A NEW STUDY TO FIND OUT THE BEST COMPUTATIONAL METHOD FOR SOLVING THE NONLINEAR EQUATION

Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
DOI : 10.5121/mathsj.2019.6302
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A NEW STUDY TO FIND OUT THE BEST
COMPUTATIONAL METHOD FOR SOLVING THE
NONLINEAR EQUATION
Mir Md. Moheuddin1
, Md. Jashim Uddin2
and Md. Kowsher3*
1
Dept. of CSE, Atish Dipankar University of Science and Technology, Dhaka-1230,
Bangladesh.
2,3
Dept. of Applied Mathematics, Noakhali Science and Technology University,
Noakhali-3814, Bangladesh.
ABSTRACT
The main purpose of this research is to find out the best method through iterative methods for solving the
nonlinear equation. In this study, the four iterative methods are examined and emphasized to solve the
nonlinear equations. From this method explained, the rate of convergence is demonstrated among the 1st
degree based iterative methods. After that, the graphical development is established here with the help of
the four iterative methods and these results are tested with various functions. An example of the algebraic
equation is taken to exhibit the comparison of the approximate error among the methods. Moreover, two
examples of the algebraic and transcendental equation are applied to verify the best method, as well as the
level of errors, are shown graphically.
KEYWORDS
Newton-Raphson method, False Position method, Secant method, Bisection method, Non-
linear equations and Rate of convergence.
1. INTRODUCTION
In numerical analysis, the most appearing problem is to determine the equation’s root in the form
of f(x) = 0 rapidly where the equations of the types of f(x) = 0 is known as Algebraic or
Transcendental according to as
f(x) = a0xn+a1xn−1+. .... +an−1x +an,(a≠0),
where the expression of f(x) being an algebraic or transcendental or a combination of both.If f(x)
= 0
at the point x= α, then α is acquainted as the root of the given equation if and only if f(α)=0. The
equations of f(x) = 0 is an algebraic equation of degree k such as x3-3x2+12=0, 2x3-3x-6=0 are
some kinds of algebraic equations of two and three degree whereas if f(x) takes on small number
of function including trigonometric, logarithmic and exponential etc then f(x) = 0 is a
Transcendental equations, for instance, x sin x + cos x = 0, log x – x+3=0 etc are the
transcendental equations. In both cases, the coefficients are known as numerical equations if it is
pure numbers. Besides, several degrees of algebraic functions are solved by well-known methods
that are common to solve it. but problems are when we are trying to solve the higher degree or
transcendental functions for which there are no direct methods of how to get the best solution
easily. Using approximate methods are a facile way for these sorts of equations. Numerical

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methods are applied to impart formative solutions of the problems including nonlinear equations.
Various methods for discovering the nonlinear equation’s root exists. For example, the Graphical
method, Bisection method, Regula-falsi method, Secant method, Iteration method, Newton-
Raphson method, Bairstow’s method and Graeffe’s root squaring method and Newton’s-iterative
method, etc. Several formulae of the numerical method is described in the books of S.S. Sastry
[3], R.L. Burden [1], A.R. Vasishtha [2] as well as there are lots of paper involving nonlinear
Algebraic, Transcendental equations like improvement in Newton-Rapson method for nonlinear
equations applying modified Adomian Decomposition method [4], on modified Newton method
for solving a nonlinear algebraic equations by mid-point [10], New three-steps iterative method
for solving nonlinear equations [7], A New two-step method for solving nonlinear equations [9],
as well as a new method for solving nonlinear equations by Taylor expansion [8]. Besides, some
investigations [5, 6, 16] have shown a better way to interpret the algebraic and transcendental
equation’s root whereas Mohammad Hani Almomani et al. [11] presents a method for selecting
the best performance systems. What is more, in [12], Zolt ´an Kov´acs et al. analyzed in
understanding the convergence and stability of the Newton-Raphson method. In [13], Fernando
Brambila Paz et al. calculated and demonstrated the fractional Newton-Raphson method
accelerated with Aitken’s method. Robin Kumar et al. [14] examined five numerical methods for
their convergence. In [15] Nancy Velasco et al. implemented a Graphical User Interface in Matlab
to find the roots of simple equations and more complex expressions. However, Kamoh Nathaniel
Mahwash et al. [16] represents the two basic methods of approximating the solutions of nonlinear
systems of algebraic equations. In [17], Farooq Ahmed Shah et al. suggest and analyze a new
recurrence relation which creates the iterative methods of higher order for solving nonlinear
equation f (x) = 0. Also, M. A. Hafiz et al. [18] propound and analyse a new predictor-corrector
method for solving a nonlinear system of equations using the weight combination of mid-point,
Trapezoidal and quadrature formulas.
In our working procedure, we have discussed and also compared with the existing some iterative
methods, for example, Newton’s method, False Position method, Secant method as well as
Bisection method to achieve the best method among them. This research paper based on analyzing
the best method to impart the best concept for solving the nonlinear equations. With the help of
this paper, we can easily perceive how to determine the approximate roots of non-linear equations
speedily. Moreover, we demonstrated some function to obtain an effective method which gives a
smaller level of iteration among the mentioned methods.
SCOPES AND OBJECTIVES OF THE STUDY
The main scopes of the study are to evaluate the best method for solving nonlinear equations
applying numerical methods. The objectives are given below;
 Finding the rate of convergence, proper solution, as well as the level of errors of the
methods.
 Comparing the existed methods to achieve the best method for solving nonlinear
equations.
2. MATERIALS AND METHODS
Due to these works, the following methods is discussed such as Newton-Raphson method, False
Position method, Secant method and Bisection method.
2.1. Newton-Raphson Method
Newton-Raphson method is the famous numerical method that is almost applied to solve the

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equation’s root and it helps to evaluate the roots of an equation f(x) = 0 where f(x) is considered
to have a derivative of f'(x). The basic idea of Newton-Raphson
Figure 1. Newton-Raphson method
Also, Newton method can be achieved from the Taylor’s series expansion of the function f(x) at
the point 𝑥0 as follows:
Since (𝑥−𝑥0) is small, we can neglect second, third and higher degree terms in (𝑥−𝑥0) and thus
we obtain
(𝑥)=𝑓(𝑥0)+(𝑥−𝑥0)𝑓′(𝑥0)……..(1)
Putting f(x) = 0 in (1) we get,
Generalizing, we get the general formula
Which is called Newton-Raphson formula.
Example: Solve the equation f(x) = 𝑥3+2𝑥2+10𝑥−20=0 using Newton’s method.
Given that, f(x)= 𝑥3+2𝑥2+10𝑥−20=0
∴𝑓′(𝑥)=3𝑥2+4𝑥+10
Apply Newton’s method to the equation, correct to five decimal places.
Now we can see that f(1) = - 7 < 0 and f(2) = 16 > 0.
Therefore the root lies between 1 and 2.
We know that, Newton’s formula as follows,

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Solving the right hand side yields
Convergence of Newton-Raphson method:
Newton’s method is given below,
This is really an iteration method where
2.2. False Position method
False position method, which is a very old algorithm for solving a root finding problem. It is
called trial and error technique that exerts test values for the sake of variable and after that
adjusting this test value in imitation of the outcome. This formula is also called as "guess and
check". The basic concept of this method is demonstrated in the following figure,

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Figure 2. False Position method
Consider the equation f(x) = 0, where f(x) is continuous. In this method we choose two points a
and b such that f(a) and f(b) be of opposite signs (a < b). So the curve of y = f(x) will meet the x-
axis between A(a, f(a)) and B(b, f(b)). Hence a root lies between these two points. Now the
equation of the chord joining the points A(a, f(a)) and B(b, f(b)) is given below
The point of intersection of the chord equation for the x-axis imparts the first approximation 𝑥0
for the root of f(x) = 0, that is achieved by substituting y = 0 in (1) we get,
If f(a) and f(𝑥0) be of opposite signs, so its root is between a and 𝑥0. Then replacing b by 𝑥0 in
(2), we obtain the next approximation 𝑥1. But if f(a) and f(𝑥0) are same sign, so f(𝑥0) and f(b) be
of opposite signs, therefore its root is between 𝑥0 and b. Again, replacing a by 𝑥0 in (2) we find
𝑥1. This work is continued till the root is achieved its desired accuracy.The genearl formula is
Example: Solve the equation f(x) = 𝑥3−3𝑥−5=0 by the method of False Position method.
Solution:
Let f(x) = 𝑥3−3𝑥−5=0
Since f(2) = -3<0
F(3) = 13>0
The root lies between 2.1875 and 3 such that a = 2.1875 and b = 3 etc
.
Convergence of False Position method:
We know that,

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In the equation f(x) = 0, if the function f(x) is convex in the interval (𝑥0,𝑥1) that takes a root then
one of the points 𝑥0, or 𝑥1 is always fixed and the other point varies with n. If the point 𝑥0 is fixed
, then the function f(x) is approximated by the straight line passing through the points (𝑥0,𝑓0) as
well as (𝑥 𝑛,𝑛), n = 1,2,……
Now, the error equation (2) from the convergence of secant method becomes
From equation (1), we get
Where C = C𝜀0 is the asymptotic error constant. Hence, the false Position method has Linear rate
of Convergence.
2.3. Secant Method
The secant method is just a variation from the Newton’s method. The basic idea is shown below,
Figure 3. Secant method
The Newton method of solving a nonlinear equation f(x) = 0 is given by the iterative formula.
One of the drawbacks of the Newton’s method is that one has to evaluation the derivative of the
function. To overcome these drawbacks, the derivative of the function f(x) is approximated as
Putting equation (2) into (1), we get,

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The above equation is called the secant method. This is like Newton-Raphson method but
requires two initial guess.
Example: Solve the equation f(x) = 𝑥3−2𝑥−5=0 using secant method.
Let the two initial approximations be given by
𝑥−1=2 and 𝑥0= 3
The root lies between 2 and 3.
Convergence of Secant method:
We know that,
Where C = ½ 𝑓′′(𝜁)/𝑓′(𝜁) and the higher powers of 𝜀 𝑛 are neglected.

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Neglecting the negative sign, we get the rate of convergence for the Secant method is p = 1.618.
2.4. Bisection Method
The bisection method is very facile and perhaps always faithful to trace a root in numerical
schemes for solving non-linear equations. It is frequently applied to acquire of searching a root
value of a function and based on the intermediate value theorem (IVT). This method always
succeeds due to it is an easier way to perceive. The basic concept of this method is given below;
Figure 4. Bisection method
This Bisection method states that if f(x) is continuous which is defined on the interval [a, b]
satisfying the relation f(a) f(b) < 0 with f(a) and (b) of opposite signs then by the intermediate
value theorem(IVT) ,it has at least a root or zero of f(x) = 0 within the interval [a, b]. It consists of
finding two real numbers a and b, having the interval [a, b], as well as the root, lies in a ≤ x ≤ b
whose length is, at each step, half of its original interval length. This procedure is repeated until
the interval imparts required accuracy by virtue of the requisite root is achieved. While it has
more than one root on the interval (a, b) ,as a result, the root must be unique. Moreover, the root
converges linearly and very slowly. But when f(𝑥0) is very much nearly zero or is very small
then we can stop the iteration. Therefore, the general formula is

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So at least one root of the given equation lies between 1 and 2.
The root lies between 1 and 1.5 etc.
Convergence of Bisection method:
2.5. Comparison Of The Rate Of Convergence Among The Four Method
The rate of convergence is a very essential issue for the sake of solution of algebraic and
transcendental equations, due to the rate of convergence of any numerical method evaluates the
speed of the approximation to the solution of the problems.
Following table demonstrates the comparison of rate of convergence.
Table 1. Comparison of the rate of convergence
From this table, we see that Newton-Raphson has a better rate of convergence than other methods
i.e. 2.

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3. ANALYSIS AND COMPARISON
3.1. Graphical Development Of The Approximation Root To Achieve The Best
Method
The graphical development of the four method is illustrated as follows,
Figure 5. Graphical development among the four methods. The function used is 𝑥2-30 with the interval [5
6] and the desired error of 0.0001.
From the above graph, for the Newton-Raphson method, in each iteration, the graph demonstrates
the approximate root with only 3 iterations as it converges more rapidly and accurately whereas
other method delays to give desire roots of the equation. In the Bisection method, it requires a
large number of iterations to meet the desired root as convergent slowly. For the Secant method,
it imparts a level of convergence but not as compared to the Newton method. The False Position
method converges more slowly due to it needs at least 10 iterations to compare to the Newton and
secant method respectively.
Both algorithms are tested with several functions taking as data for error 0.0001 and the desired
accuracy. The results gained for the approximate root, error, and iteration are summarized in table
2.6.1 for the Newton-Raphson method, in table 2.6.2 for False-Position method, in table 2.6.3 for

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Secant method and in table 2.6.4 for Bisection method. The Newton-Raphson imparts the small
number of iteration correctly as long as the initial values enclose the desired root.
Table 2. Data obtained with the Newton-Raphson Method
In the False position method, it imparts a lot of iteration number to meet the desired root.
Table 3. Data obtained with the False Position Method
We apply the secant method to obtain the result with a few numbers of iterations.
Table 4. Data obtained with the Secant Method
Similarly, the Bisection method demonstrates the roots slowly as long as the initial values enclose
the desired root as well as gives a large number of iteration.

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Table 5. Data obtained with the Bisection Method
3.2. Comparison Of The Approximate Error
Figure 6. The approximate error is plotted against the number of iterations (1-20).
From the above graphical development and comparison, it is recommended that Newton-
Raphson’s method is the fastest in the rate of convergence compared with the other methods.
4. TEST AND RESULTS TO VERIFY THE BEST METHOD
Following examples interpret the result achieved by the four methods to solve nonlinear equations
such as algebraic and transcendental equations.
Problem 1. Consider the following equation [2 3],

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Table 7. Number of iterations and the results obtained by the four method
Moreover, we can look at the following approximate error graph gained from the above four
methods which illustrate the change of errors from one method to another method.
Figure 7. The graph of NRM Figure 8. The graph of FPM

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Figure 9. The graph of SM Figure 10. The graph of BM
Comparing the above graph, we see that Newton-Raphson Method is a better method than others.
Similarly,
Problem 2. Consider the following equation [-1 -2],

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Table 8. Number of iterations and the results obtained by the four method
Moreover, we can also look at the following approximate error graph gained from the above four
methods which illustrate the change of errors from one method to another method.
Figure 11. The graph of NRM Figure 12. The graph of FPM
Figure 13. The graph of SM Figure 14. The graph of BM
Comparing the above graph, we see that Newton-Raphson Method is a better method than others.
5. CONCLUSION
In this research works, we have analyzed and compared the four iterative methods for solving
non-linear equations that assist to present a better performing method. The MATLAB results are
also delivered to check the appropriateness of the best method. With the help of the approximate
error graph, we can say that the Newton’s method is the most robust and capable method for the
sake of solving the nonlinear equation. Besides, Newton’s method gives a lesser number of
iterations concerning the other methods and it shows less processing time. Ultimately, it is
consulted steadily that Newton’s method is the most effective and accurate owing to solving the
nonlinear problems.

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REFERENCES
[1] Richard L. Burden, J.douglas Faires, Annette M. burden, “Numerical Analysis”, Tenth Edition, ISBN:
978-1-305-25366-7.
[2] R. Vasishtha, Vipin Vasishtha, “Numerical Analysis”, Fourth Edition.
[3] S.S. Sastry, “Introduction Methods of Numerical Analysis”, Fourth Edition, ISBN: 81-203-2761-6.
[4] Shin Min Kang, Waqas Nazeer, “Improvements in Newton-Rapson Method for Nonlinear equations
using Modified Adomain Decomposition Method”, International Journal of Mathematical Analysis,
Vol. 9, 2015, no. 39, 1919.1928.
[5] Muhammad Aslam Noor, Khalida Inayat Noor, Eisa Al-Said and Muhammad Waseem, “Some New
Iterative methods for Nonlinear Equations, Hindawi Publishing Corporation.
[6] Okorie Charity Ebelechukwu, ben Obakpo Johnson, Ali Inalegwu Michael, akuji Terhemba fideli,
“Comparison of Some Iterative methods of Solving Nonlinear Equations”, International Journal of
Theroretical and Applied Mathematics.
[7] Ogbereyivwe Oghovese, Emunefe O. John, “New Three-Steps Iterative Method for Solving Nonlinear
Equations, IOSR Journal of Mathematics.
[8] Masoud Allame, Nafiseh azad, “A new method for solving nonlinear equations by Taylor expansion”,
Application of Mathematics and Computer engineering.
[9] Jishe Feng, “A New Two-step Method for solving Nonlinear equations”, International Journal of
Nonlinear Science”, Vol.8(2009) No.1pp.40-44.
[10] Masoud Allame and Nafiseh Azad, “On Modified Newton Method for solving a Nonlinear Algebraic
equation by Mid-Point”, World applied science Journal 17(12): 1546-1548, 2012.
[11] Mohammad Hani Almomani, Mahmoud Hasan Alrefaei, Shahd Al Mansour, “A method for selecting
the best performance systems”, International Journal of Pure and Applied Mathematics.
[12] Zolt ´an Kov´acs, “Understanding convergence and stability of the Newton-Raphson method”,
https://www.researchgate.net/publication/277475242.
[13] Fernando Brambila Paz, Anthony Torres Hernandez, Ursula Iturrarán-Viveros, Reyna Caballero Cruz,
“Fractional Newton-Raphson Method Accelerated with Aitken’s Method”.
[14] Robin Kumar and Vipan, “Comparative Analysis of Convergence of Various Numerical Methods”,
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[15] Nancy Velasco, Dario Mendoza, Vicente Hallo, Elizabeth Salazar-Jácome and Victor Chimarro,
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[16] Kamoh Nathaniel Mahwash, Gyemang Dauda Gyang, “Numerical Solution of Nonlinear Systems of
Algebraic Equations”, International Journal of Data Science and Analysis2018; 4(1): 20-23
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[17] Farooq Ahmed Shah, and Muhammad Aslam Noor, “Variational Iteration Technique and Some
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[18] M. A. Hafiz & Mohamed S. M. Bahgat, “An Efficient Two-step Iterative Method for Solving System
of Nonlinear Equations”, Journal of Mathematics Research; Vol. 4, No. 4; 2012 ISSN 1916-9795 E-
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A NEW STUDY TO FIND OUT THE BEST COMPUTATIONAL METHOD FOR SOLVING THE NONLINEAR EQUATION

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