Research Article  Open Access
Tianbao Liu, Hengyan Li, Zaixiang Pang, "A New Family of Iterative Methods Based on an Exponential Model for Solving Nonlinear Equations", Journal of Applied Mathematics, vol. 2013, Article ID 547438, 12 pages, 2013. https://doi.org/10.1155/2013/547438
A New Family of Iterative Methods Based on an Exponential Model for Solving Nonlinear Equations
Abstract
We present two new families of iterative methods for obtaining simple roots of nonlinear equations. The first family is developed by fitting the model to the function and its derivative , at a point . In order to remove the second derivative of the first methods, we construct the second family of iterative methods by approximating the equation around the point by the quadratic equation. Analysis of convergence shows that the new methods have thirdorder or higher convergence. Numerical experiments show that new iterative methods are effective and comparable to those of the wellknown existing methods.
1. Introduction
In this paper, we consider iterative methods to find a simple root, that is,and, of a nonlinear equation: wherefor an open intervalis a scalar function.
Many of the complex problems in science and engineering contains the function of nonlinear and transcendental nature in (1), so finding the simple roots of the nonlinear equation is one of the most important problems in numerical analysis. Numerical iterative methods are often used to obtain the approximate solution of such problems because it is not always possible to obtain its exact solution by usual algebraic process. We all know that Newton’s method is an important and basic approach for solving nonlinear equations [1, 2], its formulation is given by and this method converges quadratically. Earlier, many investigations [3–21] have been made to explain the root of nonlinear algebraic and transcendental equations.
The outline of the paper is as follows. In Section 2, we firstly describe a oneparameter family of thirdorder methods by fitting the modelto the functionand its derivative,at a point, and then we use a quadratic equation for approximating the equationto obtain a fourparameter family of secondderivativefree iterative methods. 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 the exponential model: where,,, andare parameters. We construct a new iteration scheme by fitting model (3) to the function and its derivative andat a point. Imposing the conditions as follows at the point (,) and then solving (3) and (4) for,, and , we have From (3), (4), and (5), we take the values of,, andinto (3) and give At the root estimate, it follows that. We consider reducing (8) as follows Thus from (9) we develop the iteration formula: where The square root is required in (10); however, this may cost expensively and 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 Taylor approximation [5].
It is easy to know that Taylor approximation ofis
Using (12) in (10), we can obtain the following form:
We have the convergence analysis of the methods by (13).
Theorem 1. Letbe a simple zero of sufficiently differentiable functionfor an open interval. Ifis sufficiently close to, for, the methods defined by (13) are cubically convergent.
Proof. Let, and we use the following Taylor expansions: where; furthermore, we have Since (15), we obtain Since (14), we get From (14) and (15), we also easily have We obtain the following expression by taking into account (14) and (16): From (18), (19), and (20), we obtain From (17) and (21), we have From (14) and (15), we have Using (14), (15), and (16), we have We know that From (15) and (25), we obtain From (23) and (26), we have Using (13), (24), and (27), we have From, we have which completes the proof.
The family methods given by (13) are novel thirdorder methods, but the methods depend on the second derivatives in computing process, and therefore their practical applications are restricted in some cases. In recent years, several methods with free second derivatives have been developed; see [4–15] and references therein.
In order to avoid the calculation of the second derivatives, we consider approximating the equation around the pointby the quadratic equation inandin the following form [8]: where, , are parameters. We impose the tangency conditions whereisth iterate and From (30) and (31), we have where.
From (33) we can approximate Usinginstead of(11), we obtain a new fourparameter family of methods free from second derivative: where (), ,.
We also have the convergence analysis of the methods by (35).
Theorem 2. Letbe a simple zero of sufficiently differentiable functionfor an open interval. Ifis sufficiently close to, for, (), , the methods defined by (35) are at least cubically convergent; as particular cases, if, , and the methods have convergence order four.
Proof. Let, and we use the following Taylor expansions:
where; furthermore, we have
Dividing (36) by (38)
From (39), we get
Expandingin Taylor’s Series aboutand using (40), we get
From (36) and (41), we have
From (38), we obtain
From (38), (41), and (43) we also easily obtain
Substituting (36), (37), (38), (41), and (43) in the denominator of, we obtain
Using (44) and (45), we have
From (36), (37), and (46), we have
Since (43) and (47), we get
Using (48), we write
Using (36), (38), and (46), we obtain
Taking into account (36), (38), (49), and (50), we finally obtain
Taking into account the last expression (51) and, we have
This means that the methods defined by (35) are at least of order three for any (), . Furthermore, we consider that if , , then the methods defined by (35) are shown to converge the order four.
3. Some Special Cases
From (33)–(35), we have where. From (33) we can approximate Let where.:If and , from (55), we obtain a thirdorder method (LM1): :If, , and , from (55) we also obtain a thirdorder method (LM2): :If, , , and , from (55) we obtain a new thirdorder method (LM3): :If, , , and , from (55) we obtain a thirdorder method (LM4): :If , , , and , from (55) we obtain a new thirdorder method (LM5):
4. Numerical Examples
In this section, some numerical examples commonly used in the literature are presented in Table 1 to check the effectiveness of the new methods. The following methods were compared: Newton method (NM), the method of Weerakoon and Fernando [10] (WF), the method of Potra and Pták (PP) [11], Chebyshev’s method (CHM) [12, 13], Halley’s method (HM) [12], and our new methods (56) (LM1), (57) (LM2), (58) (LM3), (59) (LM4), and (60) (LM5). Displayed in Table 1 are the number of iterations (IT), the number of function evaluations (NFE) counted as the sum of the number of evaluations of the function itself plus the number of evaluations of the derivative, the value, the computing time (TIME, the unit of time is one second), and the distance of two consecutive approximations. All computations were done using Matlab 7.1 environment with a ADM athlon (tm) II X2 2503.01 GHz based PC. We accept an approximate solution rather than the exact root, depending on the precisionof the computer. We use the following stopping criteria for computer programs:, we used the fixed stopping criterion.
