New Iterative Methods for Solving Nonlinear Equations and Their

Basins of Attraction

O.ABABNEH

Zarqa university

Department of Mathematics

JORDAN

Key–Words: Nonlinear equations, Newton’s method, Basins of Attraction, Order of convergence.

Received: March 15, 2021. Revised: November 10, 2021. Accepted: December 16, 2021. Published: January 3, 2022.

1 Introduction

Solving the nonlinear equation is a non-trivial task

that has nice applications in various branches of

physics and engineering. One of the most important

techniques in order to approximate nonlinear equa-

tions are iterative methods [2, 3, ?,?, 20]. This is

an area of research that has grown exponentially over

the last few years. Sometimes the applications of the

numerical methods to solve non-linear equations de-

pending on the second derivatives are restricted in

physics and engineering. In this paper, we consider

iterative methods to ﬁnd a simple root of a nonlinear

equation f(x)=0, where f is a scalar function. The

classical Newtons method is given by

xn+1 =xn−f(xn)

f0(xn),n = 0,1,2,... (1)

This is an important and basic method [?], which con-

verges quadratically. To improve the local order of

convergence of Newtons method, many third-order

methods are developed. A family of third-order meth-

ods, called Chebyshev Halley methods [14], is deﬁned

as [15, 9]

xn+1 =xn−1 + Lf(xn)

1−αLf(xn)f(xn)

f0(xn), α ∈R. (2)

where

Lf(xn) = 1

f00 (xn)f(xn)

f0(xn)2.(3)

This family includes the classical Chebyshevs method

(α= 0), Halleys method (α= 1) and Super-Halley

method(α= 2). It is clear that to implement (2),

one has to evaluate the second derivative of the

function. This can create some problems. In order

to overcome this drawback, several techniques have

been developed [7, 12, 27, 28, 32, 21, 31]. On the

other hand, the super-Halley method can be viewed as

particular one of the following general form deﬁned

by:

xn+1 =xn− 1 + Lf(xn) + βLf(xn)2

1−αLf(xn)!f(xn)

f0(xn),

α, β ∈R. (4)

This family is known to be third order, but it de-

pends on second derivative, so its use is severely re-

stricted in applications from a practical point of view.

Therefore, it is important and interested to develop it-

erative methods which are free from second deriva-

tive and whose order is higher than three if possible.

The aim, in the present paper to improves the third-

order method in (4) to obtain a family of fourth-order

method free from second derivative. Moreover, per it-

eration in these new methods require two evaluations

of the function and just one of its ﬁrst derivative.The

remaining part of the paper is organized as follows.

In Section 2 and 3, the Description of the new

methods are given. in section 4, the convergence anal-

ysis is given. Furthermore, some examples of the new

methods are suggested. In Section 5, the proposed

methods are tested on some functions, and the results

are compared with other methods. at the end of this

work,in section 6, the basins of attraction are also pre-

sented.

Abstract: The purpose of this paper is to propose new modified Newton’s method for solving nonlinear

equations and free from second derivative. Convergence results show that the order of convergence is four.

Several numerical examples are given to illustrate that the new iterative algorithms are effective.In the end, we

present the basins of attraction to observe the fractal behavior and dynamical aspects of the proposed algorithms.

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2 Description of the methods

In [8], Chun Suggestes new fourth order modiﬁcations

of Newton’s method by using any two existing fourth-

order methods to give new fourth-order methods. As

an illustrated example, Chun consider the double-

Newton method with fourth-order convergence [43]

given by

xn+1 =xn−f(xn)

f0(xn)−f(yn)

f0(yn),(5)

the fourth-order method given by

xn+1 =xn−f(xn)

f0(xn)−"1+2f(yn)

f(xn)+f(yn)2

f(xn)2#f(yn)

f0(xn),

(6)

and King’s fourth-order family of methods [26] given

xn+1 =xn−f(xn)

f0(xn)−f(xn) + βf(yn)

f(xn)+(β−2)f(yn)

f(yn)

f0(xn),

(7)

Where β∈R. Approximately equaling the cor-

rection terms of the methods (5) and (6), he obtained

an approximation to f0(yn)

f0(yn)≈f0(xn)f(xn)2

f(xn)2+ 2f(xn)f(yn) + f(yn)2(8)

Approximately equaling the correction terms of

the methods (5) and (7), he obtained another approxi-

mation to f0(yn)

f0(yn)≈f0(xn) [f(xn)+(β−2)f(yn)]

f(xn) + βf(yn).(9)

Finally, he applied the respective

approximation(8) and (9) to the ﬁfth-order method

proposed in [18] given by

xn+1 =xn−f(xn)

f0(xn)−f0(yn)+3f0(xn)

5f0(yn)−f0(xn)

f(yn)

f0(xn).

(10)

Per iteration this method requires two evalua-

tions of the function and two evaluations of its ﬁrst-

derivative, Using (8) in (10) to obtain the new method

xn+1 =xn−f(xn)

f0(xn)

−4f(xn)2+ 6f(xn)f(yn)+3f(yn)2

4f(xn)2−2f(xn)f(yn)−f(yn)2

f(yn)

f0(xn).(11)

Using (9) in (10), to obtain the new family of

methods

xn+1 =xn−f(xn)

f0(xn)

−2f(xn) + (2β−1)f(yn)

2f(xn) + (2β−5)f(yn)

f(yn)

f0(xn),(12)

whereβ∈R.

Noor [36] proposed new fourth order method de-

ﬁned by

xn+1 =yn−f(yn)

f0(xn)−f(yn)

f0(xn)1−f0(yn)

f0(xn)

−1

2f(yn)

f0(xn)2f00(yn)

f0(xn),(13)

where yn=xn−f(xn)/f0(xn).

It is clear that to implement (13), one has to evalu-

ate the second derivative of the function. This can cre-

ate some problems. In [34], a second-derivative-free

method is obtained through approximating the second

derivative f00(yn)in (14) by

f00(yn)∼

=f0(yn)−f0(xn)

yn−xn

.(14)

Noor and Khan [35] have used the same approxi-

mation of the second derivative (14) in (13) to suggest

the following Iterative methods

xn+1 =yn−2f(yn)

f0(xn)+f(yn)f0(yn)

f02(xn)

+f0(yn)−f0(xn)

2f(xn)f(yn)

f0(xn)2

.(15)

In [1], we rederive the method in (15) to ob-

tain a family of fourth-order method free from second

derivative. Moreover, per iteration in these new meth-

ods require two evaluations of the function and just

one of its ﬁrst derivative as follow:

Combining (8) and (15), we get the new iterative

method

xn+1 =yn−2f(yn)

f0(xn)+ f(xn)2f(yn)

f0(xn)(f(xn) + f(yn))2!

+ −f(yn)f0(xn)(2f(xn) + f(yn))

2f(xn)(f(xn) + f(yn))2!f(yn)

f0(xn)2

.(16)

Using (9) in (15), we get a new family of iterative

method

xn+1 =yn−2f(yn)

f0(xn)+f(yn) (f(xn) + f(yn)(β−2))

f0(xn)(f(xn) + βf(yn)) 

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+−f0(x)f(y)

f(x)(f(x) + βf(y)) f(yn)

f0(xn)2

.(17)

Numerical results show that the number of iterations

of the new method are always less than that of the

classical Newton’s method and the method in (15).

2.1 New approximation of the second

derivative and derive new methods

let us consider the third-order methods deﬁned by

xn+1 =xn−f(xn)2

f0(xn) (f(xn)−f(yn)),(18)

where

yn=xn−f(xn)

f0(xn),(19)

which is third order method often called NewtonStef-

fensen method and

xn+1 =xn−f(xn)

f0(xn)−f(xn)2f00(xn)

2f0(xn)3,(20)

which is known as the Householder iterative method

[19]. We approximately equate the correcting terms

of both methods to obtain the following approximate

expression:

f(xn)2

f0(xn) (f(xn)−f(yn)) ≈f(xn)

f0(xn)+f(xn)2f00(xn)

2f0(xn)3

(21)

this gives a new approximation

f(xn)00 ≈2f02(xn)f(yn)

f(xn) (f(xn)−f(yn)).(22)

A family of third-order methods, called Cheby-

shev Halley methods [14], is deﬁned as [15, 9]

xn+1 =xn−1 + Lf(xn)

1−αLf(xn)f(xn)

f0(xn), α ∈R.

(23)

where

Lf(xn) = 1

f00 (xn)f(xn)

f0(xn)2.(24)

This family includes the classical Chebyshevs

method (α= 0), Halleys method (α= 1) and Super-

Halley method(α= 2). It is clear that to implement

23, one has to evaluate the second derivative of the

function. On the other hand, the super-Halley method

can be viewed as particular one of the following

general form deﬁned by:

xn+1 =xn− 1 + Lf(xn) + βLf(xn)2

1−αLf(xn)!f(xn)

f0(xn)

, α, β ∈R. (25)

This family is known to be third order, but it depends

on second derivative, so that its use is severely re-

stricted in applications from a practical point of view.

Therefore, it is important and interesting to develop it-

erative methods which are free from second derivative

and whose order is higher than three if possible, this

is main motivation of this paper.

Using 22 in 24 we obtain

Lf(xn) = 1

f00(xn)f(xn)

f0(xn)2≈f(yn)

f(xn)−f(yn),(26)

using 26 in 25 we obtain a new family of the second-

derivative -free method deﬁned as following

xn+1 =yn−1 + βf(yn)

f(xn)−f(yn)(1 + α)

f(xn)f(yn)

f0(xn) (f(xn)−f(yn)),(27)

where

yn=xn−f(xn)

f0(xn).(28)

For the methods deﬁned by 27, we have the following

convergence result.

3 Convergence analysis

In order to establish the order of convergence and

other properties of the new methods, we state

some of the important deﬁnitions . A sequence

x0, x1, x2, ..., xn, ... of approximations to the zero α,

generated by an iterative function φ, presumably con-

verges to α. More precisely, if there holds lim

n→∞ xn=

α, we say that the sequence of approximations {xn}

is convergent. Convergence conditions depend on the

form of the iteration function, its properties and the

chosen initial approximation .

Deﬁnition 1 [40] Let φ:R→Rbe an iterative

function which deﬁnes the iterative process Xn+1 =

φ(xn). If there exists a real number pand a nonzero

constant λsuch that lim

n→∞

|φ(xn)−α|

|xn−α|p=λ, then pis

called the order of convergence and λis the asymp-

totic error constant . If p= 2 or 3, the convergence is

said to be quadratic or cubic, respectively [39, 40].

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Let en=xn−αbe the error of the approximation

in the nth iterative step. Then en+1 =xn+1 −α=

φ(xn)−α. Hence, lim

n→∞

|φ(xn)−α|

|xn−α|p= lim

n→∞

|xn+1−α|

|xn−α|P=

lim

n→∞

|en+1|

|en|P=λ

Deﬁnition 2 : [39] Let rbe the number of func-

tion evaluations of the method . The efﬁciency of the

method is measured by the concept of Efﬁciency Index

and deﬁned as r

√p=p1

r, where pis the order of

convergence .

Theorem 3 Let α∈Ibe a simple zero of sufﬁciently

differentiable function f:I→Rfor an open interval

I. If xois sufﬁciently close to α, where en=xn−α

and ck=f(k)(α)/k!. Then the family deﬁned by 27

is of fourth-order convergence if β= 1 and for any

α∈R.

Proof Using Taylor expansion of f(xn)about α

and taking into account that f0(α)6= 0 we have

f(xn) = f0(α)en+c2e2

n+c3e3

n+c4e4

n+O(e5

n).

(29)

Furthermore, we have

f0(xn) = f0(α)1+2c2en+ 3c3e2

n+ 4c4e3

n+O(e4

n).

(30)

and

f(xn)

f0(xn)=en−c2e2

n+2(c2

2−c3)e3

n+(7c2c3−4c3

2−3c4)e4

(31)

+O(e5

n).

Substituting 31 in 28 yields

yn−α=c2e2

n−2(c2

2−c3)e3

−(7c2c3−4c3

2−3c4)e4

n+O(e5

n).(32)

Expanding f(yn)about αand using 32, we have

f(yn) = f0(α)c2e2

n−2(c2

2−c3)e3

−(7c2c3−4c3

2−3c4)e4

n+O(e5

n).(33)

Using Eqs. 29-33 in method 27 we have the

following error equation:

en+1 = (1 −β)c2

2e3

−c2(3c2

2−6βc2

2+βαc2

2−3c3+ 4βc3)e4+O(e5

n).

(34)

Therefore, the iteration function deﬁned by (27) is of

order at least four for βmaking the coefﬁcient of e3

in (34) zero. Thus, we need β= 1, this means that

the method deﬁned by (27) is fourth order. This com-

pletes the proof of the theorem.

If we consider the deﬁnition of efﬁciency index

[13] as p1

w, where pis the order of the method and

wis the number of function evaluations per iteration

required by the method, then the new family method

(27) has the efﬁciency index equal to 3

√4'1.5874.

3.1 Some examples

As an example of family (27), we choose some differ-

ent values of α, to get some new fourth order methods

as follow:

for α= 1, we obtain a new fourth order method

(yn=xn−f(xn)

f0(xn),

xn+1 =yn−f(xn)

f(xn)−2f(yn)

f(yn)

f0(xn),(35)

for α= 0, we obtain a new fourth order method







yn=xn−f(xn)

f0(xn),

xn+1 =yn−f2(xn)

(f(xn)−f(yn))2

f(yn)

f0(xn),(36)

for α=−1, we obtain a new fourth order method

(yn=xn−f(xn)

f0(xn),

xn+1 =yn−f(xn)+f(yn)

f(xn)−f(yn)

f(yn)

f0(xn).(37)

4 Numerical Results

In this section, we present the results of some nu-

merical tests to compare the efﬁciencies of the new

methods . Numerical computations reported here

have been carried out in a MTHEMATICA environ-

ment . The stopping criterion has been taken as

|xn+1 −xn|< ε, We used the ﬁxed stopping crite-

rion ε= 10−14 . In Table 1, the test functions has

been used. The test results in Table 2, Show that for

most of the functions we tested the new methods pro-

posed in this paper have better performance than the

other existing methods.

In some cases, we consider the present methods

(OA1), (OA2) and (OA3) works better than (TM)

method. Note that we used NC in Table 2 to mean that

the method does not converge to the root. As a con-

clusion, we can infer that the new methods has better

performance in accordance with the theoretical analy-

sis of the order. However, it should be noted that for

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Table 1: Test function

xRoot(α)

f1(x) = x3+ 4x2−10 α= 1.365230013414096

f2(x) = sin2x−x2+ 1 α= 1.404491648215341

f3(x) = x2−ex−3x+ 2 α= 0.257530285439860

f4(x) = cos x−x α = 0.739085133215160

f5(x)=(x−1)3−2α= 2.259921049894873

f6(x) = xex2−sin2x+ 3 cos x+ 5 α=−1.207647827130919

f7(x) = (x+ 2)ex−1α=−0.442854401002388

f8(x) = e(x2+7x−30) −1α= 3.000000000000000

per iteration the new methods require just two func-

tions evaluation and one of its ﬁrst derivative. Thus,

the present methods can be of practical interest.

Table 2: Comparison of various iterative schemes and

the new methods, the number of iterations to approxi-

mate the zero. Newton method (NM), Chebysheves

method (2)(CM), a modiﬁed super Halley fourth

order (MH) optimal method [9], Traub-Ostrowski

method (TM)[38] and the new methods in (18-20)

(OA1),(OA2) and (OA3) respectively.

f(x)NM CM MH TM OA1 OA2 OA3

f1, x0=−0.353 8 75 60 31 11 7

f2, x0= 2.05 4 3 3 3 3 3

f2, x0= 1.06 5 3 3 3 3 4

f3, x0= 1.04 3 3 2 2 2 2

f3, x0= 2.05 3 4 3 3 3 3

f4, x0= 1.04 3 3 2 2 2 2

f4, x0= 1.74 4 3 3 3 3 3

f5, x0=−0.39 30 5 10 10 3 5

f5, x0= 0.0NC 5NC 9 9 6 7

f6, x0=−1.05 4 3 3 3 3 3

f6, x0=−3.014 9 8 6 6 7 8

f7, x0= 2.08 6 5 4 4 4 5

f7, x0=−0.54 3 3 2 2 2 2

f8, x0= 3.512 8 7 5 5 6 7

f8, x0= 1.019 13 11 8 8 10 11

5 Basins of Attraction

The basin of attraction is a region of the phase space

where iterations are deﬁned. such that any point in

that region will eventually be iterated into the attrac-

tor. It can be used to compare the efﬁciency of various

algorithms. See, for example[4, 5]

The aim of this section is to represent the basins

for the proposed algorithms by using graphical tools

as Mathematica. The colors of basins of attraction

vary depending on the number of iterations neces-

sary to obtain the approximate solution of polynomi-

als with given accuracy. the detailed study of basins

of attraction, its theoretical background, and applica-

tions are discussed in [16, 17, 22, 23].

We assign a color to the attraction of a root. We

make the same color lighter or darker depending on

the number of iterations needed to reach the root with

the ﬁxed precision required.

we will apply the iterative methods proposed in

the previous section to get the complex roots of func-

tions.

f(z) = z3−1and f(x) = exp sin z

100 z3−1.

We start by taking a rectangle D⊂Cand we

apply the iterative methods. starting in z0∈D. In

practice, we take a grid of 1024 ×1024 points in D

and we use these points as z0.

All the ﬁgures have been generated using the

computer program Mathematica 10.0 by taking a tol-

erance ε= 10−8and a maximum of 40 iterations.

We denote the three roots as ζ=e2kπi/3, k = 0,1,2

Then, we take z0in the corresponding rectangle and

we iterate zn+1 up to |Szn−ζk|< ε for k= 0,1,2

[NM] [MH]

[CM] [TM]

[OA1] [OA2]

[OA3]

Figure 1: Dynamical planes on f(z) = z3−1

In the ﬁrst example, we have apply all the pro-

posed algorithms to obtain the simple roots of the cu-

bic complex polynomial. The results of the basin of

attractions are presented in Figure 1. For each root of

the considered polynomial, there exists a unique color

on the corresponding basins of attraction, that can be

easily seen from Figure 1. In the next example, which

has three roots also. The basins of attraction are pre-

sented in Figure 2. three unique colors corresponding

to the distinct roots can be seen in Figure 2.

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[NM] [MH]

[CM] [TM]

[OA1] [OA2]

[OA3]

Figure 2: Dynamical planes on f(x) =

exp sin z

100 z3−1

6 Conclusion

In this paper new optimal fourth order root-ﬁnding al-

gorithms for solving nonlinear equations have been

proposed.

Table 2 show the best performance of the new

methods in terms of number of iterations, and order

of convergence as compared to other well-known ex-

isting methods. We have also presented the basins of

attraction by using some complex functions through

newly developed methods which describe the fractal

behavior and dynamical aspects of the proposed meth-

ods.

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