Iterative Methods for Nonlinear Equations or Systems and Their Applications
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Tianbao Liu, Hengyan Li, "Some New Variants of Cauchy's Methods for Solving Nonlinear Equations", Journal of Applied Mathematics, vol. 2012, Article ID 927450, 13 pages, 2012. https://doi.org/10.1155/2012/927450
Some New Variants of Cauchy's Methods for Solving Nonlinear Equations
Abstract
We present and analyze some variants of Cauchy's methods free from second derivative for obtaining simple roots of nonlinear equations. The convergence analysis of the methods is discussed. It is established that the methods have convergence order three. Per iteration the new methods require two function and one first derivative evaluations. Numerical examples show that the new methods are comparable with the wellknown existing methods and give better numerical results in many aspects.
1. Introduction
In this paper, we consider iterative methods to find a simple root ; that is, and , of a nonlinear equation where for an open interval is a scalar function.
Finding the simple roots of the nonlinear equation (1.1) is one of the most important problems in numerical analysis of science and engineering, and iterative methods are usually used to approximate a solution of these equations. We know that Newton’s method is an important and basic approach for solving nonlinear equations [1, 2], and its formulation is given by this method converges quadratically.
The classical Cauchy’s methods [2] are expressed as where This family of methods given by (1.3) is a wellknown thirdorder method. However, the methods depend on the second derivatives in computing process, and therefore their practical applications are restricted rigorously. In the recent years, several methods with free second derivatives have been developed; see [3–9] and references therein.
In this paper, we will improve the family defined by (1.3) and obtain a threeparameter family of secondderivativefree variants of Cauchy’s methods. The rest of the paper is organized as follows. In Section 2, we describe new variants of Cauchy’s methods and analyze the order of convergence. In Section 3, we obtain some different iterative methods by taking several parameters. In Section 4, different numerical tests confirm the theoretical results, and the new methods are comparable with other known methods and give better results in many cases. Finally, we infer some conclusions.
2. Development of Methods and Convergence Analysis
Consider approximating the equation around the point by the quadratic equation in and in the following form: We impose the tangency conditions where is th iterate and By using the tangency conditions from (2.2), we obtain the value of is determined in terms of in the following: From (2.1), we have Substituting (2.4) into (2.5) yields Using (2.6) we can approximate We define Using instead of , we obtain a new threeparameter family of methods free from second derivative where . Similar to the classical Cauchy’s method, a square root is required in (2.9). However, this may cost expensively, even fail in the case . In order to avoid the calculation of the square roots, we will derive some forms free from square roots by the Taylor approximation [10].
It is easy to know that the Taylor approximation of is Using (2.10) in (2.9), we can obtain the following form: where , .
On the other hand, it is clear that Then, Using (2.12) in (2.9), we also can construct a new family of iterative methods as follows: where , .
We have the convergence analysis of the methods by (2.13).
Theorem 2.1. Let be a simple zero of sufficiently differentiable function for an open interval . If is sufficiently close to , for , the methods defined by (2.13) are at least cubically convergent; as particular cases, if or the methods have convergence order four.
Proof. Let ; we use the following Taylor expansions:
where . Furthermore, we have
Dividing (2.14) by (2.15),
From (2.16), we get
Expanding , in Taylor’s series about and using (2.17), we get
From (2.14) and (2.18), we have
Because of (2.15), we obtain
From (2.20) and (2.21), we have
Because of (2.15) and (2.20), we get
From (2.20), (2.22), and (2.23), we also easily obtain
Because of (2.14) and (2.18), we get
Furthermore, we have
Because of (2.21) and (2.27), we have
From (2.14) and (2.15), we also easily have
By a simple manipulation with (2.26) and (2.29), we obtain
Substituting (2.25), (2.29), and (2.30) in the denominator of , we obtain
Dividing (2.24) by (2.31) gives us
where
Since
If we consider , from (2.12) we obtain
Because of (2.13), we have
From , we have
This means that if , the methods defined by (2.13) are at least of order three for any . Furthermore, we consider that if
then the methods defined by (2.9) are shown to converge of the order four. From (2.37) and (2.38), it is obvious that the methods defined by (2.13) are of order four by taking .
If considering , we from (2.32) have
From (2.13), we can obtain
From (2.40) and , we have
This means that the methods defined by (2.13) are at least of order three for any . Furthermore, we consider that if
then the methods defined by (2.13) are shown to converge of the order four. From (2.41) and (2.42), it is obvious that the methods defined by (2.13) are of order four by taking .
Similar to the proof of Theorem 2.1, we can prove that for , the methods defined by (2.9) and (2.11) are at least cubically convergent; as particular cases, if , , or , , the methods have convergence order four.
3. Some Special Cases
(1^{0}) If , from (2.8) we obtain For , we obtain from (2.13) a thirdorder method (LM1) For , we obtain from (2.13) a thirdorder method (LM2) (2^{0}) If , , from (2.8) we obtain For , we obtain from (2.13) a fourthorder method [11] For , we obtain from (2.13) a thirdorder method (LM3) (3^{0}) If , from (2.8) we obtain For , we obtain from (2.13) a thirdorder method (LM4) For , we also obtain the fourthorder method which was obtained in [10].
For , we obtain a fourthorder method as follows [11]: For , we obtain the new fourthorder method (4^{0}) If , , for , we obtain a fourthorder method from (2.11) and (3.4) For , we obtain from (2.11) a thirdorder method (5^{0}) If , , from (2.11) we obtain some iterative methods as follows:
For , we obtain a thirdorder method (LM5) where is defined by (3.7).
For , we obtain a fourthorder method (LM6) For , we obtain the fourthorder method as follows [10]:
4. Numerical Examples
In this section, firstly, we present some numerical test results about the number of iterations () for some cubically convergent iterative methods in Table 1. The following methods were compared: Newton’s method (NM), the method of Weerakoon and Fernando [12] (WF), Halley’s method (HM), Chebyshev’s method (CHM), SuperHalley’s method (SHM), and our new methods (3.2) (LM1), (3.3) (LM2), (3.6) (LM3), (3.8) (LM4), and (3.14) (LM5).

Secondly, we employ our new fourthorder methods defined by (3.15) (LM6) and the super cubic convergence method by (3.2) (LM1), to solve some nonlinear equations and compare them with Newton’s method (NM), Newtonsecant method [13] (NSM), and Ostrowski’s method [14] (OM). Displayed in Table 2 are the number of iterations () and the number of function evaluations (NFEs) counted as the sum of the number of evaluations of the function itself plus the number of evaluations of the derivative.

All computations were done using Matlab7.1. We accept an approximate solution rather than the exact root, depending on the precision of the computer. We use the following stopping criteria for computer programs: , we used the fixed stopping criterion . In table, “−” is divergence.
We used the following test functions and display the computed approximate zero [15]:
5. Conclusions
In this paper, we presented some variants of Cauchy’s methods free from second derivative for solving nonlinear equations. Per iteration the methods require twofunction and one firstderivative evaluations. These methods are at least threeorder convergence, if , , or , , the methods have convergence order four, respectively, and if , , the method has super cubic convergence. We observed from numerical examples that the proposed methods are efficient and demonstrate equal or better performance as compared with other wellknown methods.
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Copyright
Copyright © 2012 Tianbao Liu and Hengyan Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.