International Journal of Analysis and Applications

Volume 18, Number 6 (2020), 920-938

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-18-2020-920

SOME MULTISTEP ITERATIVE METHODS FOR NONLINEAR EQUATION USING

QUADRATURE RULE

GUL SANA∗, MUHAMMAD ASLAM NOOR, KHALIDA INAYAT NOOR

Department of Mathematics, COMSATS University Islamabad,

Park Road, Islamabad, Pakistan

∗Corresponding author: gulsana123@yahoo.com

Abstract. We introduce a sequence of third and fourth order iterative schemes to determine the roots of

nonlinear equations by applying quadrature formula and decomposition approach. We also examined the

convergence of suggested iterative methods under varied constraint. Various numerical test examples are

presented to exhibit the validity, efficiency and implementaion of our algorithms.

1. Introduction

An extensive range of problems that appeared in diverse directions of pure and applied sciences can

be considered by mean of nonlinear equations in the framework of novel and inventive techniques. Many

authors proposed, analyzed and modified diversity of numerical methods using various approaches such as

Taylor series, decomposition methods, quadrature formulas, modified homotopy perturbation method and

variational iterative method, and. A vast literature is available that highlight different methods used to solve

nonlinear equations for example see, [1, 3–33] and references therein. In Adomian decomposition method

solution is considered in terms of an infinite series, which converges to towards exact solution. Chun [6]

and Abbasbandy [1] constructed and investigated different higher order iterative methods by applying the

decomposition technique of Adomian [2]. Darvishi and Barati [9] also applied the adomian decomposition

Received June 22nd, 2020; accepted July 21st, 2020; published September 3rd, 2020.

2010 Mathematics Subject Classification. 41A55.

Key words and phrases. convergence analysis; quadrature formula; iterative method; decomposition technique.

©2020 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

920

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-920


Int. J. Anal. Appl. 18 (6) (2020) 921

technique to develop Newton type methods that are cubically convergent for the solution of system of non-

linear equation. Implementation of this Adomian decomposition technique required higher order derivatives

evaluation, which is major pitfall of this method. To overcome this drawback, several new techniques have

been suggested and analyzed by many researchers. Daftardar-Gejji and Jafari [10] used different modifi-

cations of adomian decomposition method and introduced simple technique which do not need derivative

evaluation of the Adomian polynomial. Numerous authors used the decomposition technique [10] and de-

rived different higher order iterative methods. Numerical computation of univariate integral is one of the

most notable problem in numerical analysis. Weerakon and Fernando cite32 improved the convergence of

Newton method using quadrature rule. Later on, Ozban [27] investigated some new variant forms of Newton

method by using the concept of harmonic mean and mid-point rule. Frontini and Sormani [11] modified

the P-dimensional case using quadrature formula and developed new iterative methods for system of non-

linear equations that are better than classical methods. Noor [25] developed fifth order convergent iterative

method using Gaussian quadrature formula and investigated its efficacy as compared to the methods already

existed in literature. Ali et al.[3] introduced a family of iterative method using the quadrature formula

as well as the fundamental theorem of calculus and decomposition technique and checked the validity and

performance of these methods by examining two mathematical models. Also in [21, 29, 30], decomposition

technique [10] is merged sophistically with coupled system of equation to investigate various order iterative

methods. Motivated and inspired by the on-going research activities in this direction in this work we con-

sider the well-known fixed point iterative method in which we rewrite the nonlinear equation f(u) = 0 as

u = g(u). In section2, we use fundamental law of calculus, along with the functional equation u = g(u) as

coupled system of equation and applying the decomposition technique [10]. We present and introduce some

new iterative methods. Section 3 consists of convergence analysis of proposed methods. Some numerical

examples are presented to make a comparative study of newly constructed methods with known third and

fourth order convergent iterative algorithms.

2. Creation of iterative methods

This section comprises of some new multistep third and fourth order convergent iterative methods in view of

mid-point, Trapezoidal rule and decomposition technique [10].

2.1. Mid-Point Rule. Consider the nonlinear equation

f(u) = 0 (2.1)

which is equivalent to

u = g(u). (2.2)



Int. J. Anal. Appl. 18 (6) (2020) 922

Assume that α is simple root of nonlinear equation (2.1) and γ is initial guess sufficiently close to root. Using

fundamental theorem of calculus and mid-point quadrature formula, we have

u = g(γ) + (u−γ)g′(
u + γ

2
). (2.3)

Now using the technique of He [16], the nonlinear equation (2.1) can be written as an equivalent coupled

system of equations

u = g(γ) + (u−γ)g′(
u + γ

2
) + H(u)

H(u) =g(u) −g(γ) − (u−γ)g′(
u + γ

2
)

=u
(

1 −g′(
u + γ)

2

)
+ γg′(

u + γ

2
) −g(γ), (2.4)

from which, it follows that

u =
H(u)

1 −g′
(
u+γ
2

) + g(γ) −γg′(u+γ2 )
1 −g′

(
u+γ
2

)
=c + M(u), (2.5)

where

c =γ (2.6)

M(u) =
H(u)

1 −g′(u+γ
2

)
+

g(γ) −γ
1 −g′(u+γ

2
)
. (2.7)

It is clear that M(u) is nonlinear operator.Now we establish sequence of higher order iterative methods

implementing decomposition technique presented by Daftardar-Gejji and Jafari [10]. In this technique, the

solution of (2.1) can be represented as in terms of infinite series.

u =

∞∑
i=0

ui. (2.8)

here the operator M(u) can be decomposed as

M(u) = M(uo) +

∞∑
i=1

{M(
i∑

j=0

uj) −M(
i−1∑
j=0

uj)}. (2.9)

Thus from equation (2.5),(2.8) and (2.9), we have

∞∑
i=1

ui = c + M(uo) =

∞∑
i=1

M(

i∑
j=0

uj) −M(
i−1∑
j=0

uj), (2.10)



Int. J. Anal. Appl. 18 (6) (2020) 923

which generates the following iterative scheme

uo = c

u1 = M(uo)

u2 = M(uo + u1) −M(uo)

.

.

.

un+1 = M(

n∑
j=0

uj) −M(
n−1∑
j=0

uj) n = 1, 2, ...

(2.11)

Consequently, it follows that

u1 + u2 + ... + un+1 = M(uo + u1 + u2 + ... + un),

and

u = c +

∞∑
i=1

ui. (2.12)

It is noted that u is approximated by

Un = (uo + u1 + u2 + ... + un),

and

lim
x→∞

Un = u.

For n=0,

u ≈ Uo = uo = c = γ. (2.13)

From (2.4), it can easily be computed as

H(uo) = 0.

Using(2.7), we get

u1 = M(uo) =
H(u0)

1 −g′(uo+γ
2

)
+

g(γ) −γ
1 −g′(uo+γ

2
)

=
g(γ) −γ

1 −g′(uo+γ
2

)

For n=1, u ≈ U1 = uo + u1 = uo + M(uo),



Int. J. Anal. Appl. 18 (6) (2020) 924

u ≈ U1 = uo + u1 = γ +
g(γ) −γ

1 −g′(uo+γ
2

)
.

Using(2.13), we have

u =
g(γ) −γg′(γ)

1 −g′(γ)
. (2.14)

This fixed point formulation is used to suggest the following algorithm.

Algorithm2A. For a given u0 (initial guess), approximate solution un+1 is computed by the following

iterative scheme

un+1 =
g(un) −ung′(un)

1 −g′(un)
. (2.15)

Kang et al. [28] developed this algorithm and proved that Algorithm 2A has quadratic convergence.

From(2.4) and (2.7),we have

H(uo + u1) = g(uo + u1) −g(γ) − (uo + u1 −γ)g′(
uo + u1 + γ

2
).

Thus

u1 + u2 = M(uo + u1) =
H(uo + u1)

1 −g′(uo+u1+γ
2

)
+

g(γ) −γ
1 −g′(uo+u1+γ

2
)

=
g(uo + u1) −g(γ) − (uo + u1 −γ)g′(uo+u1+γ2 )

1 −g′(uo+u1+γ
2

)
+

g(γ) −γ
1 −g′(uo+u1+γ

2
)

=
g(uo + u1)

1 −g′(uo+u1+γ
2

)
−

(uo + u1 −γ)g′(uo+u1+γ2 )
1 −g′(uo+u1+γ

2
)

−
γ

1 −g′(uo+u1+γ
2

)

For n=2, u ≈ U2 = uo + u1 + u2 = c + M(uo + u1).

= γ −
γ

1 −g′(uo+u1+γ
2

)
+

g(uo + u1)

1 −g′(uo+u1+γ
2

)
−

(uo + u1 −γ)g′(uo+u1+γ2 )
1 −g′(uo+u1+γ

2
)

=
g(uo + u1)

1 −g′(uo+u1+γ
2

)
−

(uo + u1)g
′(uo+u1+γ

2
)

1 −g′(uo+u1+γ
2

)

Take

uo + u1 = v =
g(γ) −γg′(γ)

1 −g′(γ)
=

g(v)

1 −g′(v+γ
2

)
−

vg′(v+γ
2

)

1 −g′(v+γ
2

)
.



Int. J. Anal. Appl. 18 (6) (2020) 925

This relation yields the following two step method for solving nonlinear equation

Algorithm 2B. For a given initial guess u0, un+1 the approximated solution un+1 can be computed by the

following iterative schemes.

vn =
g(un) −ung′(un)

1 −g′(un)
(2.16)

un+1 =
g(vn) −vng′(

un+vn)
2

1 −g′(un+vn)
2

)
, n = 0, 1, 2, .... (2.17)

It is noted that

u0 + u1 + u2 = w =
g(uo + u1)

1 −g′(uo+u1+γ
2

)
−

(uo + u1)g
′(

(uo+u1+γ
2

)

1 −g′(uo+u1+γ
2

)
. (2.18)

From (2.4) and (2.7), we can write

H(uo + u1 + u2) = g(uo + u1 + u2) −g(γ) − (uo + u1 + u2 −γ)g′(
uo + u1 + u2 + γ

2
).

and

u1 + u2 + u3 = M(uo + u1 + u2) =
H(uo + u1 + u2)

1 −g′(uo+u1+u2+γ
2

)
+

g(γ) −γ
1 −g′(uo+u1+u2+γ

2
)

=
g(uo + u1 + u2) −g(γ) − (uo + u1 + u2 −γ)g′(uo+u1+u2+γ2 )

1 −g′(uo+u1+u2+γ)
2

+
g(γ) −γ

1 −g′(uo+u1+u2+γ
2

)

=
g(uo + u1 + u2)

1 −g′(uo+u1+u2+γ
2

)
−

(uo + u1 + u2 −γ)g′(uo+u1+u2+γ2 )
1 −g′(uo+u1+u2+γ

2
)

−
γ

1 −g′(uo+u1+u2+γ
2

)
.

For n=3, u ≈ U3 = uo + u1 + u2 + u3 = uo + M(uo + u1 + u2).

= γ −
γ

1 −g′(uo+u1+u2+γ
2

)
+

g(uo + u1 + u2)

1 −g′(uo+u1+u2+γ
2

)
−

(uo + u1 + u2 −γ)g′(uo+u1+u2+γ2 )
1 −g′(uo+u1+u2+γ

2
)

.

Using (2.18), we have

= γ +
g(w)

1 −g′(w+γ
2

)
−

(w −γ)g′(w+γ
2

)

1 −g′(w+γ
2

)
−

γ

1 −g′(w+γ
2

)

=
g(w)

1 −g′(w+γ
2

)
−

wg′(w+γ
2

)

1 −g′(w+γ
2

)
.

Using this relation, we suggest the following three step method for solving nonlinear equation (2.1).



Int. J. Anal. Appl. 18 (6) (2020) 926

Algorithm 2C. For a given initial guessu0, compute the approximate solution un+1 by the following

iterative scheme

vn =
g(un) −ung′(un)

1 −g′(un)

wn =
g(vn) −vng′(

un+vn)
2

1 −g′(un+vn)
2

)
(2.19)

un+1 =
g(wn) −wng′(

un+wn)
2

1 −g′(un+wn)
2

)
, n = 0, 1, 2, ... (2.20)

2.2. Trapezoidal rule. Again using the technique of He [16] and fundamental law of calculus along with

Trapezoidal rule, we can obtain .

u = g(γ) + (u−γ)
(g′(u) + g′(γ)

2

)
+ H(u)

H(u) =g(u) −g(γ) − (u−γ)
(g′(u) + g′(γ)

2

)
=u
[
1 −

(g′(u) + g′(γ)
2

)]
+ γ
(g′(u) + g′(γ)

2

)
−g(γ), (2.21)

from which it follows that

u =
H(u)

1 −
(
g′(u)+g′(γ)

2

) + g(γ) −γ
(
g′(u)+g′(γ)

2

)
1 −

(
(g′(u)+g′(γ)

2

)
=c + M(u), (2.22)

where

c =γ (2.23)

M(u) =
H(u)

1 −
(
g′(u)+g′(γ)

2

) + g(γ) −γ
1 −

(
g′(u)+g′(γ)

2

). (2.24)
Now applying decomposition technique of Daftardar-Gejji and Jafari [10], we have

For n=0,

u ≈ Uo = uo = c = γ. (2.25)

From (2.21), it can easily be computed as H(uo) = 0.

Using(2.24), we get

u1 = M(uo) =
H(uo)

1 −
(
g′(uo)+g′(γ)

2

) + g(γ) −γ
1 −

(
g′(uo)+g′(γ)

2

) = g(γ) −γ
1 −

(
g′(uo)+g′(γ)

2

).



Int. J. Anal. Appl. 18 (6) (2020) 927

For n=1,

u ≈ U1 = uo + u1 = uo + M(uo) (2.26)

u ≈ U1 = uo + u1 = γ +
g(γ) −γ

1 −
(

(g′(uo)+g′(γ)
2

).
Using(2.26),we have

u =
g(γ) −γg′(γ)

1 −g′(γ)
. (2.27)

This formulation determines the algorithm 2A.

From(2.21) and (2.24),we have

H(uo + u1) = g(uo + u1) −g(γ) − (uo + u1 −γ)
(g′(uo + u1) + g′(γ)

2

)
.

and

u1 + u2 = M(uo + u1) =
H(uo + u1)

1 −
(
g′(uo+u1)+g′(γ)

2

) + g(γ) −γ
1 −

(
g′(uo+u1)+g′(γ)

2

)
=
g(uo + u1) −g(γ) − (uo + u1 −γ)

(
g′(uo+u1)+g

′(γ)
2

)
1 −

(
(g′(uo+u1)+g′(γ)

2

) + g(γ) −γ
1 −

(
g′(uo+u1)+g′(γ)

2

)
=

g(uo + u1)

1 −
(
g′(uo+u1)+g′(γ)

2

) − (uo + u1 −γ)
(
g′(uo+u1)+g

′(γ)
2

)
1 −

(
g′(uo+u1)+g′(γ)

2

)
−

γ

1 −
(
g′(uo+u1)+g′(γ)

2

).
For n=2, u ≈ U2 = uo + u1 + u2 = c + M(uo + u1).

=γ −
γ

1 −
(
g′(uo+u1)+g′(γ)

2

) + g(uo + u1)
1 −

(
g′(uo+u1)+g′(γ)

2

) − (uo + u1 −γ)
(

(g′(uo+u1)+g
′(γ)

2

)
1 −

(
g′(uo+u1)+g′(γ)

2

) .
Take

uo + u1 = v =
g(γ) −γg′(γ)

1 −g′(γ)
.

=γ −
γ

1 −
(
g′(v)+g′(γ)

2

) − g(v)
1 −

(
g′(v)+g′(γ)

2

) − (v −γ)
(

(g′(v)+g′(γ)
2

)
1 −

(
g′(v)+g′(γ)

2

)
=

g(v)

1 −
(
g′(v)+g′(γ)

2

) − v
(
g′(v)+g′(γ)

2

)
1 −

(
g′(v)+g′(γ)

2

).



Int. J. Anal. Appl. 18 (6) (2020) 928

This relation yields the following two step method for solving nonlinear equation

Algorithm 2D. For a given initial guessu0, the approximate solution un+1 can be computed by the following

iterative schemes

vn =
g(un) − (un)g′(un)

1 −g′(un)
(2.28)

un+1 =
g(vn) −vn

(
g′(vn)+g

′(un)
2

)
1 −

(
g′(vn)+g′(un)

2

) n = 0, 1, 2, ... (2.29)
It is noted that

uo + u1 + u2 = w =
g(uo + u1)

1 −
(
g′(uo+u1)+g′(γ)

2

) − (uo + u1)
(
g′(uo+u1)+g

′(γ)
2

)
1 −

(
g′(uo+u1)+g′(γ)

2

) . (2.30)
From (2.21) and (2.24),we can write

H(uo + u1 + u2) = g(uo + u1 + u2) −g(γ) − (uo + u1 + u2 −γ)
(g′(uo + u1 + u2) + g′(γ)

2

)
and

u1 + u2 + u3 =M(uo + u1 + u2) =
H(uo + u1 + u2)

1 −
(
g′(uo+u1+u2)+g′(γ)

2

) + g(γ) −γ
1 −

(
g′(uo+u1+u2)+g′(γ)

2

)
=
g(uo + u1 + u2) −g(γ) − (uo + u1 + u2 −γ)

(
g′(uo+u1+u2)+g

′(γ)
2

)
1 −

(
(g′(uo+u1+u2)+g′(γ)

2

)
+

g(γ) − (γ)(
1−g′(uo+u1+u2)+g′(γ)

2

)
=

g(uo + u1 + u2)

1 −
(
g′(uo+u1+u2)+g′(γ)

2

) − (uo + u1 + u2 −γ)
(

(g′(uo+u1+u2)+g
′(γ)

2

)
1 −

(
g′(uo+u1+u2)+g′(γ)

2

)
−

γ

1 −
(
g′(uo+u1+u2)+g′(γ)

2

).
For n=3, u ≈ U3 = uo + u1 + u2 + u3 = uo + M(uo + u1 + u2)

= γ −
γ

1 −
(
g(uo+u1+u2)+g′(γ)

2

) + g(uo + u1 + u2)
1 −

(
g′(uo+u1+u2)+g′(γ)

2

)
−

(uo + u1 + u2 −γ)
(

(g′(uo+u1+u2)+g
′(γ)

2

)
1 −

(
g′(uo+u1+u2)+g′(γ)

2

) .



Int. J. Anal. Appl. 18 (6) (2020) 929

Using (2.29), we obtain

=γ −
γ

1 −
(
g′(w)+g′(γ)

2

) − g(w)
1 −

(
g′(w)+g′(γ)

2

) − (w −γ)
(
g′(w)+g′(γ)

2

)
1 −

(
g′(w)+g′(γ)

2

)
=

g(w)

1 −
(
g′(w)+g′(γ)

2

) − w
(
g′(w)+g′(γ)

2

)
1 −

(
g′(w)+g′(γ)

2

).
This formulation yeilds the following three step method for solving nonlinear equation(2.1).

Algorithm 2E. For a given initial guessu0, the approximated solution un+1 is computed by the follow-

ing iterative schemes.

vn =
g(un) −ung′(un)

1 −g′(un)

wn =
g(vn) −vn

(
g′(vn)+g

′(un)
2

)
1 −

(
g′(vn)+g′(un)

2

) (2.31)
un+1 =

g(wn) −wn
(
g′(wn)+g

′(un)
2

)
1 −

(
g′(wn)+g′(un)

2

) , n = 0, 1, 2, ... (2.32)

3. Convergence Analysis of proposed Iterative methods

This section comprises of convergence analysis of proposed algorithms 2B, 2D, 2C and 2E and it shown that

these methods are third and fourth order convergent respectively.

Theorem 3.1.Let I ⊂ R be an open interval and f : I → R is differential function.If β ∈ I be root of

f(u) = 0 and u0 is sufficiently close to β then multistep method defined by Algorithm 2B and Algorithm 2C

has at least third and fourth order of convergence and satisfies the error equation.

Poof. Let β be the root of nonlinear equation f(u) = 0 or equivalently u = g(u). Let en and en+1 be the

errors at nth and (n+1) iterations respectively.

Now expanding g(u) and g′(u) by using taylor series about α, we have

g(u) = β + eng
′(β) +

e2n
2
g′′(β) +

e3n
6
g′′′(β) + O(e4n)

g′(u) = g′(β) + eng
′′(β) +

e2n
2
g′′′(β) +

e3n
6
giv(β) + O(e4n)

g(un) −ung(un) = β −βg′(β) −βg′′(β)en −
1

2

(
g′′(β) + βg′′′(β)

)
e2n +

1

6

(
2g′′′(β) + βgiv(β)

)
+ O(e4n)

1 −g′(un) = 1 −g′(β) −g′′(β)en −
e2n
2
g′′′(β) −

e3n
6
giv(β) + O(e4n)

g(un) −ung(un)
1 −g′(un)

= β +
g′′(β)

2(−1 + g′(β))
e2n −

2g′′′(β) − 2g′′′(β)g′(β) + 3g′′2(β)
6(−1 + g′(β))2

e3n + O(e
4
n).



Int. J. Anal. Appl. 18 (6) (2020) 930

From (2.16), we obtain

vn = β +
g′′(β)

2(−1 + g′(β))
e2n +

(−2g′′′(β) + 2g′′′(β)g′(β) − 3g′′2(β)
6(−1 + g′(β))2

)
e3n

+
1

12(−1 + g′(β))3
(

2giv(β) − 4giv(β)g′(β) + 2giv(β)g′2(β) + 7g′′(β)g′′′(β) − 7g′′(β)g′′′(β)g′(β)

+ 6g′′3(β)
)
e4n + O(e

5
n).

g(vn) = β+
g′(β)g′′(β)

2(−1 + g′(β))
e2n +

(
g′(β)

(
− 2g′′′(β) + 2g′′′(β)g′(β) − 3g′′2(β)

))
6(−1 + g′(β))2

e3n

+
1

24(−1 + g′(β))3
(

4giv(β)g′(β) − 8giv(β)g′2(β) + 4giv(β)g′3(β) − 14g′′(β)g′′′(β)g′2(β)

+14g′′(β)g′′′(β)g′(β) + 15g′′3(β)g′(β) − 3g′′3(β)
)
e4n + O(e

5
n). (3.1)

Expanding g′
(
vn+un

2

)
in terms of Taylors series about α, we have

g′
(vn + un

2

)
= g′(β) +

g′′(β)

2
en +

(
2g′′2(β) + g′′′(β)g′(β) −g′′′(β)

)
8(−1 + g′(β))

e2n

+
(−14g′′(β)g′′′(β) + 14g′′(β)g′′′(β)g′(β) + giv(β) − 12g′′3(β) − 2giv(β)g′(β) + giv(β)g′2(β)

48(−1 + g′(β))2
)
e3n

+
1

96(−1 + g′(β))3
(

11giv(β)g′′(β) − 22giv(β)g′(β)g′′(β) + 11giv(β)g′2(β)g′′(β) + 37g′′2(β)g′′′(β)

− 37g′′2(β)g′′′(β)g′(β) + 24g′′4(β) + 8g′′′2(β) − 16g′′′3(β)g′(β) + 8g′2(β)g′′′2(β)
)

+ O(e5n). (3.2)

g(vn) −vng′
(vn + un

2

)
= −β(−1 + g′(β)) −

1

2
g′′(β)en −

1

8(−1 + g′(β)

(
β
(

2g′′2(β) −g′′′(β)

−g′′′(β)g′(β)
)
e2n −

1

48(−1 + g′(β))2
(
− 14βg′′(β)g′′′(β) + 14βg′(β)g′′(β)g′′′(β) − 12βg′′′2(β)

+βgiv(β) − 2(β)giv(β)g′(β) + βgiv(β)g′2(β) − 12g′′2(β) + 12g′(β)g′′2(β)
)
e3n

+
1

96(−1 + g′(β))3
(
− 44g′′2(β)g′′′(β)g′(β) + 22g′′′(β)g′′(β)g′2(β) + 8βg′′′2(β) + 24g′′3(β)

−24g′(β)g′′3(β) + 22g′′′(β)g′′(β) + 11βg′′(β)giv(β) + 37βg′′2(β)g′′′(β) − 16βg′′′2(β)g′(β)

+8g′′′2(β)g′2(β) + 24βg′′4(β) − 22βg′(β)g′′(β)giv(β) + 11βg′2(β)g′′(β)giv(β)

− 37βg′(β)g′′′(β)g′′2(β)
)
e4n + O(e

5
n).

(3.3)



Int. J. Anal. Appl. 18 (6) (2020) 931

1 −g′
(vn + un

2

)
= 1 −g′(β) −

g′′(β)

2
en +

(
2g′′2(β) + g′′′(β)g′(β) −g′′′(β))

)
8(−1 + g′(β))

e2n

−
(−14g′′(β)g′′′(β) + 14g′′(β)g′′′(β)g′(β) + giv(β) − 12g′′3(β) − 2giv(β)g′(β) + giv(β)g′2(β)

48(−1 + g′(β))2
)
e3n

−
1

96(−1 + g′(β))3
(

11giv(β)g′′(β) − 22giv(β)g′(β)g′′(β) + 11giv(β)g′2(β)g′′(β) + 37g′′2(β)g′′′(β)

−37g′′2(β)g′′′(β)g′(β) + 24g′′4(β) + 8g′′′2(β) − 16g′′′3(β)g′(β) + 8g′2(β)g′′′2(β)
)
e4n + O(e

5
n)

(3.4)

Dividing (3.3),(3.4) and from (2.19), we have

wn =
g(vn) −vng′

(
vn+un

2

)
1 −g′

(
vn+un

2

)
= β +

( g′′2(β)
4(−1 + g′(β))2

)
e3n +

1

48(−1 + g′(β))3
(
g′′(β)

(
11g′′′(β)g′(β) − 11g′′′(β) − 18g′′2(β)

))
e4n

+ O(e5n).

Expanding g(wn) in terms of Taylor series about α, we obtain

g(wn) = β +
( g′(β)g′′2(β)

4(−1 + g′(β))2
)
e3n

−
1

48(−1 + g′(β))3
(
g′(β)g′′(β)

(
− 11g′′′(β)g′(β) + 11g′′′(β) + 18g′′2(β)

))
e4n + O(e

5
n).

(3.5)

Expanding g
(
wn+un

2

)
in terms of Taylor series about α, we get

g′
(wn + un

2

)
= g′(β) +

g′′(β)

2
en +

g′′′(β)

8
e2n +

1

48(−1 + g′(β))2
(

6g′′3(β) + giv(β) − 2giv(β)g′(β)

+giv(β)g′2(β)
))
e3n −

1

96(−1 + g′(β))3
(
g′′2(β) − 17g′′′(β)g′(β) + 17g′′′(β) + 18g′′2(β)

)
e4n + O(e

5
n).

1 −g
(
wn + un

2

)
= 1 −g′(β) −

g′′(β)

2
en −

g′′′(β)

8
e
2
n

−

(
6g′′3(β) + giv(β) − 2giv(β)g′(β) + giv(β)g′2(β))

)
48(−1 + g′(β))2

e
3
n

−
(
g
′′2

(β)
(−17g′′(β)g′(β) − 17g′′′(β) + 18g′′2(β)

96(−1 + g′(β))3
))
e
4
n + O(e

5
n).

(3.6)

Now using (2.20)and dividing (3.5),(3.6), simplifying

un+1 =
g(wn) −wng′( wn+un2 )

1 −g′( wn+un
2

)

=β +
(

g′′3(β)

8(−1 + g′(β))3
)
e
4
n + O(e

5
n)

en+1 =β +
(

g′′3(β)

8(−1 + g′(β))3
)
e
4
n + O(e

5
n).



Int. J. Anal. Appl. 18 (6) (2020) 932

Theorem 3.2.Let f : I → R is differential function and I ⊂ R be an open interval . If β ∈ I be root of f(u) = 0

and u0 is sufficiently close to β then multistep method defined by Algorithm 2D and Algorithm 2E has at least third

and fourth order of convergence and satisfies the error equation.

Proof. From equation (3.1), we have

g(vn) = β +
g′(β)g′′(β)

2(−1 + g′(β))
e
2
n +

(
g′(β)

(
− 2g′′′(β) + 2g′′′(β)g′(β) − 3g′′2(β)

))
6(−1 + g′(β))2

e
3
n

+
1

24(−1 + g′(β))3
(

4g
iv

(β)g
′
(β) − 8giv(β)g′2(β) + 4giv(β)g′3(β) − 14g′′(β)g′′′(β)g′2(β)

+14g
′′
(β)g

′′′
(β)g

′
(β) + 15g

′′3
(β)g

′
(β) − 3g′′3(β)

)
e
4
n + O(e

5
n).

g
′
(vn) = g

′
(β) +

g′′2(β)

2(−1 + g′(β))
e
2
n +

(
− 2g′′′(β) + 2g′′′(β)g′(β) − 3g′′2(β)

)
6(−1 + g′(β))2

e
3
n

+
1

24(−1 + g′(β)3)

(
g
′′
(β)
(

4g
iv

(β) − 8giv(β)g′(β) + 4giv(β)g′2(β) + 11g′′(β)g′′′(β)

−11g′′(β)g′′′(β)g′(β) + 12g′′3(β)
))
e
4
n + O(e

5
n).

Expanding
(
g′(vn)+g

′(un)
2

)
in terms of Taylors series about α, we have

(
g′(vn) + g

′(un)

2

)
= g
′
(β) +

g′′(β)

2
en +

(
g′′2(β) + g′′′(β)g′(β) −g′′′(β)

)
4(−1 + g′(β))

e
2
n

+
1

12(−1 + g′(β))2
(
− 2g′′(β)g′′′(β) + 2g′′(β)g′′′(β)g′(β) + giv(β) − 3g′′3(β) − 2giv(β)g′(β)

+g
iv

(β)g
′2

(β)
)
e
3
n +

1

48(−1 + g′(β))3
(
g
′′
(β)
(

4g
iv

(β) − 8giv(β)g′(β) + 4giv(β)g′2(β)

+11g
′′
(β)g

′′′
(β) − 11g′′(β)g′′′(β)g′(β) + 12g′′3(β)

))
+ O(e

5
n).

g(vn) −vn
(
g′(vn) + g

′(un)

2

)
= −β(−1 + g′(β)) −

1

2
βg
′′
(β)en −

1

4(−1 + g′(β)

(
β
(
−g′′′(β) −g′′2(β)

+g
′′′

(β)g
′
(β)
)
e
2
n −

1

12(−1 + g′(β))2
(
− 2βg′′(β)g′′′(β) + 2βg′(β)g′′(β)g′′′(β) + βgiv(β) − 2βgiv(β)g′(β)

+βg
iv

(β)g
′2

(β) − 3(β)g′′2(β) − 3g′′2(β) + 3g′(β)g′′2(β)
)
e
3
n

−
1

48(−1 + g′(β))3
(
g
′′
(β)
(

4βg
iv

(β) + 11βg
′′′

(β)g
′′
(β) − 8βgiv(β)g′(β) + 4βgiv(β)g′2(β)

−11βg′(β)g′′(β)g′′′(β) − 28g′(β)g′′′(β) + 14g′′(β)g′2(β) + 12g′′2(β) − 12g′(β)g′′2(β)

+14g
′′′

(β) + 12g
′′3

(β)
)
e
4
n + O(e

5
n).

(3.7)

1 −
(
g′(vn) + g

′(un)

2

)
= 1 −g′(β) −

g′′(β)

2
en −

(
g′′2(β) + g′′′(β)g′(β) −g′′′(β))

)
4(−1 + g′(β))

e
2
n

−
(−2g′′(β)g′′′(β) + 2g′′(β)g′′′(β)g′(β) + giv(β) − 3g′′3(β) − 2giv(β)g′(β) + giv(β)g′2(β)

12(−1 + g′(β))2
)
e
3
n

−
1

48(−1 + g′(β))3
(
g
′′
(β)
(

4g
iv

(β) − 8giv(β)g′(β) + 4giv(β)g′2(β) + 11g′′(β)g′′′(β)

−11g′′(β)g′′′(β)g′(β) + 12g′′3(β)
))

+ O(e
5
n).

(3.8)



Int. J. Anal. Appl. 18 (6) (2020) 933

Subsituting the values (3.7),(3.8) in (2.30) and simplifying, we obtain

wn =
g(vn) −vn

(
g′(vn)+g

′(un)
2

)
1 −

(
g′(vn)+g′(un)

2

)
=β +

(
g′′2(β)

4(−1 + g′(β))2
)
e
3
n +

1

24(−1 + g′(β))3
(
g
′′
(β)
(

7g
′′′

(β) − 7g′′′(β) − 9g′′2(β)
))
e
4
n

+ O(e
5
n).

Expanding g(wn) in terms of Taylor series about α, we obtain

g(wn) = β +
(

g′(β)g′′2(β)

4(−1 + g′(β))2
)
e
3
n +

1

24(−1 + g′(β))3
(
g
′′
(β)
(

7g
′′′

(β)g
′
(β) − 7g′′′(β) − 9g′′2(β)

))
e
4
n

+ O(e
5
n).

(3.9)

Expanding
(
g′(wn)+g

′(un)
2

)
in terms of Taylor series about α, we have

(
g′(wn) + g

′(un)

2

)
= g
′
(β) +

g′′(β)

2
en +

g′′′(β)

4
e
2
n +

1

24(−1 + g′(β))2
(

2g
iv

(β) − 4giv(β)g′(β)

+ 2g
iv

(β)g
′2

(β) + 3g
′′3

(β))
)
e
3
n +

(g′′2(β)(7g′′′(β)g′(β) − 7g′′′(β) − 9g′′2(β)
48(−1 + g′(β))3

)
e
4
n + O(e

5
n).

(3.10)

g(wn) −wn
(
g′(wn) + g

′(un)

2

)
= −β(−1 + g′(β)) −

1

2
βg
′′
(β)en −

1

4
βg
′′′

(β)e
2
n

−
1

24(−1 + g′(β))2
(
β
(

2g
iv

(β) − 4giv(β)g′(β) + 2giv(β)g′2(β) + 3g′′3(β)
))
e
3
n

−
1

48(−1 + g′(β))3
(
g
′′2

(β)
(

7βg
′′′

(β)g
′
(β) − 9βg′′2(β) − 7βg′′′(β) − 6g′′(β) + 6g′(β)g′′(β)

))
e
4
n

+O(e
5
n).

(3.11)

1 −
(
g′(wn) + g

′(un)

2

)
= 1 −g′(β) −

g′′(β)

2
en −

g′′′(β)

4
e
2
n −

1

24(−1 + g′(β))2
(

2g
iv

(β)

−4giv(β)g′(β) + 2giv(β)g′2(β) + 3g′′3(β)
)
e
3
n

−
1

48(−1 + g′(β))3
(
g
′′2

(β)
(

7g
′′′

(β)g
′
(β) − 7g′′′(β) − 9g′′2(β)

))
e
4
n

+O(e
5
n).

(3.12)

Substituting the values from (3.11) and (3.12) in (2.31) and simplifying

un+1 =
g(wn) −wng′( g

′(wn)+g
′(un)

2
)

1 − ( g
′(wn)+g′(un)

2
)

=β +
(

g′′2(β)

8(−1 + g′(β))3
)
e
4
n + O(e

5
n)

en+1 =β +
(

g′′2(β)

8(−1 + g′(β))3
)
e
4
n + O(e

5
n).

This shows that Algoithm 2E has fourth order convergence. �



Int. J. Anal. Appl. 18 (6) (2020) 934

4. Numerical Results and Applications

This section elaborated the efficacy of algorithms introduced in this paper with the support of examples. We obtain

estimated simple root rather than the exact based on the exactness � of the computer. We utilize �=10−15,For

computational work we use the following stopping criteria

(i). |un+1 −un| < � and (ii). |f(un+1| < �

We make a comparative representation of the Chun method (CM)[6], Newton method (NM), the Abbasbandy method

(AM)[1],Halley method (HM) [12,13], Weerakoon and Fernando(WF) [30], with the Algorithm 2B (AG1), Algorithm

2C (AG2) and Algorithm 2D (AG3) and Algorithm 2E (AG4).As for the convergence criteria, it was required that

the distance of two successive estimation for the zero was less than 10−15. In Table 1 we also displayed, the number

of iterations (IT), the approximate root un and the valuef(un)

The examples are the same as in Chun [4]

f1(u) = u
3 − 10, g(u) =

√
10

u

f2(u) = cosu−u, g(u) =cosu

f3(u) = (u− 1)3 − 1, g(u) =1 +
√

1

u− 1

f4(u) = e
u2+7u−30 − 1, g(u) =

1

7
(30 −u2)

f5(u) = sin
2
u−u2 + 1, g(u) =sinu +

1

sinu + u

f6(u) = u
2 −eu − 3u, g(u) =

u2 −eu + 2
3

Table1.Numerical Examples

f1(u) = u
3 − 10, g(u) =

√
10
u

, u0=1.5

Methods IT un f(un) δ = |un −un−1|

NM 6 2.1544346900318837 7.857615e-27 3.486728e-14

HM 6 2.1544346900318837 6.573851e-30 1.008516e-15

AM 4 2.1544346900318837 3.071853e-26 1.972938e-09

CM 9 2.1544346900318837 1.399647e-29 1.040560e-15

WF 4 2.1544346900318837 7.066989e-31 5.866631e-11

AG1 3 2.1544346900318837 8.936935e-27 3.625733e-09

AG2 3 2.1544346900318837 9.833722e-66 1.458067e-16

AG3 3 2.1544346900318837 4.856920e-26 6.374604e-09

AG4 3 2.1544346900318837 7.081450e-63 7.553158e-16

f2(u) = cosu−u, g(u) = cosu , u0=1.7



Int. J. Anal. Appl. 18 (6) (2020) 935

NM 4 0.7390851332151609 3.924473e-16 3.258805e-08

HM 5 0.7390851332151606 2.373589e-27 8.014391e-14

AM 4 0.7390851332151606 8.935133e-33 2.845706e-11

CM 5 0.7390851332151606 7.035845e-25 9.756878e-13

WF 3 0.7390851332151606 1.735664e-21 4.084953e-07

AG1 3 0.7390851332151606 5.258162e-26 8.637471e-09

AG2 3 0.7390851332151606 3.458981e-60 3.722340e-15

AG3 3 0.7390851332151606 1.117701e-27 2.392676e-09

AG4 3 0.7390851332151606 2.655872e-66 1.101871e-16

f3(u) = (u− 1)3 − 1, g(u) = 1 +
√

1
u−1 , u0=3.5

NM 7 2 2.484117e-21 2.877567e-11

HM 8 2 8.422670e-24 1.675577e-12

AM 6 2 2.626229e-20 6.615927e-11

CM 5 2 7.505295e-35 2.657272e-12

WF 5 2 9.839450e-37 6.550899e-13

AG1 3 2 1.956948e-17 4.708259e-06

AG2 3 2 1.845148e-41 1.408551e-10

AG3 3 2 2.746096e-20 5.271145e-07

AG4 3 2 9.745968e-50 1.200801e-12

f4(u) = e
u2+7u−30 − 1, g(u) = 1

7
(30 −u2) , u0=3.5

NM 12 3 5.475744e-24 2.530687e-13

HM Diverge

AM 7 3 4.775960e-18 2.353603e-07

CM 8 3 9.665693e-21 7.518279e-12

WF 8 3 5.290016e-24 1.916154e-09

AG1 4 3 1.938304e-114 2.931715e-38

AG2 3 3 2.118685e-21 2.446184e-05

AG3 3 3 3.811229e-37 1.704784e-12

AG4 3 3 4.174513e-90 1.629758e-22



Int. J. Anal. Appl. 18 (6) (2020) 936

f5(u) = sin
2u−u2 + 1, g(u) = sinu + 1

sinu+u
, u0= -1

NM 6 1.4044916482153412 1.819126e-25 3.058088e-13

HM 6 1.4044916482153412 9.561689e-28 2.217103e-14

CM 6 1.4044916482153412 1.669602e-16 6.551037e-09

AM 4 1.4044916482153412 6.138132e-21 1.329320e-07

WF 4 1.4044916482153412 9.413532e-30 1.793023e-10

AG1 4 1.4044916482153412 2.097175e-88 9.878333e-30

AG2 3 1.4044916482153412 7.391875e-68 3.273071e-17

AG3 3 1.4044916482153412 1.895915e-31 9.551650e-11

AG4 3 1.4044916482153412 3.473304e-75 4.818950e-19

f6(u) = u
2 −eu − 3u, g(u) = u

2−eu+2
3

, u0=2

NM 5 1.4044916482153412 3.439576e-27 9.869210e-14

HM 8 1.4044916482153412 1.163256e-29 5.739414e-15

CM 4 1.4044916482153412 7.664653e-31 1.280280e-10

AM 5 1.4044916482153412 9.348485e-20 3.638191e-10

WF 4 1.4044916482153412 6.103947e-34 1.630507e-11

AG1 3 1.4044916482153412 4.442861e-18 5.125169e-06

AG2 3 1.4044916482153412 7.862110e-56 7.105574e-14

AG3 4 1.4044916482153412 6.398407e-50 6.398407e-501

AG4 3 1.4044916482153412 5.810512e-38 2.083378e-09

5. Conclusion

We have developed the Algorithms 2A,2B,2C,2D,2E by approximating the definite integral
∫u
γ
g(u)du = g(u)−g(γ)

by mean of quadrature rule along with writing the nonlinear equation as coupled system of equation and using the

decomposition method. We have determined the convergence analysis of the newly suggested iterative schemes.

With the help of test examples, computational comparison has been made with well known third and fourth order

convergent iterative methods. Moreover it is observed that multistep algorithms suggested in this article requires

only first derivative evaluation of function and our results can be considered as an alternative to already known third

and fourth order convergent iterative method.



Int. J. Anal. Appl. 18 (6) (2020) 937

Acknowledgements

The authors would like to thank the Rector, COMSATS University Islamabad,Islamabad Pakistan, for providing

excellent research and academic environments.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this

paper.

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	1. Introduction
	2. Creation of iterative methods
	2.1. Mid-Point Rule
	2.2. Trapezoidal rule 

	3. Convergence Analysis of proposed Iterative methods
	4. Numerical Results and Applications
	5. Conclusion
	Acknowledgements
	References