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Some Iterative Methods for Solving Nonlinear Equations

Rostam K. Saeed*, Karwan H.F.Jwamer, Delan O. Salem

Department of Mathematics, College of Science-Salahaddin University/Erbil, Halwer-Kurdistan Region, Iraq

Abstract: In this paper, three iteration methods are introduced to solve nonlinear equations. The convergence criteria for these methods are also discussed. Several examples are presented and compared to other well-known methods, showing the accuracy and fast convergence of the proposed methods.

Keywords: nonlinear equation, order of convergence, taylor series expansion, iterative methods

Citation: Rostam K. Saeed, Karwan H.F.Jwamer, Delan O. Salem, Some Iterative Methods for Solving Nonlinear Equations. American Journal of Numerical Analysis, 3 (2). 49-51. January 2015. doi:10.12691/ajna-3-2-3

1. Introduction

One of the oldest and most basic problems in mathematics is that of solving an nonlinear equation . This problem has motivated many theoretical developments including the fact that solution formulas do not in general exist.

Thus, the development of algorithms for finding solution has historically been an important enterprise. Newton-Raphson method [11] is the most popular technique for solving nonlinear equations. Many topics related to Newton ̓s method still attract attention from researchers. As is well known, a disadvantage of the method is that the initial approximation , must be chosen sufficiently close to a true solution in order to guarantee their convergence.

Finding a criterion for choosing is quite difficult and difficult and therefore effective and globally convergent algorithms are needed. In recent years, several methods have been developed to solve the nonlinear equation by Newton method and their modifications [2,4-11]. Suppose is a sequence that convergent to with for all If positive constants and exist with

then converges to of order with asymptotic error constant . In general, a sequence with a high order of convergence more rapidly than sequence with a lower order.

In this work, a cubic iterative methods based on Taylor ̓s series expansion are introduced as follows:

Consider a nonlinear equation The Taylor ̓s series expansion around a given initial point assuming being close enough to the simple root , is given as follows:

[picture] | (1)

where HOT denotes the higher order terms. Then the nonlinear equation becomes,

[picture] | (2)

when is close enough to equation (2) can be approximated as

[picture] | (3)

Thus we have

[picture] | (4)

By assuming which yields the one-step iteration method called Newton method [3] with second-order convergence

[picture] | (5)

2. New Iterative Methods

Depending on the relations (1)-(5), a new one-step iteration method can be constructed with third-order convergence. For this reason, we rewrite equation (2) as

[picture] | (6)

Equation (6) can be approximated as

[picture] | (7)

Substituting (4) into the bracket of equation (7),

We obtain

That is

[picture] | (8)

From equation (8),we obtain

[picture] | (9)

Thus we can solve equation (9) by assuming

as

[picture] | (10)

This suggests that following one-step iteration method:

Algorithm 1.1: For a given compute the approximate solution by the one-step iteration scheme:

[picture] | (11)

It will be shown that the proposed method (11) has third order convergence, and this will be done by applying the following Maple13 program:

To derive another iteration method, again, substituting (9) into the bracket of equation (7), we obtain

That is

From the above equation, we get

[picture] | (12)

Thus we can solve equation (12) by assuming

as

[picture] | (13)

This leads to the following new algorithm:

Algorithm 1.2: For given compute the approximate solution by the one-step iteration scheme:

[picture] | (14)

for .

It will be shown that the proposed method (14) has third order convergence, and this will be done by applying the following Maple13 program:

Also, based on relation (4), another new one-step iteration method can be constructed with third-order convergence. For this reason, we rewrite equation (2) as follows:

[picture] | (15)

Equation (15) can be approximate as

[picture] | (16)

Substituting (4) into the brackets of equation (16), we find

That is

[picture] | (17)

Using the above equation, we can suggest the following new one-step iteration method as follows:

Algorithm 1.3: For a given compute the approximate solution by the one-step iteration scheme:

It will be shown that the proposed method (18) has third order convergence, and this will be done by applying the following Maple13 program:

where

3. Numerical Examples

We present some examples to illustrate the efficiency of the new proposed methods in this paper. We compare the Newton’s method (NM), the method of Saeed and Aziz [8] (SA), the methods of Saeed and Khthr [9] (SK),the method of Saeed and Khthr [10] (SKh) and the method proposed in this paper by the algorithms 1.1-1.3. We use the following stopping criteria for computer program:

Displayed in Table 1 is the number of iterations (IT).

Table 1. Comparison of various iterative methods

4. Conclusion

We present a new method with three-order convergence for solving nonlinear equations. Analysis of efficiency and the number of iterations shows that the new algorithm is more efficient and it performs better than classical Newton’s and similar or better that the methods proposed by [8,9,10]. Also, we see Algorithm 1.3 diverge for because is a symmetric function on the interval which contain

References

[1]   | Burden, R. L. and Faires, J. D., Numerical Analysis, 9th edition, Brooks/Cole Publishing Company, 2011.
[2]   | Chun, C., Some fourth-order iterative methods for solving non-linear equations, Appl. Math. Lett., 195, (2008), 454-456.
[3]   | Kincaid, D. and Cheney, W., Numerical Analysis-Mathematics of Scientific Computing, Brooks/Cole Publishing Company, 1991.
[4]   | Kou, J.; Li, Y. and Wang, X., A family of fifth-order iterations composed of Newton and third-order methods, Appl. Math. Lett., 186, (2007), 1258-1262.
[5]   | Ostrowski, A. M., Solution of Equation in Euclidean and Banach Space, 3rd edition, Academic Press, New York, 1973.
[6]   | Porta, F. A. and Pták, V., Nondiscrete induction and iterative processes, Research Notes in Mathematics, Vol. 103, Pitman, Boston, 1994.
[7]   | Saeed, R. K. and Ahmed, S. O., A New Fourth-order Iterative Method for Solving Nonlinear Equations, Zanco, Journal of Pure and Applied Sciences, 21(5), (2010), 1-5.
[8]   | Saeed, R. K. and Aziz, K. M., Iterative methods for solving nonlinear equations by using quadratic spline function, Math. Sci. Lett., 2(1), (2013), 37-43.
[9]   | Saeed and Khthr, F. W., Three new iterative methods for solving nonlinear equations, Australian Journal of Basic and Applied Sciences, 4(6), (2010), 1022-1030.
[10]   | Saeed, R. K. and Khthr, F. W., New Third-order Iterative Method for Solving Nonlinear Equations. Journal of Applied Sciences Research, 7(6), (2011), 916-921.
[11]   | Zhou, X., Modified Chebychev-Halley methods free from second derivative, Appl. Math. Comput., 203, (2008), 824-827.