (16), f∈Cma,b,m≥4, f′x≠0, f′′x≠0, 0≤Lfx<1/2 and Lf′x≤0 on an interval a,b containing the root α of f. Starting from the same initial point x0∊a,b, the convergence’s rate of the sequence xnp+1 is higher than one’s of the sequence (xnp).Proof.
Supposing that the initial valuex0 satisfies fx0f′x0>0, so x0>α. According to Corollary. 4 given above, we know that if 0≤Lfx<1/21/2 and Lf′x≤0 in a,b, the sequences (xnp) and (xnp+1) are decreasing and converge to α from any point x0∈a,b.Letun and vn be defined, respectively, by (xnp+1) and (xnp). Since u0=v0=x0 and the two sequences are decreasing, we expect that un≤vn for all n∈ℕ. This can be shown by induction. Let n=1, then from (46),
(51)u1−v1=−fx0f′x0L0/2p+2Sp+1L0Sp+2L0,where L0=Lfx0.We know thatSp+1L0>0 and Sp+2L0>0 for L0<1/2.So,Sp+1L0Sp+2L0>0. As L0>0 and fx0/f′x0>0, then u1≤v1.Now, we assume thatun≤vn. Since, under the above hypotheses, Fp+1 is an increasing function in a,b, we obtainFp+1un≤Fp+1vn.On the other hand, we have(52)Fp+1vn−Fpvn=−fvnf′vnLfvn/2p+2Sp+1LfvnSp+2Lfvn≤0.We deduce thatFp+1un≤Fpvn. So un+1≤vn+1 and the induction is completed. The case fx0f′x0<0 is similar to the previous one.Consequently, the power of the proposed family has been shown analytically by justifying that, under certain conditions, the convergence’s speed of its methods increases with the parameterp, where p is a nonnegative integer. As the methods of Halley and super-Halley are particular cases of this family, in which the parameters p are the smallest, their convergence rates are smaller than the ones of the other new higher parameter methods.


## 5. Numerical Results

For the numerical results, we use a fixed stopping criterion ofε=10−15 and fxn≤10−15. The computations were performed using MATLAB R2015b.In order to compare two methods, we give the number of iterations (N) required to satisfy the stopping criterion, the number d of function (and derivatives) evaluations per step, and the order of convergence q of the method. Based on q and d, there is an efficiency measure defined by E=q1/d (efficiency index).Unfortunately, for the methods of the same orderq and demanding the same number of function evaluations d, the efficiency index E is the same for these methods. In this case, the comparison is made on the basis of the number of iterations (N). This number depends on how far the starting point x0 is from α and the value of the asymptotic constant. For two methods of the same order q, the one having the smallest asymptotic constant will converge faster than the other having the higher asymptotic constant, for a starting point x0 sufficiently close to α. But if x0 is too far from α (and x0 in the basin of attraction of α), it is possible that a method with a higher asymptotic constant converge faster [18]. Thus, in order to make the comparison more realistic and fairer, it is preferable to use an approximate asymptotic constant at the step n, defined by
(53)C~=xn+1−αxn−αq,where xn and xn+1 are two consecutive iterations. In general, choosing x0 close enough to α and a very high precision (300 significant digits or more), then, taking xn and xn+1 closer to the root, C~ will tend towards the theoretical asymptotic constant.Furthermore, we cannot compare two methods of differentq-order and demanding the same number of function evaluations d, on the basis of asymptotic constant. It is quite obvious that the method with the highest q is the fastest, for a starting point x0 sufficiently close to the solution. But, if x0 is too far from α (and x0 in the basin of attraction of α), the”order” of convergence is not necessarily q, especially for the first iterations of the method [10]. Thus, it is more correct and judicious to use the computational order of convergence ρ at the step n, which can be approximated using the formula [37]:
(54)ρ≅lnxn+1−α/xn−αlnxn−α/xn−1−α,where xn−1,xn, and xn+1 are three consecutive iterations. In general, choosing x0 close enough to α and a very high precision (300 significant digits or more), then, taking xn−1,xn, and xn+1 closer to the root, ρ will tend towards the theoretical order of convergence q.Here, the values ofρ and C~ will be calculated by using the same total number of function evaluation iterations (or, if not possible, the same total number of iterations) for all methods.The tests functions, used in Tables1–7, and their roots α are displayed in Table 2.Table 1
Comparison between some methods of Halley’s family in the case of monotonic convergence.
H0H1H2H3H4H5H6H7x02020202020202020x16.8593872741555.3550260820523.0447924412754.0119640112033.6878063706933.4736228130693.3289904352803.229883661414x23.3289904352803.0282259514313.0000000000723.0000284221073.0000003211763.0000000018203.0000000000053.0x33.0006332200753.0000000018203.03.03.03.03.0x43.0000000000053.0x53.0N54333332ρ1.66062.23822.93693.71474.54475.41166.3054__C~8.5e-022.2e-033.6e-042.7e-059.9e-071.7e-081.5e-10__Table 2
Test functions and their roots.
Test functionsRoot (α)f1x=x3+x2−5x−72.365230013414097f2x=x−12−lnx+12.057103549994738f3x=sinx+3−x3−1-0.7211373399241717f4x=ex−1−4x2+8x−41.714805912362778f5x=1105cos2x+5−2x1.085982678007472f6x=x2+x−123.000000000000000f7x=e−x2+x+2−1-1.000000000000000f8x=4cosπ4−x22−30.5235987755982989f9x=x−24−13.0000000000000000f10x=x3−102.154434690031884f11x=1−x2−2sinπ4−x221.895494267033981f12x=x−13−12.000000000000000f13x=cosx−x0.7390851332151607f14x=x−3ex+12.947530902542285Table 3
Comparison with other third order methods.
TestNN: number of iterations/C~: approximate asymptotic constantsf,x0C~NMCBCHSHH0H1H2H3H4H5f11.68N6777443333C~2.332.062.060.200.050.060.060.060.06f24.5N7555443333C~0.230.230.220.183e-27e-36e-36e-36e-3f3-1.3N5544433333C~2.740.440.270.300.110.110.110.110.11f4N66544333331.4C~7.413.332.770.880.100.090.090.090.09f51.4N5544433333C~3.911.280.910.950.540.530.530.530.53f67N5444433222C~3e-23e-23e-22e-23e-48e-7___f7-0.485N6444444433C~1.741.681.610.810.390.470.470.470.47f81.1N5544433333C~1.260.490.320.290.170.160.170.170.17f92.7N6555443333C~4.116.825.571.340.861.011.031.481.02Table 4
Comparison of methods of various orders in terms of the number of iterations and number of functions evaluations.
Testfct°x0N: number of iterationsd: number of functions evaluationsFGCCKBH2H3H4H5FGCCKBH2H3H4H5f60.5753333322820121299962543333333281212129999f24.53332333312121289999f41.3444333333161612129999-13332333312121289999f121.55853333333220121299992.9333244451212121212121215f101.265333434242012129129123.833324444121212812121212f132.5533324444121212812121212-0.5543344342016121212129120332233331212889999f143.43322444412128812121212f111.38D33433332D121212999Table 5
Comparison of methods of various orders in terms of the error in the consecutive iterations, the absolute value of the error, the absolute value of the function, and the computational order of convergence.
FGCCKBiH2H3H4H5Orderq56683333E1.495351.565081.565081.681791.442251.442251.442251.44225x0=25Test functionf6xn−xn−14.6e-022.351.653.9e-012.341.651.188.6e-01xn−α3.0e-105.3e-033.4e-047.2e-117.0e-033.4e-049.3e-061.5e-07fxn2.1e-093.7e-022.4e-35.1e-104.9e-022.4e-036.5e-051.0e-06ρ2.19292.72573.285.53692.59873.28004.02334.8121x0=4.5Test functionf2xn−xn−12.0e-011.3e-017.3e-022.441.4e-018.6e-025.8e-024.3e-02xn−α6.4e-057.8e-078.8e-092.441.9e-053.9e-061.2e-065.1e-07fxn1.2e-041.4e-061.6e-088.1e-033.4e-057.0e-062.2e-069.0e-07ρ3.20534.03364.54352 it3.08182.98672.88032.8115x0=−1Test functionf4xn−xn−13.5e-031.3e-021.5e-021.598.0e-023.5e-027.7e-039.6e-03xn−α8.7e-137.2e-124.2e-121.591.2e-051.2e-061.3e-082.5e-08fxn3.2e-122.6e-111.5e-116.3e-034.6e-054. 7e-065.0e-089.8e-08ρ3.15063.15064.51902 it2.94782.69822.49532.5142x0=2.9Test functionf12xn−xn−17.4e-023.7e-021.2e-029.0e-016.5e-048.3e-33.8e-025.2e-04xn−α5.1e-067.0e-095.1e-139.0e-019.2e-111.9e-071.8e-054.5e-11fxn1.5e-052.1e-081.5e-129.8e-032.8e-105.7e-075.4e-051.4e-10ρ3.83954.87305.50502 it2.99233.04452.91422.2690f10, x0=3.8Test functionf10xn−xn−11.0e-014.7e-021.3e-02D4.9e-043.5e-031.2e-024.4e-02xn−α1.0e-066.8e-102.1e-14D8.5e-123.0e-091.3e-076.1e-06fxn1.9e-059.4e-093.0e-13D1.2e-104.2e-081.9e-068.5e-05ρ4.05055.09405.6205D2.99692.26573.01193.0580x0=2.55Test functionf13xn−xn−14.3e-022.5e-013.9e-021.8e-011.9e-031.5e-031.6e-031.6e-03xn−α1.4e-092.0e-122.1e-121.8e-014.4e-102.4e-102.8e-102.7e-10fxn2.4e-093.3e-123.5e-124.06e-037.3e-103.9e-104.7e-104.8e-10ρ4.58935.41806.17422 it3.03363.03133.03193.0317f14, x0=3.4Test functionf14xn−xn−11.3e-023.7e-034.5e-014.5e-012.2e-047.1e-041.5e-032.9e-03xn−α1.3e-096.7e-154.5e-014.5e-015.4e-121.8e-101.9e-092.9e-03fxn2.4e-081.2e-135.7e-037.4e-039.7e-113.3e-093.4e-081.3e-08ρ4.56985.59982 it2 it2.98602.98152.97312.9649Table 6
Computational order of convergenceρ, depending on the initial point x0, for the test functions f4 and f6.
ρFGCCKBiH2H3H4H5Orderq56683333x0Test functionf40.34.74292.74092.93182.95902.96460.14.45705.54552.85812.81892.90242.9244-0.14.17735.15885.30152.92002.63432.82332.8702-0.53.66274.58675.01692.94842.67802.61132.7280-13.15063.15064.51902.94782.69822.49532.5142-1.52.76973.50234.08802.95062.66272.52072.3521-32.10472.61123.16982.80102.93782.97983.0175-51.70922.04532.48914.24212.93472.73742.91532.9550x0Test functionf654.70397.74.13545.10864.396584.07345.04095.34874.34985.350093.87394.81984.19575.1853103.68764.60825.02284.04625.0228123.35754.22154.71033.76704.71035.6757152.96063.73604.28763.40444.28765.20656.1486202.49713.13813.71476.19802.93693.71474.54475.4116301.98362.43242.94734.99994.62322.94733.61264.3288451.60301.92582.31773.93734.19162.31772.80573.3488Table 7
Computational order of convergenceρ, depending on the initial pointx0, for the test function f14.
Functionf14Computational order of convergenceρWGFGNRCCKBiH5Orderq4556683x03.23.81264.81863.13813.43.56014.56984.70595.59982.96493.73.92224.07804.36045.17933.198743.75603.56683.92114.6659-9.94723.10954.33.48103.11343.41384.15524.92955.92413.08374.53.25752.85863.05723.83864.75995.93453.061952.68762.38102.20613.17354.32385.48993.008161.92431.86731.24392.35223.84023.78142.8965


### 5.1. Numerical Comparison of the Six Proposed Methods

Example: Given the functionf6x=x2+x−12 in the interval I=3,10, we have ∈Cma,b,m≥4, f′x≠0,f′′x≠0 in I, Lfx=2x2+x−12/2x+12 and Lf′x=0. It is easy to prove that 0≤Lfx<1/2 in I. Since in addition, Lf′x≤0 in I, then the conditions of Theorem 6 are satisfied for the sequences Hp given by Equations (4), (10), (17), (18), (19), and (20). By taking x0=20, we have Fpx0∊I for all p∈0,1,⋯,7, and fx0f′x0>0.In this case, it is not easy to compare the chosen methods because they have the same order of convergence (q=3), the same number of function evaluations (d=3) by step, the same efficiency index E=33, and the same theoretical asymptotic constant (see formula (21)). In Table 1, the comparison will be, thus, made on the basis of the number of iterations (N), the approximate asymptotic constants C~ defined by (53), and/or the computational order of convergence ρ defined by (54). As we know that the method with the smallest values of N and C~ and/or the higher values of ρ will be locally faster.We note in Table1 that
(i)
All the sequencesHp0≤p≤7 are decreasing and converge to the solution α=3 of the equation fx=0 in I(ii)
By increasing the value of the parameterp, where p is a nonnegative integer, the values of N and C~ decrease and the value of ρ increases. Thus, the convergence speed of these methods increases with the parameter p(iii)
The convergence rates of the new methods, in which the parameterp is high p=2,⋯,7, are higher than those of Halley’s p=0 and super-Halley’s p=1, in which the parameter p is low.The given example illustrates the great importance of the Theorem6 which stipulates that, under certain conditions, the higher the parameter p is (p is a nonnegative integer), the faster the convergence of methods Hp becomes.


### 5.2. Comparison with Other Third-Order Methods

In Table3, we shall present the numerical results obtained by employing various cubically convergent iterative and Newton’s methods. Comparing Newton’s method (NM) defined by formula (3), Chebyshev’s method (CB) defined by (16) in [16], Chun’s method (CH) defined by Equation (30) with an=1 in [16], Sharma’s method (SM) defined by Equation (26) with an=1 in [17], Halley’s method (H0) defined by Equation (4) given before, and super-Halley’s method (H1) given by Equation (10) above, with the four new methods designated as H2,H3,H4, and H5 defined above, respectively, by Equations (17), (18), (19), and (20).In Table3, all the methods have the same order of convergence (q=3) and require the same number of function evaluations (d=3). Consequently, they have the same efficiency index E=33. Thus, the comparison in Table 3 can be made on the basis of the number of iterations (N) and the approximate asymptotic constants C~ defined by (53). We know that the method with the smallest values of N and C~ is locally faster.From the numerical results given in Table3, we see that the four proposed methods H2,H3,H4,andH5 of the new family appear more interesting and effective than the other chosen third-order methods, because in the majority of the selected examples, our methods converge with fewer iterations and smaller approximate asymptotic constants.


### 5.3. Comparison with Higher Order Methods

Now, we compare the four selected methods of the new familyH2,H3,H4,andH5 with some higher order methods: B, an eighth-order method, denotes for Bi et al. [37] (formula (36) with α=1); FG and NR, a fifth-order methods, respectively, denote for Fang et al. [32] (formula (2)) and Muhammad et al. [42] (algorithm 2.4.); CC represents the Chun and Ham method [34] (formulas (12), (13), (14)) and K denotes for Kou [33] (first formula), both last methods are from the sixth order; WG, a fourth-order iterative method, denotes for Wang [43] (formula (23)).Table4 shows the number of iterations (N) and the number of function evaluations (d) required to approximate the rootα.In Table5, we exhibit the absolute values of the error en=xn−α, the error in the consecutive iterations xn+1−xn, the absolute values of the function fxn, the computational order of convergence ρ, and the efficiency index E.In Table4, the comparison, above, with several fifth-, sixth-, and even eighth-order methods, confirms the efficiency and power of the new proposed family. In fact, in most of the considered examples, Table 4 shows that our methods behave in a similar way as the higher order ones, as they require an equal or smaller number of function evaluations Table 5 confirms the power of higher order methods which shows, generally, higher values of computational order of convergence (ρ) if the starting pointx0 is sufficiently close to α. However, for the first case (f6), in which the starting point x0 is far from the root α, the higher order methods (FG, CC, and K) show low values of computational order of convergence ρ, contrary to our methods. This remark leads us to test some functions f4,f6,andf14 to study the variation of ρ as a function of x0. These results are displayed in Tables 6 and 7 that show that, for our methods H2,H3,H4,andH5, the choice of an x0, far from the root α only results a small variation of ρ comparing to its theoretical value (q=3). Contrary to high-order methods, the further away from x0, the more the value of ρ decreases during the first iterations. This leads us to think that these high-order methods would start the first iteration with a speed lower than the normal one; then, in the line of iterations, would progressively regain speed to reach its maximum in the last iteration. Thus, the delay that would be in the first iterations could lead to a decrease in the average speed of convergence, and consequently to an increase in the number of iterations N. This would explain why, in some cases (such as the example f6), our methods, which are of order 3, show numbers N and d similar or even smaller than higher order methods, contrary to the predictions. Having said that, the power of high-order methods is confirmed on several levels and it manifests itself especially when x0 is close enough to the root and the required precision is very high.


## 5.1. Numerical Comparison of the Six Proposed Methods

Example: Given the functionf6x=x2+x−12 in the interval I=3,10, we have ∈Cma,b,m≥4, f′x≠0,f′′x≠0 in I, Lfx=2x2+x−12/2x+12 and Lf′x=0. It is easy to prove that 0≤Lfx<1/2 in I. Since in addition, Lf′x≤0 in I, then the conditions of Theorem 6 are satisfied for the sequences Hp given by Equations (4), (10), (17), (18), (19), and (20). By taking x0=20, we have Fpx0∊I for all p∈0,1,⋯,7, and fx0f′x0>0.In this case, it is not easy to compare the chosen methods because they have the same order of convergence (q=3), the same number of function evaluations (d=3) by step, the same efficiency index E=33, and the same theoretical asymptotic constant (see formula (21)). In Table 1, the comparison will be, thus, made on the basis of the number of iterations (N), the approximate asymptotic constants C~ defined by (53), and/or the computational order of convergence ρ defined by (54). As we know that the method with the smallest values of N and C~ and/or the higher values of ρ will be locally faster.We note in Table1 that
(i)
All the sequencesHp0≤p≤7 are decreasing and converge to the solution α=3 of the equation fx=0 in I(ii)
By increasing the value of the parameterp, where p is a nonnegative integer, the values of N and C~ decrease and the value of ρ increases. Thus, the convergence speed of these methods increases with the parameter p(iii)
The convergence rates of the new methods, in which the parameterp is high p=2,⋯,7, are higher than those of Halley’s p=0 and super-Halley’s p=1, in which the parameter p is low.The given example illustrates the great importance of the Theorem6 which stipulates that, under certain conditions, the higher the parameter p is (p is a nonnegative integer), the faster the convergence of methods Hp becomes.


## 5.2. Comparison with Other Third-Order Methods

In Table3, we shall present the numerical results obtained by employing various cubically convergent iterative and Newton’s methods. Comparing Newton’s method (NM) defined by formula (3), Chebyshev’s method (CB) defined by (16) in [16], Chun’s method (CH) defined by Equation (30) with an=1 in [16], Sharma’s method (SM) defined by Equation (26) with an=1 in [17], Halley’s method (H0) defined by Equation (4) given before, and super-Halley’s method (H1) given by Equation (10) above, with the four new methods designated as H2,H3,H4, and H5 defined above, respectively, by Equations (17), (18), (19), and (20).In Table3, all the methods have the same order of convergence (q=3) and require the same number of function evaluations (d=3). Consequently, they have the same efficiency index E=33. Thus, the comparison in Table 3 can be made on the basis of the number of iterations (N) and the approximate asymptotic constants C~ defined by (53). We know that the method with the smallest values of N and C~ is locally faster.From the numerical results given in Table3, we see that the four proposed methods H2,H3,H4,andH5 of the new family appear more interesting and effective than the other chosen third-order methods, because in the majority of the selected examples, our methods converge with fewer iterations and smaller approximate asymptotic constants.


## 5.3. Comparison with Higher Order Methods

Now, we compare the four selected methods of the new familyH2,H3,H4,andH5 with some higher order methods: B, an eighth-order method, denotes for Bi et al. [37] (formula (36) with α=1); FG and NR, a fifth-order methods, respectively, denote for Fang et al. [32] (formula (2)) and Muhammad et al. [42] (algorithm 2.4.); CC represents the Chun and Ham method [34] (formulas (12), (13), (14)) and K denotes for Kou [33] (first formula), both last methods are from the sixth order; WG, a fourth-order iterative method, denotes for Wang [43] (formula (23)).Table4 shows the number of iterations (N) and the number of function evaluations (d) required to approximate the rootα.In Table5, we exhibit the absolute values of the error en=xn−α, the error in the consecutive iterations xn+1−xn, the absolute values of the function fxn, the computational order of convergence ρ, and the efficiency index E.In Table4, the comparison, above, with several fifth-, sixth-, and even eighth-order methods, confirms the efficiency and power of the new proposed family. In fact, in most of the considered examples, Table 4 shows that our methods behave in a similar way as the higher order ones, as they require an equal or smaller number of function evaluations Table 5 confirms the power of higher order methods which shows, generally, higher values of computational order of convergence (ρ) if the starting pointx0 is sufficiently close to α. However, for the first case (f6), in which the starting point x0 is far from the root α, the higher order methods (FG, CC, and K) show low values of computational order of convergence ρ, contrary to our methods. This remark leads us to test some functions f4,f6,andf14 to study the variation of ρ as a function of x0. These results are displayed in Tables 6 and 7 that show that, for our methods H2,H3,H4,andH5, the choice of an x0, far from the root α only results a small variation of ρ comparing to its theoretical value (q=3). Contrary to high-order methods, the further away from x0, the more the value of ρ decreases during the first iterations. This leads us to think that these high-order methods would start the first iteration with a speed lower than the normal one; then, in the line of iterations, would progressively regain speed to reach its maximum in the last iteration. Thus, the delay that would be in the first iterations could lead to a decrease in the average speed of convergence, and consequently to an increase in the number of iterations N. This would explain why, in some cases (such as the example f6), our methods, which are of order 3, show numbers N and d similar or even smaller than higher order methods, contrary to the predictions. Having said that, the power of high-order methods is confirmed on several levels and it manifests itself especially when x0 is close enough to the root and the required precision is very high.


## 6. Conclusion

In this paper, we built a new family of Halley’s method with third-order convergence for solving nonlinear equations with simple roots. The proposed scheme is interesting because it regenerates Halley’s method, super-Halley’s method, and infinity of new methods. The originality of this family lies, on the one hand, in the fact that these sequences are governed by a single formula depending on a parameterp, where p is a nonnegative integer, and on the other hand, under certain conditions, the convergence speed of its methods improves when the p value increases. In order to reveal the quality of the new family, we focused on four of its methods. A first study on the global convergence of these selected methods was carried out. To test the new techniques, several digital examples were produced. The performance of our methods is compared with well-known methods of similar or higher order. The numerical results clearly illustrated the efficiency of the techniques of the new family proposed in this article.



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