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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 972537, 9 pages
http://dx.doi.org/10.1155/2011/972537
Research Article

Geometrically Constructed Families of Newton's Method for Unconstrained Optimization and Nonlinear Equations

1Department of Mathematics, Maharishi Markandeshwar University, Sadopur, Ambala, Haryana 134007, India
2University Institute of Engineering and Technology, Panjab University, Chandigarh 160 014, India
3Department of Mathematics, Panjab University, Chandigarh 160 014, India
4Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Punjab 148106, India

Received 11 December 2010; Revised 8 June 2011; Accepted 15 June 2011

Academic Editor: Naseer Shahzad

Copyright © 2011 Sanjeev Kumar et al. 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.

Abstract

One-parameter families of Newton's iterative method for the solution of nonlinear equations and its extension to unconstrained optimization problems are presented in the paper. These methods are derived by implementing approximations through a straight line and through a parabolic curve in the vicinity of the root. The presented variants are found to yield better performance than Newton's method, in addition that they overcome its limitations.