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International Journal of Mathematical Education in Science and Technology Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tmes20
A family of ellipse methods for solving non-linear equations a
b
K.C. Gupta , V. Kanwar & Sanjeev Kumar
a
a
Department of Applied Sciences , Indo Global College of Engineering , Abhipur, Mohali, Punjab, India b
University Institute of Engineering and Technology, Panjab University , Chandigarh-160 014, India Published online: 14 May 2009.
To cite this article: K.C. Gupta , V. Kanwar & Sanjeev Kumar (2009) A family of ellipse methods for solving non-linear equations, International Journal of Mathematical Education in Science and Technology, 40:4, 571-575, DOI: 10.1080/00207390902825336 To link to this article: http://dx.doi.org/10.1080/00207390902825336
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International Journal of Mathematical Education in Science and Technology
571
A family of ellipse methods for solving non-linear equations K.C. Guptaa, V. Kanwarb* and Sanjeev Kumara a
Department of Applied Sciences, Indo Global College of Engineering, Abhipur, Mohali, Punjab, India; bUniversity Institute of Engineering and Technology, Panjab University, Chandigarh-160 014, India (Received 21 July 2008)
This note presents a method for the numerical approximation of simple zeros of a non-linear equation in one variable. In order to do so, the method uses an ellipse rather than a tangent approach. The main advantage of our method is that it does not fail even if the derivative of the function is either zero or very small in the vicinity of the required root.
Downloaded by [Panjab University] at 00:54 05 March 2014
Keywords: non-linear equations; Newton’s method; order of convergence
1. Introduction Solving non-linear equations is one of the most important problems in numerical analysis. Perhaps, the most celebrated of all such methods is the classical Newton’s method, given by xnþ1 ¼ xn
fðxn Þ , f 0 ðxn Þ
n 0:
ð1Þ
Newton’s formula (1) is both simple and converges quadratically. However, it may fail to converge if either the initial guess is far from the required root or the derivative of the function is very small or even zero in the vicinity of the required root. Therefore, the requirement of f 0 ðxÞ 6¼ 0 is an essential condition for the convergence of Newton’s method. For a more detailed description of this method, see [1–3]. In this article, we present a family of ellipse methods in which f 0 ðxÞ ¼ 0 is permitted at some points in the vicinity of the required root. We prove that the new methods are quadratically convergent and moreover, have the same error equation as Newton’s method.
2. Derivation of the method Assume that the equation fðxÞ ¼ 0,
ð2Þ
has a simple root r which is to be found and let x0 be our initial approximation to this root. If, on the same graph of y ¼ fðxÞ,
ð3Þ
2 y fðx0 Þ ðx x0 Þ2 ¼ 1, þ p2 f 2 ðx0 Þ
ð4Þ
we sketch the ellipse
*Corresponding author. Email:
[email protected] DOI: 10.1080/00207390902825336
572
Classroom Notes
for some p 2 1. For any initial guess x0 < 0, Newton’s method converges to the root very efficiently. For any initial guess x0 > 1, the Newton iterates move away from zero. For example, x0 ¼ 2, then x1 ¼ 4, x2 ¼ 5:3333 and so on. On the other hand, our method can give the required root if the sign in (8) is chosen suitably. Example 5:
4x4 4x2 ¼ 0.
In applying Newton’s method to solve this equation, problems arise if the starting points pffiffiffi give horizontal tangents or cycle back pffiffiffiffiffi and forth from one to another. The points ð 2=2Þ give horizontal tangents and ð 21=7Þ cycle, each leading to the other and back [5]. Our method does not exhibit this behaviour. Example 6: Example 7:
ex
2
þ7x30
1 ¼ 0.
6
ðx 1Þ 1 ¼ 0.
Acknowledgement The authors would like to thank the referee for his valuable comments and suggestions.
International Journal of Mathematical Education in Science and Technology
575
References
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[1] J.F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, NJ, 1964. [2] A.M. Ostrowski, Solution of Equations in Euclidean and Banach Space, Academic Press, New York (NY), 1973. [3] J.E. Dennis and R.B. Schnable, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, NJ, 1983. [4] Adi Ben-Israel, Newton’s method with modified functions, Contemp. Math. 204 (1997), pp. 39–50. [5] G.B. Thomas Jr, R.L. Finney, M.D. Wier, and F.R. Giordano, Thomas’ Calculus, AddisonWesley Publishing company, Inc., Massachusetts (MA), 2001, p. 301.
