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On new sixth and seventh order iterative methods for solving nonlinear equations using homotopy perturbation technique
BMC Research Notes volumeÂ 15, ArticleÂ number:Â 267 (2022)
Abstract
Objectives
This paper proposes three iterative methods of order three, six and seven respectively for solving nonlinear equations using the modified homotopy perturbation technique coupled with system of equations. This paper also discusses the analysis of convergence of the proposed iterative methods.
Results
Several numerical examples are presented to illustrate and validation of the proposed methods. Implementation of the proposed methods in Maple is discussed with sample computations.
Introduction
The applications of nonlinear equations of the type \(f(x)=0\) arise in various branches of pure and applied sciences, engineering and computing. In resent time, several scientists and engineers have been focused to solve nonlinear equations numerically as well as analytically. In the literature, there are several iterative methods/algorithms available which are derived from different techniques such as homotopy, interpolation, Taylorâ€™s series, quadrature formulas, decomposition etc., and also available various modifications and improvements of the existing methods, and different hybrid iterative methods, see, for exampleÂ [1, 4,5,6,7, 9,10,11,12,13,14,15,16, 28,29,30,31,32, 36,37,38]. In general, the roots of nonlinear or transcendental equations cannot be expressed in closed form or cannot be computed analytically. The rootfinding algorithms provide us to compute approximations to the roots, these approximations are expressed either as small isolating intervals or as floating point numbers. In this paper, we use the modified homotopy perturbation technique (HPT) to create a number of iterative methods for solving the given nonlinear equations with converging order more than or equal to three. The given nonlinear equations are expressed as an equivalent coupled system of equations with help of the Taylorâ€™s series and technique of HeÂ [4]. This enables us to express the given nonlinear equation as a sum of linear and nonlinear equations. The Maple implementation of the proposed algorithm is also discussed, and various Maple implementations for differential and transcendental equations are available in the literature, see, for exampleÂ [17,18,19,20,21,22,23,24,25,26,27].
The rest of paper is organized as follows: SectionÂ recalls the preliminary concepts related to the topic; In SectionÂ , we present the methodology and steps involving in the proposed algorithms; SectionÂ discuses the analysis of convergence to show the order of proposed methods are more than or equal to three; SectionÂ presents several numerical examples to illustrate and validate the proposed methods/algorithms; and finally SectionÂ presents the Maple implementation of the proposed algorithms with sample computations.
Preliminaries
In this paper, we consider the nonlinear equation of the type
Iterations techniques are a common approach widely used in various numerical algorithms/methods. It is a hope that an iteration in the general form of \(x_{n+1}=g(x_n)\) will eventually converge to the true solution \(\alpha\) of the problemÂ (1) at the limit when \(n\rightarrow \infty\). The concern is whether this iteration will converge, and, if so, the rate of convergence. Specifically we use the following expression to represent how quickly the error \(e_n=\alpha x_n\) converges to zero. Let \(e_n=\alpha x_n\) and \(e_{n+1} = \alpha x_{n+1}\) for \(n \ge 0\) be the errors at nth and \((n+1)\)th iterations respectively. If two positive constants \(\mu\) and p exist, and
then the sequence is said to converge to \(\alpha\). Here \(p\ge 1\) is called the order of convergence, the constant \(\mu\) is the rate of convergence or asymptotic error constant. This expression may be better understood when it is interpreted as \(\vert e_{n+1}\vert =\mu \vert e_n\vert ^p\) when \(n\rightarrow \infty\). Obviously, the larger p and the smaller \(\mu\), the more quickly the sequence converges.
Theorem 1
[3] Suppose that \(\phi \in C^p[a,b]\). If \(\phi ^{(k)}(x)=0\), for \(k=0, 1, 2, \ldots , p1\) and \(\phi ^{(p)}(x) \ne 0\), then the sequence \(\{x_n\}\) is of order p.
This paper focus on developing iterative methods/algorithms that are having the order of converges three, six and seven respectively. The following section presents the proposed methods using Taylorâ€™s series and modified HPT.
Main text
In this section, we present new iterative methods and its order of convergences with numerical examples, maple implementation and sample computations using maple mathematical software tool.
New iterative methods
We assume that \(\alpha\) is an exact root of the equationÂ (1) and let a be an initial approximation (sufficiently close) to \(\alpha\). We can rewrite the nonlinear equationÂ (1) using Taylorâ€™s series expansion as coupled system
We have, from Newtonâ€™s method, that
We can writeÂ (6) in the following form
It can be expressed in the form of
where
Here T(x) is a nonlinear operator. It is clear, from relationÂ (4), that
Note that the equationÂ (10) will play important role in the derivation of the iteration methods, see for exampleÂ [2]. We use the technique of homotopy perturbation to develop the proposed iterative algorithms to solve the given nonlinear equationÂ (1). Using the HPT, we can construct a homotopy \(H(\upsilon ,p,m) : \mathbb {R} \times [0,1] \times \mathbb {R} \rightarrow \mathbb {R}\) satisfying
where \(p\in [0,1]\) is embedding parameter and \(m \in \mathbb {R}\) is unknown number. Clearly, fromÂ (11), we have
Hence, the parameter p is monotonically increases on [0,Â 1]. The solution of equationÂ (11) can be expressed as a power series in p
Now the approximate solution ofÂ (1) is
One can express the equationÂ (11), as follows, by expanding T(x) using Taylorâ€™s series expansion around \(x_0\),
By PuttingÂ (12) inÂ (14), we get
By comparing the coefficients of powers of p, we get
FromÂ (17), we have \(x_1 = T(x_0) + m\). To obtain the value of m, assume \(x_2=0\). Now fromÂ (18)
Now, \(x_0,x_1,x_2,x_3,\ldots\) are obtained as follows. FromÂ (16), we have
From the assumption \(x_2=0\) and fromÂ (19), we get
FromÂ (6),Â (10) andÂ (9), we have
The approximate solution is obtained as
This formulation allows us to form the following iterative methods.
Algorithm 1
For \(i=0\), we have
Hence, for a given \(x_0\), we have the following iterative formula to find the approximate solution \(x_{n+1}\).
Algorithm 2
For \(i=1\), we have
Hence, for a given \(x_0\), we have the following iterative schemes to find the approximate solution \(x_{n+1}\).
Note: Since \(x_2=0\), we have the formulaÂ (29) for \(i=2\). i.e., \(x \approx x_0 + x_1 = x_0 + x_1 + x_2\).
Algorithm 3
For \(i=3\), we have
Hence, for a given \(x_0\), we have the following iterative formula to find the approximate solution \(x_{n+1}\).
Order of convergence
In this section, we show, in the following theorems, that the orders of converges of AlgorithmsÂ 1,Â 2 and 3 are three, six and seven respectively. Let \(I \subset \mathbb {R}\) be an open interval. To prove this, we follow the proofs ofÂ [9, TheoremÂ 5, TheoremÂ 6].
Theorem 2
Let \(f:I \rightarrow \mathbb {R}\). Suppose \(\alpha \in I\) is a simple root ofÂ (1) and \(\theta\) is a sufficiently small neighborhood of \(\alpha\). Let \(f''(x)\) exist and \(f'(x) \ne 0\) in \(\theta\). Then the iterative formulaÂ (28) given in AlgorithmÂ 1 produces a sequence of iterations \(\{x_n:n=0,1,2,\ldots \}\) with order of convergence three.
Proof
Let
Since \(\alpha\) is a root of f(x), hence \(f(\alpha ) = 0\). One can compute that
Hence the AlgorithmÂ 1 has third order convergence, by TheoremÂ 1. \(\square\)
One can also verify that the order of convergence of AlgorithmÂ 1 as in the following example.
Example 1
Consider the following equation. It has a root \(\alpha =\sqrt{30}\). We show, as discussed in proof of TheoremÂ 2, that the AlgorithmÂ 1 has third order convergence.
Following TheoremÂ 2, we have
Now
Hence, by TheoremÂ 2, the AlgorithmÂ 1 has third order convergence.
Theorem 3
Let \(f:I \rightarrow \mathbb {R}\). Suppose \(\alpha \in I\) is a simple root ofÂ (1) and \(\theta\) is a sufficiently small neighborhood of \(\alpha\). Let \(f''(x)\) exist and \(f'(x) \ne 0\) in \(\theta\). Then the iterative formulaÂ (29) given in AlgorithmÂ 2 produces a sequence of iterations \(\{x_n:n=0,1,2,\ldots \}\) with order of convergence six.
Proof
Let
Since \(\alpha\) is a root of f(x), hence \(f(\alpha ) = 0\). One can compute that
Hence the AlgorithmÂ 2 has sixth order convergence, by TheoremÂ 1. \(\square\)
We can also verify the order of convergence of AlgorithmÂ 2 as in the following example.
Example 2
Consider the equationÂ (31). Using TheoremÂ 3, similar to ExampleÂ 1, we have
Now, we can check that
Hence, by TheoremÂ 3, the AlgorithmÂ 2 has sixth order convergence.
Theorem 4
Let \(f:I \rightarrow \mathbb {R}\). Suppose \(\alpha \in I\) is a simple root ofÂ (1) and \(\theta\) is a sufficiently small neighborhood of \(\alpha\). Let \(f''(x)\) exist and \(f'(x) \ne 0\) in \(\theta\). Then the iterative formulaÂ (30) given in AlgorithmÂ 3 produces a sequence of iterations \(\{x_n:n=0,1,2,\ldots \}\) with order of convergence seven.
Proof
Let
Since \(\alpha\) is a root of f(x), hence \(f(\alpha ) = 0\). One can compute that
Hence the AlgorithmÂ 3 has seventh order convergence, by TheoremÂ 1. \(\square\)
Again, one can verify the order of convergence of AlgorithmÂ 3 using the following example.
Example 3
Consider the equationÂ (31). Following TheoremÂ 4, similar to ExampleÂ 1 and ExampleÂ 2, we have
Now, we can check that
Hence, by TheoremÂ 4, the AlgorithmÂ 3 has seventh order convergence.
Numerical example
This section presents several numerical examples to illustrate the proposed algorithms, and comparisons are made to confirm that the proposed algorithms give solution faster than existing methods.
Example 4
Consider a nonlinear equation
Suppose the initial approximation is \(x_0 = 2\) with tolerance error \(10^{10}\) correct to ten decimal places. Following the proposed algorithms (in equationsÂ 28,Â 29 andÂ 30), we have
Iteration1 using AlgorithmÂ 1:
Iteration2 using AlgorithmÂ 1:
Now,
Iteration3 using AlgorithmÂ 1:
Now,
Similarly, the Iteration4 using AlgorithmÂ 1 is \(x_4 = 0.2575302855\). One can observe that Iteration3 and Iteration4 are same up to ten decimal places and also the tolerance error is \(10^{10}\). Hence the required approximate root of the given equationÂ (32) is 0.2575302855.
Now, we compute the iterations using AlgorithmÂ 2 as follows.
Iteration1 using AlgorithmÂ 2:
Iteration2 using AlgorithmÂ 2:
Similarly, the Iteration3 using AlgorithmÂ 2 is \(x_3 = 0.2575302853\). One can observe that Iteration2 and Iteration3 are same up to ten decimal places and also the tolerance error is \(10^{10}\).
Now, the iterations using AlgorithmÂ 3 are as follows.
Iteration1 using AlgorithmÂ 3:
Iteration2 using AlgorithmÂ 3:
Example 5
Consider the following equations with corresponding initial approximations to compare results of the proposed three methods with other existing methods. We take tolerance error \(10^{15}\) with correct to 15 decimal places.
 (a):

\(\cos x  x =0\) with initial approximation \(x_0=1.7\),
 (b):

\(xe^{x}  0.1 =0\) with initial approximation \(x_0=0.1\),
 (c):

\(\sin ^2 x  x^2 + 1 =0\) with initial approximation \(x_0=1\),
 (d):

\(xe^{\sin x}+1 =0\) with initial approximation \(x_0=4\),
 (e):

\(x^310 =0\) with initial approximation \(x_0=1.5\).
TableÂ 1 gives a comparison of iterations number with different methods. In the table, ER, NR, NM, A1, A2, A3 and DIV indicate Exact Root, NewtonRaphson method, Noor MethodÂ [2], AlgorithmsÂ 1, Â 2 and Â 3 and diverges respectively.
From TableÂ 1, it is clear that the numerical results show that the proposed methods are more efficient than other existing methods.
Mapleimplementation
In this section, we present implementation of the proposed AlgorithmsÂ 1, Â 2 and Â 3 in Maple. Various maple implementations for differential and transcendental equations are available, see, for exampleÂ [17,18,19,20,21,22,23,24,25,26,27, 35]. One can also implement the proposed algorithms in Microsoft Excel similar to the implementation of existing algorithms inÂ [33, 34].
Pseudo code
Input: Given f(x); initial approximation x[0]; tolerance \(\epsilon\); correct to decimal places \(\delta\); maximum number of iterations n.
Output: Approximate solution

I.
for i from 0 to n do
Maple code
We present the maple code of the proposed algorithms as follows, and sample computations presented in Section.
AlgorithmÂ 1 in Maple
AlgorithmÂ 2 in Maple
AlgorithmÂ 3 in Maple
Sample computations
Consider the following function for sample computations using the Maple implementation.
with initial approximation \(x[0]=3.5\), tolerance \(\epsilon = 10^{5}\), correct to decimal places \(\delta =10^{10}\) (i.e., up to 10 decimal places);, and maximum number of iterations \(n=10\).
AlgorithmÂ 1 sample computations using Maple
AlgorithmÂ 2 sample computations using Maple
Similarly, one can apply the AlgorithmÂ 3 using Maple code.
Conclusion
In this paper, we presented three iterative methods of order three, six and seven respectively for solving nonlinear equations. With the help of modified homotopy perturbation technique, we obtained coupled system of equations which gives solution faster than existing methods. The analysis of convergence of the proposed iterative methods are discussed with example for each proposed method. Maple implementations of the proposed methods are discussed with sample sample computations. Numerical examples are presented to illustrate and validation of the proposed methods.
Limitations
The proposed algorithms are implemented in Maple only. However, we can also implement these algorithms in Mathematica, SCILab, Matlab, Microsoft Excel etc.
Availability of data and materials
The datasets generated and analyzed during the current study are presented in this manuscript.
Abbreviations
 HPT:

Homotopy Perturbation Technique
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The authors are thankful to the reviewers and editor for providing valuable inputs to improve the quality and present format of manuscript.
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ST involved in creation of the proposed algorithms for solving nonlinear equations using the modified HPT, the convergence analysis, and Maple implementation. PS is involved in suggestion and verification of the numerical examples in the present paper. Both authors read and approved the final manuscript.
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Prof. Srinivasarao Thota completed his M.Sc. in Mathematics from Indian Institute of Technology (IIT) Madras, India and Ph.D. in Mathematics from Motilal Nehru National institute of Technology (NIT) Allahabad, India. Prof. Thotaâ€™s area of research interests are Computer Algebra (symbolic methods for differential equations), Numerical Analysis (root finding algorithms), Mathematical Modeling (ecology). He has published more than 40 research papers in various international journals and presented his research work at several international conferences as oral presenter and invited/keynote/guest speaker in different countries. Presently working at department of mathematics, SR University, Warangal, India.
Prof. P. Shanmugasundaram completed M.Sc. in Mathematics from Sri Vasavi College, M.Phill in Mathematics from Madurai Kamraj University, Madurai, India and Ph.D. in Mathematics Anna University, Chennai, India. He has published more than 25 research papers in various international journals. Presently, working at Department of Mathematics, College of Natural & Computational sciences, Mizanâ€“Tepi University, Ethiopia.
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Thota, S., Shanmugasundaram, P. On new sixth and seventh order iterative methods for solving nonlinear equations using homotopy perturbation technique. BMC Res Notes 15, 267 (2022). https://doi.org/10.1186/s13104022061545
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DOI: https://doi.org/10.1186/s13104022061545
Keywords
 Iterative methods
 Nonlinear equations
 Order of convergence
 Homotopy perturbation technique
Mathematics subject classification
 65N30
 49M37