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Journal of Applied Mathematics
Volume 2012, Article ID 167927, 12 pages
Research Article

Choosing Improved Initial Values for Polynomial Zerofinding in Extended Newbery Method to Obtain Convergence

1Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia 81310, Skudai, Johor, Malaysia
2Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran

Received 29 April 2012; Accepted 19 August 2012

Academic Editor: Ram N. Mohapatra

Copyright © 2012 Saeid Saidanlu 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.


In all polynomial zerofinding algorithms, a good convergence requires a very good initial approximation of the exact roots. The objective of the work is to study the conditions for determining the initial approximations for an iterative matrix zerofinding method. The investigation is based on the Newbery's matrix construction which is similar to Fiedler's construction associated with a characteristic polynomial. To ensure that convergence to both the real and complex roots of polynomials can be attained, three methods are employed. It is found that the initial values for the Fiedler's companion matrix which is supplied by the Schmeisser's method give a better approximation to the solution in comparison to when working on these values using the Schmeisser's construction towards finding the solutions. In addition, empirical results suggest that a good convergence can still be attained when an initial approximation for the polynomial root is selected away from its real value while other approximations should be sufficiently close to their real values. Tables and figures on the errors that resulted from the implementation of the method are also given.