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Mathematical Problems in Engineering
Volume 2013, Article ID 523895, 4 pages
http://dx.doi.org/10.1155/2013/523895
Research Article

Eigenvalue Decomposition-Based Modified Newton Algorithm

Department of Mathematics and System, Science College, National University of Defense Technology, Changsha 410073, China

Received 2 November 2012; Revised 3 February 2013; Accepted 19 February 2013

Academic Editor: Jian Guo Zhou

Copyright © 2013 Wen-jun Wang 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

When the Hessian matrix is not positive, the Newton direction may not be the descending direction. A new method named eigenvalue decomposition-based modified Newton algorithm is presented, which first takes the eigenvalue decomposition of the Hessian matrix, then replaces the negative eigenvalues with their absolute values, and finally reconstructs the Hessian matrix and modifies the searching direction. The new searching direction is always the descending direction. The convergence of the algorithm is proven and the conclusion on convergence rate is presented qualitatively. Finally, a numerical experiment is given for comparing the convergence domains of the modified algorithm and the classical algorithm.