Logarithmic mean Newton’s method for simple and multiple roots of Nonlinear Equations

K. L. Verma

NSCBM Govt. Post Grdauate College, Hamirpur (HP) INDIA

*Corresponding Author E-mail: klverma@netscape.net

ABSTRACT:

In this paper the convergence behavior of a variant of Newton’s method based on the Logarithmic mean for solving nonlinear equations is proposed. The convergence properties of this method for solving equations which have simple or multiple roots have been discussed and it is shown that it converges cubically to simple roots and linearly to multiple roots. Moreover, the values of the corresponding asymptotic error constants of convergence are determined. Theoretical results have been verified on the relevant numerical problems. A comparison of the efficiency of this method with other mean-based Newton’s methods, based on the others means, is also included.

KEYWORDS: Newton’s method; Iterative methods; Mean-based Newton’s method; Logarithmic mean: Order of convergence; Asymptotic error constant

1. INTRODUCTION:

Solving nonlinear equations is one of the most important challenges and interesting task in applied mathematics, numerical analysis, engineering sciences, optimization theory, control theory, economic models and other related disciplines. Finding the exact solution of the nonlinear equation is unlikely, hence the devise of iterative formulae for solving non linear transcendental equations of the type

(1.1)

To find the exact solution of such nonlinear equation (1.1) is very complicated and mostly unworkable, therefore to overcome this difficulty numerical methods have been developed for finding the numerical solutions such transcendental nonlinear equation. D´ıez, [1] proved the convergence properties of the Newton method and performs an accurate analysis of the secant method in front of multiple roots. Weerakoon and Fernando [2] derived trapezoidal Newton's or arithmetic mean Newton's (AN) method using the trapezoid rule. Hasanov et al [3] modified Newton’s method by using Simpson’s rule and obtain a third order convergent method. Frontini and Sormani [4] made modifications in the Newton’s method to derive the iterative method with efficiency index of 1.442 and the order of convergence of three. With the same efficiency index, Özban [5] also derived new variants of Newton’s method, and Chen [6] described some new iterative formulae having third order convergence. The algorithm used in classical Newton’s iterative method

CONCLUSIONS:

All numerical tests agree with the theoretically result of this paper. The most important characteristics of Logarithm Newton's method (LNM) are: Third order of convergence (for simple roots), does not require the computation of second or higher order derivatives, by the numerical results (Table 1.) it is evident that the total number of functional evaluation required is less than of Newton's method. It is interesting to consider the behavior of tested methods for multiple roots. The test function (f) has a multiple roots and the COC is linear. This is in accordance with the theoretically properties of Newton's method for multiple roots. The AEC of the LNM method for multiple roots is always less than the AEC of the AN method, but it is always greater than the AEC of the HN method. In Table 2, we show the computational results of some numerical tests made to compare the efficiencies of the LMN with other MBN methods and with the CN method

REFERENCES:

[1] P. D´ıez, A note on the convergence of the secant method for simple and multiple roots, Appl. Math. Lett. 16 (2003) 1211–1215.

[2] S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000) 87-93.

[3] M. Frontini, E. Sormani, Some variants of Newton’s method with third order convergence, Appl. Math. Comput. 140 (2003) 419-426.

[4] A.Y. Özban, Some new variants of Newton’s method, Appl. Math. Lett. 17 (2004) 677-682.

[5] M. Frontini, E. Sormani, Modified Newtons method with third-order convergence and multiple roots, J. Comput. Appl. Math. 156 (2003), 345-354.

[6] Mamta, V. Kanwar , V.K. Kukreja, Sukhjit Singh, On some third-order iterative methods for solving nonlinear equations, Applied Mathematics and Computation 171 (2005) 272-280.

[7] T. Lukić, N. M. Ralević, Geometric mean Newton’s method (GN) for simple and multiple roots, Applied Mathematics Letters 21 (2008) 30–36.

8] M. Igarashi, A termination criterion for iterative methods used to find the zeros of polynomials, Math. Comp. 42 (1984) 165–171.

Received on 10.01.2014 Accepted on 31.01.2014

Int. J. Tech. 4(1): Jan.-June. 2014; Page 197-202