A new three step derivative free method using weight function for numerical solution of non-linear equations arises in application problems

Abstract

In this paper a three-step numerical method, using weight function, has been derived for finding the root of non-linear equations. The proposed method possesses the accuracy of order eight with four functional evaluations. The efficiency index of the derived scheme is 1.682. Numerical examples, application problems are used to demonstrate the performance of the presented schemes and compare them to other available methods in the literature of the same order. Matlab, Mathematica 2021 & Maple 2021 software were used for numerical results.

Full text

VFAST Transactions on Mathematics
A new three step derivative free
method using weight function for
numerical solution of non-linear
equations arises in application
problems
Sanaullah Jamali1, Zubair Ahmed Kalhoro1, Abdul Wasim Shaikh1, Muhammad
Saleem Chandio1, Sanaullah Dehraj2
1Institute of Mathematics and Computer Science, University of Sindh, Jamshoro,
Sindh-Pakistan; 2Department of Mathematics and Statistics, Quaid-e-Awam University of
Engineering, Science and Technology, 67480, Nawabshah, Sindh-Pakistan
Keywords: Non-linear
Equation, Convergence
Analysis, Efficiency
Index, Derivative free,
weight function
Subject Classification:
35J05, 35J10, 35K05,
35L05
Journal Info:
Submitted:
November 17, 2022
Accepted:
December 27, 2022
Published:
December 31, 2022
Abstract In this paper a three-step numerical method, using weight function,
has been derived for finding the root of non-linear equations. The proposed
method possesses the accuracy of order eight with four functional evaluations.
The efficiency index of the derived scheme is 1.682. Numerical examples, ap-
plication problems are used to demonstrate the performance of the presented
schemes and compare them to other available methods in the literature of the
same order. Matlab, Mathematica 2021 & Maple 2021 software were used for
numerical results.
*Correspondence Author Email Address:
1 Introduction
Iterative algorithms for computing approximate zeros of nonlinear equations are of significant importance
in computational and applied mathematics due to their numerous applications in many fields of modern
science, including engineering, mathematical chemistry, bio-mathematics, physics, and statistics, among
sanaullahdehraj@quest.edu.pk
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Volume 10, Number 2, July-December, 2022 pp: 164-174
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VFAST Transactions on Mathematics
others. Numerous engineering problems take the form of nonlinear functions. Analytical approaches
generally do not tackle these types of engineering problems, hence we require an iterative technique.
Newton [1] presented one of the most well-known and famous iterative algorithm around the end of the
fifteenth century. This algorithm, commonly referred as Newton’s technique, has been widely used for
many years to solve non-linear equations.
Nowadays’s scholars are intend to develop multipoint iterative, derivative free, optimal methods to
solve nonlinear equations. A. Cordero et al. [2] proposed a general procedure to obtain optimal deriva-
tive free iterative methods by using polynomial interpolation for non-linear equations f(x) = 0. A new
bracketing and derivative free method of quadratic convergence had been proposed by S. Jamali et al. [3]
& derived from the Newton backward interpolation technique, numerical tests shows the stability of the
method. S. Jamali et al. [4] presented a Stirling interpolation technique based second-order bracketing
and derivative free method for the solution of non-linear equations. For solving non-linear equations, a
procedure for design of Steffensen-type algorithms of various orders is proposed by A. Cordero & J. Tor-
regrosa [5], many iterative techniques may be transformed into derivative-free iterative systems by using
particular divided difference of first order has been suggested. For the solution of non-linear equation B.
Neta [6] proposed a higher order derivative-free method and the novelty in the method is using Traub’s
method as first step. A novel high-order derivative-free approach for solving a non-linear equation is pro-
posed. The utilization of Traub’s approach as a first step is innovative, and introduce two new family of
Chebyshev-Halley type algorithms without derivatives for solving non-linear equations numerically. I. K.
Argyros et el. [7] suggested optimal fourth and eighth order of three and four point methods, require
the three and four function evaluations only. A. Suhadolnik [8] presented several techniques that are
based on combinations of bisection, regula falsi, and parabolic interpolation named combined bracket-
ing methods for solution of the non-linear equations. [9–13] also proposed several modified methods
using algebraic and interpolation technique. A new approach for solving non-linear equations based on
Muller’s algorithm has been developed by A. Suhadolnik [14]. F. Soleymani et al.[15] presented a Modified
Jarratt Method with Twelfth-Order Without Memory using Hermite interpolation technique for numerical
solution of nonlinear equations. S. Jamali et al.[16] recently proposed an optimal two point derivative
free method for solution of nonlinear algebraic and transcendental equation in application problems. The
Muller’s approach is based on an interpolating polynomial constructed from the last three points of an
iterative sequence. For solution of non-linear equations M. S. Petkovic et al. [17] Proposed two point
derivative free family of methods. An arbitrary real parameter and suitable parameter function are used
in these methods. A. Cordero et al. [18] Transform two renowned fourth-order Ostrowski’s method and
sixth-order improved Ostrowski’s method in to derivative free methods by approximate derivatives by
central-difference quotients.
2 Derivation of proposed method
It is proposed in [19], a non-optimal eighth order method with five function evaluation (three function
and two first derivative)
i.e
Step.1 ψn=ξn–φ(ξn)
φ′(ξn)
Step.2 ηn=ψn–φ(ψn)
φ′(ξn)φ2(ξn)
φ2(ξn)–2φ(ξn)φ(ψn)+φ2(ψn)
Step.3 ξn+1 =ηn–φ(ηn)
φ′(ηn)
(1)
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In three-point formula (1) requires five function evaluation per iteration, to reduce the number of func-
tion evaluation and make it derivative free, we approximate φ′(ξn)≈φ[κn,ξn] and φ′(ηn)≈φ[ψn,ηn]using
finite difference. And also introduce the weight function (A(t1) + B(t2)+C(t3))in third step of (1) to enhance
the convergence.
Finally we got.
Step.1 ψn=ξn–φ(ξn)
φ[κn,ξn]
Step.2 ηn=ψn–φ(ψn)
φ[κn,ξn]φ2(ξn)
φ2(ξn)–2φ(ξn)φ(ψn)+φ2(ψn)
Step.3 ξn+1 =ηn–(A(t1) + B(t2)+C(t3))φ(ηn)
φ[ψn,ηn]
(2)
we replace φ′(ξn)≈φ[κn,ξn], where [5]κn=ξn+λφm(ξn)∀λ,0∧m≥2
takingλ= 1 ∧m= 3 .
Where φ[κn,ξn] = φ(κn)–φ(ξn)
κn–ξn,φ[ψn,ηn] = φ(ψn)–φ(ηn)
ψn–ηnand for weight function (A(t1) + B(t2)+C(t3))we expand
the taylor series about 1
A(t3) = A(1) + (t3– 1)A′(1) + 1
2(t3– 1)2A′′ (1) + 1
6(t3– 1)3A′′′ (1) + ....
B(t1) = B(1) + (t1– 1)B′(1) + 1
2(t1– 1)2B′′ (1) + 1
6(t1– 1)3B′′′ (1) + ....
C(t2) = C(1) + (t2– 1)C′(1) + 1
2(t2– 1)2C′′ (1) + 1
6(t2– 1)3C′′′ (1) + ....
(3)
By taking
A(1)= 2, A′(1)=A′′ (1)= 6, An(1)= 0, if n> 2, B(1)= –1, B′(1)=B′ ′ (1)= 0,Bn(1)= 0,
if n> 2, C(1)= 3, C′(1)= 2, Cn(1)= 0 if n> 1
we get
A(t1) = 1 +t2
1
B(t2) = –1
C(t3) = 1 + 2t3
(4)
and t1=φ(ψn)
φ(ξn),t2=φ(ηn)
φ(ψn)&t3=φ(ηn)
φ(ξn)
3 Convergence Analysis
Theorem I: Let a∈Dbe a simple zero of a sufficiently differentiable function φ:D⊂R→Rin an open
interval D, which contains ξ0as an initial approximation of a. Then the method (2) is of order eighth and
includes only four function evaluations per iteration, and no derivatives used.
Proof.
We write down the Taylor’s series expansion of the function φ(ξn).
φ(ξn) =
∞
X
m=0
φm(a)
m!(ξn–a)m=φ(a)+φ′(a)(ξn–a)+φ′′ (a)
2! (ξn–a)2+φ′′′ (a)
3! (ξn–a)3+... (5)
For simplicity, we assume that Ak=1
k!φk(a)
φ′(a),k≥2.
and assume that εn=ξn–a. thus, we have
φ(ξn)=φ′(a)hεn+A2ε2
n+A3ε3
n+A4ε4
n+ ...i(6)
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Furthermore, we have
φ[κn,ξn]=φ(κn)–φ(ξn)
κn–ξn
=φ′(a)





1 + 2A2εn+ 3A3ε2
n+A2φ′3(a)+ 4A4ε3
n+
3φ′3(a)A2
2+A3ε4
n+ ... + Oε9
n




(7)
Dividing (5) by (6) gives us
φ(ξn)
φ[κn,ξn]=εn–A2ε2
n+ 2 A2
2–A3ε3
n+–4A3
2+ 7A2A3–A2φ′3(a)– 3A4ε4
n+
... + Oε9
n(8)
and hence, we have
Step.1 ψn=ξn–φ(ξn)
φ[κn,ξn]=A2ε2
n+ (–2A2
2+A2φ′2(a) + 2A3)ε3
n+ (4A3
2–
A2
2φ′2(a) – 7A2A3+ 3A3φ′2(a) + 3A4)ε4
n+ ... + O(ε9
n)(9)
and
φ(ψn) = φ′(a)[A2ε2
n+ (–2A2
2+A2φ′2+ 2A3)ε3
n+ (4A3
2–A2
2φ′2(a) – 7A2A3+
3A3φ′2(a) + 3A4)ε4
n+ ... + O(ε9
n)] (10)
from equation (5) and (9) we got
φ[ξn,ψn] = φ′(a)(1 + A2εn+ (A2
2+A3)ε2
n+ (–2A3
2+ 3A2A3+A4)ε3
n+ ... + O(ε9
n)) (11)
φ(ψn)
φ[ξn,ψn]=A2ε2
n+ (2A3– 3A2
2)ε3
n+ (7A3
2– 10A2A3+A2φ′3(a) + 3A4)ε4
n+ ... + O(ε9
n) (12)
and
φ2(ξn)
φ2(ξn)–2φ(ξn)φ(ψn)+φ2(ψn)= 1 + 2A2εn+ (4A3– 3A2
2)ε2
n+ 2(A3
2– 4A2A3+
A2φ′3(a) + 3A4)ε3
n+ ... + O(ε9
n)
(13)
Step.2 ηn=ψn–φ(ψn)
φ[κn,ξn]φ2(ξn)
φ2(ξn)–2φ(ξn)φ(ψn)+φ2(ψn)=2A3
2–A2A3ε4
n+
–10A4
2+ 14A2
2A3–A2
2φ′3(a) – 2A2A4– 2A2
3ε5
n+ ... + Oε9
n(14)
And
φ(ηn) = φ′(a)




(2A3
2–A2A3)ε4
n+





–10A4
2+ 14A2
2A3–
A2
2φ′3(a) – 2A2A4– 2A2
3





ε5
n+ ... + Oε9
n




(15)
And
φ[ψn,ηn] = φ(ψn) – φ(ηn)
ψn–ηn
=φ′(a)










1 + A2
2ε2
n+ 2A2A3–A2
2ε3
n+
A26A3
2+A2φ′3(a)– 7A3+ 3A4ε4
n+
... + Oε9
n











(16)
And weight function
(A(t1) + B(t2)+C(t3))= 1 + A2
2ε2
n+ 2 A2A3–A3
2ε3
n+A2A3
2– 2A2A3+ 2A4ε4
n+
... + Oε9
n(17)
Finally
Step.3 ξn+1 =ηn–(A(t1) + B(t2)+C(t3))φ(ηn)
φ[ψn,ηn]=
A2
22A2
2–A33A3
2+A2φ′3(a) – 4A3+A4ε8
n+Oε9
n(18)
Since the order of convergence of proposed method is four, and there are three function evaluation used
in each iteration with no derivative. Its efficiency index is 41/3 = 1.587401...
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4 Application Problems
Maple 2021 and Mathematica 2021 were used for solution of following application problems, and Matlab
was used to find the COC of application problems.
Problem 1. Planck’s radiation law. See in [7,20–22]
φ1= exp(–x)–1+
x
5
(19)
Table 2. Numerical results for problem 1., error fixed at delta = 10
–3000
Method IG
NFE
Proposed 8th 4
416
SM 8th 4
520
AKKB 8th 4
520
KBK 8th 4
416
JLM 8th 4
520
Table 1. Numerical results for problem 1. for first four iterations and their absolute function values at
x0= 4
Method 1st iteration 2nd iteration 3rd iteration 4th iteration
Proposed 8th 4.96511428 ×100
9.67569418 ×10–9
4.96511423 ×100
3.50682003×10–68
4.96511423 ×100
1.04415339 ×10–543
4.96511423 ×100
6.45016629 ×10–4348
SM 8th 4.96512223 ×100
1.54477412 ×10–6
4.96511423 ×100
6.60093214×10–38
4.96511423 ×100
4.01840927 ×10–226
4.96511423 ×100
2.04525547 ×10–1355
AKKB 8th 4.96511467 ×100
8.42634074 ×10–8
4.96511423 ×100
1.84817509×10–40
4.96511423 ×100
9.38132174 ×10–204
4.96511423 ×100
3.16132118 ×10–1020
KBK 8th 4.96511410 ×100
2.56680409 ×10–8
4.96511423 ×100
3.29102226×10–64
4.96511423 ×100
2.40345205 ×10–511
4.96511423 ×100
1.94478951 ×10–4088
JLM 8th 4.96511426 ×100
6.35715655 ×10–9
4.96511423 ×100
3.07879628×10–54
4.96511423 ×100
3.97273188 ×10–326
4.96511423 ×100
1.83374553 ×10–1957
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Problem 2. (car stability). Application in mechanical engineering See in [23].
φ2(x) = x4– 1.9404x2+ 0.75 (20)
Table 3. Numerical results for problem 2. for first four iterations and their absolute function values at
x0= –0.5.
Method 1st iteration 2nd iteration 3rd iteration 4th iteration
Proposed 8th –7.29955558×10–1
9.28913981×10–10
–7.29955559×10–1
6.01242356×10–72
–7.29955559 ×10–1
1.85200488 ×10–569
–7.29955559 ×10–1
1.50100656 ×10–4549
SM 8th –7.29686598×10–1
3.43561979×10––4
–7.29955559×10–1
5.96113371×10–26
–7.29955559 ×10–1
4.96026069 ×10–200
–7.29955559 ×10–1
1.14001249 ×10–1592
AKKB 8th –7.29809213×10–1
1.86914376 ×10–4
–7.29955559×10–1
7.58360417×10–31
–7.29955559 ×10–1
5.75141912 ×10–242
–7.29955559 ×10–1
6.29462552 ×10–1931
KBK 8th –7.29954445×10–1
1.42205591 ×10–6
–7.29955559×10–1
2.25812858×10–46
–7.29955559 ×10–1
–9.12880222×10–365
–7.29955559 ×10–1
–6.51230669 ×10–2912
JLM 8th –7.29955550×10–1
1.20206197 ×10–8
–7.29955559×10–1
6.40069903×10–48
–7.29955559 ×10–1
1.45891770 ×10–283
–7.29955559 ×10–1
2.04574324 ×10–1697
Table 4. Numerical results for problem 2., error fixed at delta = 10–3000
Method IG N FE
Proposed 8th –0.5 4 16
SM 8th –0.5 5 20
AKKB 8th –0.5 5 20
KBK 8th –0.5 5 20
JLM 8th –0.5 5 20
Problem 3. (The Shockley Diode Equation and Electric Circuit). See in [24].
φ3= –0.5 + 0.1 x+ 1.4ln (x+ 1)(21)
Table 5. Numerical results for problem 3. for first four iterations and their absolute function values at
x0= 0.
Method 1st iteration 2nd iteration 3rd iteration 4th iteration
Proposed 8th 3.89992153 ×10–1
3.80387471 ×10–5
3.89977198 ×10–1
5.64267581×10–39
3.89977198 ×10–1
1.32263897 ×10–309
3.89977198 ×10–1
1.20528858 ×10–2474
SM 8th 3.90022793 ×10–1
5.04816858 ×10–5
3.89977198 ×10–1
2.79924005×10–37
3.89977198 ×10–1
2.50206521 ×10–295
3.89977198 ×10–1
1.01945125 ×10–2359
AKKB 8th 3.90071054 ×10–1
1.03914445 ×10–4
3.89977198 ×10–1
2.45440600×10–34
3.89977198 ×101
2.37750739 ×10–271
3.89977198 ×10–1
1.84300132 ×10–2167
KBK 8th 3.89596121 ×10–1
4.21985237 ×10–4
3.89977198 ×10–1
4.01193524×10–31
3.89977198 ×10–1
2.67875461 ×10–247
3.89977198 ×10–1
1.05819347 ×10–1976
JLM 8th 3.89992153 ×10–1
1.65577677 ×10–5
3.89977198 ×10–1
1.03787154×10–32
3.89977198 ×10–1
6.29492624 ×10–196
3.89977198 ×10–1
3.13381179 ×10–1175
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Table 6. Numerical results for problem 3., error fixed at delta = 10–3000
Method IG N FE
Proposed 8th 0 5 20
SM 8th 0 5 20
AKKB 8th 0 5 20
KBK 8th 0 5 20
JLM 8th 0 5 20
Problem 4. Beam Designing Model (An Engineering Problem) See in [25].
φ4=x4+ 2.87x2– 10.28
4.62 –x(22)
Table 7. Numerical results for problem 4. for first four iterations and their absolute function values at
x0= –3.5.
Method
1st iteration
2nd iteration
3rd iteration
4th iteration
Proposed 8th
–3.33038696 ×100
3.52258020 ×10–6
–3.33038866 ×100
6.26674991×10–46
–3.33038866 ×100
6.28755917 ×10–364
–3.33038866 ×100
6.45654017 ×10–2908
SM 8th
–3.33039778 ×100
1.88258338 ×10–5
–3.33038866 ×100
7.72183249×10–39
–3.33038866 ×100
6.18758077 ×10–306
–3.33038866 ×100
1.05178003 ×10–2442
AKKB 8th
–3.33044757 ×100
1.21628040 ×10–4
–3.33038866 ×100
6.36232200×10–32
–3.33038866 ×100
3.57010797 ×10–250
–3.33038866 ×100
3.50919087 ×10–1996
KBK 8th
–3.33039293 ×100
8.81113549 ×10–6
–3.33038866 ×100
3.72411245×10–43
–3.33038866 ×100
3.79307745 ×10–342
–3.33038866 ×100
4.39281694 ×10–2734
JLM 8th
–3.33038937 ×100
1.45545266 ×10–6
–3.33038866 ×100
6.70379768×10–39
–3.33038866 ×100
6.40118774 ×10–233
–3.33038866 ×100
4.85175162 ×10–1397
Table 8. Numerical results for problem 4., error fixed at delta = 10–3000
Method IG N FE
Proposed 8th –3.5 5 20
SM 8th –3.5 5 20
AKKB 8th –3.5 5 20
KBK 8th –3.5 5 20
JLM 8th –3.5 5 20
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Problem 5 Below algebraic and transcendental problems were taken from literature and tested in pro-
posed method.
Functions
φ5(x) = sin(x)+ cos (x)+x
φ6(x)=esin(x)–x+ 1
φ7(x) = x4+x3+ 8x+ 9
φ8(x)=x6+(1 – 2x)5
φ9(x)= 1 + ex2+x– cos –x2+ 1+x3
φ10 (x)= tanh(x)+ 2x
Table 9. Shows the value of |x1–x0|,|x2–x1|,|x3–x2|&COC of different methods of order eight.
|xn–xn–1|Proposed 8th SM 8th AKKB 8th KBK 8th JLM 8th
Case 1. For φ5(x), –0.2
|x1–x0|2.5663e– 1 2.5667e– 1 2.5659e– 1 2.5662e– 1 2.5662e– 1
|x2–x1|3.1377e– 6 4.1049e– 5 3.2525e– 5 4.0069e– 6 7.4256 – 7
|x3–x2|1.3122e– 47 2.0867e– 28 1.2030e– 25 5.1459e– 46 1.5995e– 41
COC 8.4228 6.1363 5.2430 8.3075 6.2592
Case 2. For φ6(x), 3
|x1–x0|3.6934e– 1 3.6932e– 1 3.6934e– 1 3.6931e– 1 3.6934e– 1
|x2–x1|2.3201e– 6 1.1058e– 5 1.6206e– 6 2.7809e– 5 1.4600e– 7
|x3–x2|1.7253e– 48 1.5512e– 32 4.1167e– 33 1.5342e– 39 2.3747e– 45
COC 8.2024 5.9360 4.9639 8.0087 5.9017
Case 3. For φ8(x), 1.01
|x1–x0|1.0000e– 2 1.0000e– 2 9.9997e– 3 1.0000e– 2 1.0000e– 2
|x2–x1|1.6348e– 12 3.3753e– 9 3.1459e– 7 5.9699e– 11 1.1601e– 8
|x3–x2|3.3124e– 90 2.5462e– 61 1.6935e– 43 8.54943 – 77 3.4369e– 44
COC 8.2003 8.0539 8.0557 8.0063 5.9857
Case 4. For φ9(x), –0.8
|x1–x0|2.0000e– 1 2.0000e– 1 2.0000e– 1 1.9999e– 1 2.0000e– 1
|x2–x1|3.1850e– 7 4.6487e– 6 1.8970e– 6 6.9811e– 6 4.1646e– 6
|x3–x2|2.8535e– 53 2.4506e– 42 4.5696e– 464 8.4973e– 42 4.0472e– 34
COC 8.2021 7.8292 7.8874 8.0579 5.9837
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Case 5. For φ10 (x), 1
|x1–x0|3.0001 3.0062 3.0075 Not Converge 3.0021
|x2–x1|7.5517e– 5 6.1713e– 3 7.4714e– 3 Not Converge 2.1370e– 3
|x3–x2|2.2396e– 47 3.8545e– 25 7.8817e– 24 Not Converge 2.6473e– 25
COC 8.5470 8.2617 8.0531 Not Converge 6.9598
5 Conclusion
In this paper, the main attention has been made to derive a three-step optimal derivative free method of
eighth order for finding the root of a non-linear equation in application problems using weight function.
Various application problems have been tested by proposed method and compared with other counter-
part methods available in the literature. It’s observed from the above tables that the proposed method is
reducing the error faster than the other methods. It’s accurate, consistent and its stability is much better
as compared to counterpart methods available in the literature, above proposed method is good addition
in literature. In the future we are interested in proposing a four-point derivative free optimal iterative
method of order sixteen.
Credit Author Statement
Sanaullah Jamali: Solution of the problem, Zubair Ahmed Kalhoro: Conceptualization, Methodology,
Abdul Wasim Shaikh: Visualization, Investigation. Muhammad Saleem Chandio: Supervision, Software
support, Sanaullah Dehraj: Software, Validation, Writing- Reviewing and Editing
Compliance with Ethical Standards
It is declared that all authors don’t have any conflict of interest. Furthermore, informed consent was
obtained from all individual participants included in the study.
Funding Information
No funding
Authors Information
ORICD:
Sanaullah Dehraj: 0000-0001-8453-3389
Sanaullah Jamali: 0000-0002-1162-5909
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