COMPARATIVE STUDY OF A NEW ITERATIVE METHOD WITH THAT OF
NEWTONS METHOD FOR SOLVING ALGEBRAIC AND TRANSCEDENTAL
EQUATIONS
AZIZUL HASAN1 , NAJMUDDIN AHMAD2
1 Department
of Mathematics, Jazan University , Jazan, KSA.
2 Department of Mathematics, Integral University, lucknow, India.
Abstract: The aim of this paper is to construct an efficient iterative method to solve
non linear equations. One new iterative method for solving algebraic and
transcendental equations is presented using a Taylor series formula . using the , the
Newton’s method and the an Improve iterative method and the result compared. It
was observed that the Newton method required more number of iteration in
comparison to improve iterative method. By the use of numerical experiments to
show that this method are more efficient than Newton – Raphson method.
Keywords: Newton’s method, Improve iterative method, Algebraic and
Transcendental equations, Numerical examples
Introduction: Solving non linear equations is one of the most important and
challenging problems in science and engineering applications.
For Solving nonlinear equations Newton’s method is one of the most pre
dominant problems in numerical analysis [1]. Some historical points on this method
can 𝛼 be found in [17,18 19,20,21, 22].
Recently, some methods have been proposed and analyzed for solving
nonlinear equations [ 2, 3, and 6]. These methods have been suggested by using
quadrature formulas, decomposition and Taylor’s series [4, , 9 and 14]. As we know,
quadrature rules play an important and significant role in the evaluation of
integrals. One of the most well – known iterative methods is Newton’s classical
method which has a quadratic convergence rate. Some authors have derived new
iterative methods which are more efficient than that of Newton’s [10,11,13, and 15].
This paper is organized as follows. Part 1 provides some preliminaries which
are needed. Part 2 is devoted to suggest one new iterative method by using a Taylor
Series expansion upto four terms. These are implicit – type methods. To implement
these methods, we use Newton’s method as a predictor and then use one new
method as a corrector. The resultant methods can be considered as two – step
iterative methods. In Part 3, a comparison between this methods with that of
Newton’s methods. Several examples are given to illustrate the efficiencies and
advantages of these methods.
Definition and Notation:
Let 𝛼 ∈ 𝑅 and 𝑥𝑁 ∈ 𝑅, 𝑁 = 0,1,2,3, … … … Then the sequence 𝑥𝑁 is said to be
convergence to 𝛼 if lim |𝑥𝑁 − 𝛼| = 0. If there exists a constant 𝑐 > 0, an integer
𝑁→∞
𝑁0 ≥ 0 and 𝑝 ≥ 0 such that for all 𝑁 > 𝑁0
we have
|𝑥𝑁+1 − 𝛼| ≤ 𝑐|𝑥𝑁 − 𝛼|𝑝
then 𝑥𝑁 is said to be converges to 𝛼 with convergence order at least p. If 𝑝 = 2, the
convergence is to be quadratic or if 𝑝 = 3 then it is cubic.
Notation: The notation 𝑒𝑛 = 𝑥𝑛 − 𝛼, is the error in the 𝑛𝑡ℎ iteration.
The equation
𝑒𝑛+1 = 𝑐𝑒𝑛𝑝 + 𝑂(𝑒𝑛𝑝+1 ) is called the error equation. By
substituting 𝑒𝑛 = 𝑥𝑛 − 𝛼 for all n in any iterative method and simplifying. We obtain
the error equation for that method. The value of p obtained is called the order of this
method.
NEWTON-RAPHSON METHOD
We consider the problem of numerical determine a real root 𝛼 of non linear
equation
𝑓(𝑥) = 0, 𝑓: 𝐷 ⊂ 𝑅 → 𝑅
(1.1)
The Newton-Raphson method finds the slope (tangent line) of the function at the
current point and uses the zero of the tangent line as the next reference point. The
process is repeated until the root is found [5-7]. The method is probably the most
popular technique for solving nonlinear equation because of its quadratic
convergence rate. But it is sometimes damped if bad initial guesses are used [8-9].It
was suggested however, that Newton’s method should sometimes be started with
Picard iteration to improve the initial guess [9.16]. Newton Raphson method is
much more efficient than the Bisection method. However, it requires the calculation
of the derivative of a function as the reference point which is not always easy or
either the derivative does not exist at all or it cannot be expressed in terms of
elementary function [6,7-8]. Furthermore, the tangent line often shoots wildly and
might occasionally be trapped in a loop [6]. The function, f(x)=0 f can be expanded
in the neighbourhood of the root xk through the Taylor expansion:
(𝑥− x )2
f(x) = f(xk ) + (𝑥 − xk ) 𝑓 ′ (𝑥𝑘 ) + 2! k 𝑓 ′′ (xk )
where x can be seen as a trial value for the root at the nth step and the approximate
value of the next step 𝑥𝑘+1 can be derived from
f(𝑥𝑘+1 ) = f(𝑥𝑘 ) + (𝑥𝑘+1 - 𝑥𝑘 ) 𝑓 ′ (𝑥𝑘 ) =0
The known numerical method for solving non linear equations is the Newton’s
method is given by
𝑓(𝑥 )
𝑘
𝑥𝑘+1 = 𝑥𝑘 − 𝑓′ (𝑘)
,
𝑘 = 0, 1, 2,3, … … …
where 𝑥0 is an initial approximation sufficiently near to 𝛼. The convergence order
of the Newton’s method is quadratic for simple roots [4]. By implication, the
quadratic convergence we mean that the accuracy gets doubled at each iteration.
Algorithm of the Newton- Raphson Method :
Inputs: f(x) –the given function, xo –the initial approximation, 𝜀-the error tolerance
and N –the maximum number of iteration.
Output: An approximation to the root x =𝛾 or a message of a failure. Assumption:
x=𝛾 is a simple root of f (x)=0
 Compute f(x) , and 𝑓 ′ (𝑥)
𝑓(𝑥𝑘 )
 Compute 𝑥𝑘+1 = 𝑥𝑘 − 𝑓′ (𝑘)
,
𝑘 = 0, 1, 2,3, … … … do until convergence or
failure. Test for convergence of failure If |𝑓(𝑥𝑘+1 )| < 𝜀, |𝑥𝑘+1 − 𝑥𝑘 |/𝑥𝑘 < 𝜀
or k>N, stop.
End.
It was remarked in [1], that if none of the above criteria has been satisfied, within a
predetermined, say, N, iteration, then the method has failed after the prescribed
number of iteration. In this case, one could try the method again with a different xo.
Meanwhile, a judicious choice of xo can sometimes be obtained by drawing the
graph of f(x), if possible. However, there does not seems to exist a clear- cut
guideline on how to choose a right starting point, xo that guarantees the
convergence of the Newton-Raphson method to a desire root.
Description of the Improve iterative method: Let us consider the algebraic and
transcendental equation is
f(x) = 0
(1.2)
Let 𝛾 be the correct root of this equation in the open interval I in which the function
f is defined and continuous and the function is also differentiable term by terms.
Following the basic assumptions and Abbasbandy and maheshwari (1, 11) and also
see (7,13) we also taking Taylor series expansion of f(x) up to four terms
f(x) = f(xk ) + (𝑥 − xk ) 𝑓 ′ (𝑥𝑘 ) +
(𝑥 − xk )3
3!
𝑓 ′′′ (xk )
(𝑥− xk )2
2!
𝑓 ′′ (xk ) +
(1.3)
where xk is the k-th approximations to the root of the given equation (1.2)
since 𝛾 is the correct root of equation ( 1.3) , so
(𝛾 − xk )2 ′′
f(γ) = f(xk ) + (𝛾 − xk ) 𝑓 ′ (𝑥𝑘 ) +
𝑓 (xk )
2!
(𝛾 − xk )3 ′′′
+
𝑓 (xk )
(1.4)
3!
and f(γ) = 0
0 = f(xk ) + (𝛾 − xk )
𝑓 ′ (𝑥𝑘
(𝛾 − xk )2 ′′
)+
𝑓 (xk )
2!
(𝛾 − xk )3 ′′′
+
𝑓 (xk )
(1.5)
3!
Taking four terms of the equation (1.5) ,the value of the root of equation (1.2) can be
Obtained if let 𝛾 = 𝑥𝑘+1
0 = f(xk ) + (𝑥𝑘+1 − xk ) 𝑓 ′ (𝑥𝑘 ) +
(𝑥𝑘+1 −xk )2
2!
𝑓 ′′ (xk ) +
(𝑥𝑘+1 − xk )3
3!
𝑓 ′′′ (xk )
(1.6)
xk+1 = f(xk ) + (𝑥𝑘+1 − xk )𝑓 ′ (𝑥𝑘 ) +
(𝑥𝑘+1 −xk )2
2!
𝑓 ′′ (xk ) +
(𝑥𝑘+1 − xk )3
3!
𝑓 ′′′ (xk )
(1.7)
Solving right hand side of the above equation. We get the desire root of the equation
upto numerical accuracy of fourteen decimal places.
Algorithm: For a given 𝑥0 , compute the approximate solution 𝑥𝑘+1 by the
following iterative scheme
(𝑥𝑘+1 − xk )2 ′′
(𝑥𝑘+1 − xk )3 ′′′
xk+1 = f(xk ) + (𝑥𝑘+1 − xk )𝑓 ′ (𝑥𝑘 ) +
𝑓 (xk ) +
𝑓 (xk )
2!
3!
where k = 0,1,2, … … … …
Numerical Experiments:
In this section, we employ the methods obtained in this paper to solve some
nonlinear equations and compare them with the Newton’s method (NM). We use the
stopping criteria |𝑥𝑛+1 − 𝑥𝑛 | < 𝜀 and |𝑓(𝑥𝑛+1 )| < 𝜀, where 𝜀 = 10−14, for computer
programs. All programs are written in Matlab.
Comparison between the New Iterative method, Newtons method and number of
Iteration for the various functions.
EXAMPLE 1.
Function
𝑓(𝑥) = 𝑥 3 − 𝑥 + 3,
Methods
Newton method
Present method
Number of iterations
7
1
EXAMPLE 2.
Function
Methods
Newton method
Present method
Number of iterations
5
1
Initial Approximation 𝑥0 = −1
𝑥𝑛
-1.67169988165716
-1.67169988165716
|𝑓(𝑥𝑛 )|
7.105427357601002E-015
7.105427357601002E-015
𝑓(𝑥) = 𝑥 3 + 4𝑥 2 − 10, Initial Approximation 𝑥0 = 1
𝑥𝑛
1.36523001341410
1.36523001341410
|𝑓(𝑥𝑛 )|
5.151434834260726E-014
5.151434834260726E-014
EXAMPLE3.Function
Methods
Newton method
Present method
𝑓(𝑥) = 𝑥 3 − 2𝑥 + 5,
Newton method
Present method
𝑓(𝑥) = 𝑥 4 − 24,
Number of
iterations
5
3
Methods
Newton method
Present method
|𝑓(𝑥𝑛 )|
2.21336383940064
2.21336383940064
-1.421085471520200E-013
-1.421085471520200E-013
Initial Approximation 𝑥0 = 1
𝑥𝑛
5
3
0.77288295914921
0.77288295914921
𝑓(𝑥) = sin 𝑥 − 0.5 𝑥,
Number of iteration
5
4
EXAMPLE7. Function
EXAMPLE8. Function
Number of iterations
Methods
Newton method
6
Present method
2
|𝑓(𝑥𝑛 )|
7.771561172376096E-016
7.771561172376096E-016
Initial Approximation
𝑥𝑛
1.88808675302834
1.88808675302834
𝑓(𝑥) = 𝑥 2 − 5,
|𝑓(𝑥𝑛 )|
-2.775557561562891E-016
-2.775557561562891E-016
Initial Approximation 𝑥0 = 1.6
𝑥𝑛
1.89549426703398
1.89549426703398
𝑓(𝑥) = 𝑥 log 𝑥 − 1.2,
Number of iterations
Methods
Newton method
5
Present method
4
𝑥0 = 2
𝑥𝑛
Number of iterations
EXAMPLE6. Function
|𝑓(𝑥𝑛 )|
-3.819167204710539E-014
7.460698725481052E-014
Initial Approximation
𝑓(𝑥) = 𝑥 3 − 𝑒 −𝑥 ,
EXAMPLE 5. Function
Methods
Newton method
Present method
𝑥𝑛
−2.09455148154233
−2.09455148154232
Number of iterations
8
2
EXAMPLE4. Function
Methods
Initial Approximation 𝑥0 = 2
|𝑓(𝑥𝑛 )|
-5.773159728050814E-015
-5.773159728050814E-015
Initial Approximation
𝑥𝑛
2.23606797749979
2.23606797749978
𝑥0 = 2
𝑥0 = 1
|𝑓(𝑥𝑛 )|
8.881784197001252E-016
-4.352074256530614E-014
EXAMPLE9. Function
𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 2.5, Initial Approximation
Number of iterations
Methods
Newton method
5
Present method
1
𝑥𝑛
2.64178352745293
2.64178352745293
𝑥0 = 2.5
|𝑓(𝑥𝑛 )|
2.131628207280301E-014
2.131628207280301E-014
Conclusion: Based on our results and discussions, we now conclude that the new
method is formally the most effective of the newton methods we have considered
here in the study. requires only a single function evaluation per iteration We derived
one improve iterative method based on Taylor series formula using upto four temrs
for solving nonlinear equations. Analysis of efficiency from the numerical compution
shows that this method are preferable to the well-known Newton’s method. From
numerical examples, we showed that these methods have great practical utilities.
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Corresponding Authors:
Najmuddin Ahmad
Department of Mathematics
Integral University lucknow India.