Choosing Improved Initial Values for Polynomial Zerofinding in Extended Newbery Method to Obtain Convergence
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 Ne...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/167927 |
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| author | Saeid Saidanlu Nor’aini Aris Ali Abd Rahman |
| author_facet | Saeid Saidanlu Nor’aini Aris Ali Abd Rahman |
| author_sort | Saeid Saidanlu |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-27527a13ce834b8f9444260ad278dde2 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-27527a13ce834b8f9444260ad278dde22025-02-03T01:11:34ZengWileyJournal of Applied Mathematics1110-757X1687-00422012-01-01201210.1155/2012/167927167927Choosing Improved Initial Values for Polynomial Zerofinding in Extended Newbery Method to Obtain ConvergenceSaeid Saidanlu0Nor’aini Aris1Ali Abd Rahman2Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia 81310, Skudai, Johor, MalaysiaDepartment of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia 81310, Skudai, Johor, MalaysiaDepartment of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia 81310, Skudai, Johor, MalaysiaIn 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.http://dx.doi.org/10.1155/2012/167927 |
| spellingShingle | Saeid Saidanlu Nor’aini Aris Ali Abd Rahman Choosing Improved Initial Values for Polynomial Zerofinding in Extended Newbery Method to Obtain Convergence Journal of Applied Mathematics |
| title | Choosing Improved Initial Values for Polynomial Zerofinding in Extended Newbery Method to Obtain Convergence |
| title_full | Choosing Improved Initial Values for Polynomial Zerofinding in Extended Newbery Method to Obtain Convergence |
| title_fullStr | Choosing Improved Initial Values for Polynomial Zerofinding in Extended Newbery Method to Obtain Convergence |
| title_full_unstemmed | Choosing Improved Initial Values for Polynomial Zerofinding in Extended Newbery Method to Obtain Convergence |
| title_short | Choosing Improved Initial Values for Polynomial Zerofinding in Extended Newbery Method to Obtain Convergence |
| title_sort | choosing improved initial values for polynomial zerofinding in extended newbery method to obtain convergence |
| url | http://dx.doi.org/10.1155/2012/167927 |
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