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Another method discovered from Borsch-Supan [6] and also described and brought in -the following form from Ehrlich [7] and Aberth [8] +the following form from Ehrlich [7] and Aberth~\cite{Aberth73}. \begin{center} $ Z_{i}=Z_{i}-\frac{1}{{\frac {P'(Z_{i})} @@ -113,67 +113,68 @@ implementation.................To be continued.................. -\bibliographystyle{alpha} -\begin{thebibliography}{2} +\bibliographystyle{plain} +\bibliography{biblio} +%% \begin{thebibliography}{2} -\bibitem [1] {1} O. Aberth, Iteration Methods for Finding -all Zeros of a Polynomial Simultaneously, Math. Comput. 27, 122 -(1973) 339–344. +%% \bibitem [1] {1} O. Aberth, Iteration Methods for Finding +%% all Zeros of a Polynomial Simultaneously, Math. Comput. 27, 122 +%% (1973) 339–344. -\bibitem [2] {2} Ilieff, L. (1948-50), On the approximations of Newton, Annual -Sofia Univ. 46, 167-171. +%% \bibitem [2] {2} Ilieff, L. (1948-50), On the approximations of Newton, Annual +%% Sofia Univ. 46, 167-171. -\bibitem [3] {3} Docev, K. (1962), An alternative method of Newton for -simultaneous calculation of all the roots of a given algebraic -equation, Phys. Math. J., Bulg. Acad. Sci. 5, 136-139. +%% \bibitem [3] {3} Docev, K. (1962), An alternative method of Newton for +%% simultaneous calculation of all the roots of a given algebraic +%% equation, Phys. Math. J., Bulg. Acad. Sci. 5, 136-139. -\bibitem [4]{4} Durand, E. (1960), Solution Numerique des Equations -Algebriques, Vol. 1, Equations du Type F(x)=0, Racines d'une -Polynome. Masson, Paris. +%% \bibitem [4]{4} Durand, E. (1960), Solution Numerique des Equations +%% Algebriques, Vol. 1, Equations du Type F(x)=0, Racines d'une +%% Polynome. Masson, Paris. -\bibitem [4] {4} Aberth, O. (1973), Iterative methods for finding all zeros of -a polynomial simultaneously, Math. Comp. 27, 339-344. +%% \bibitem [4] {4} Aberth, O. (1973), Iterative methods for finding all zeros of +%% a polynomial simultaneously, Math. Comp. 27, 339-344. -\bibitem [5] {5} Kerner, I.O. (1966), Ein Gesamtschritteverfahren zur -Berechnung der Nullstellen von Polynomen, Numer. Math. 8, 290-294. +%% \bibitem [5] {5} Kerner, I.O. (1966), Ein Gesamtschritteverfahren zur +%% Berechnung der Nullstellen von Polynomen, Numer. Math. 8, 290-294. -\bibitem [6]{6} Borch-Supan, W. (1963), A posteriori error for the zeros of -polynomials, Numer. Math. 5, 380-398. +%% \bibitem [6]{6} Borch-Supan, W. (1963), A posteriori error for the zeros of +%% polynomials, Numer. Math. 5, 380-398. -\bibitem [7] {7} Ehrlich, L. W. (1967), A modified Newton method for -polynomials, Comm. Ass. Comput. Mach. 10, 107-108. +%% \bibitem [7] {7} Ehrlich, L. W. (1967), A modified Newton method for +%% polynomials, Comm. Ass. Comput. Mach. 10, 107-108. -\bibitem [10] {10}Loizon, G. (1983), Higher-order iteration functions for -simultaneously approximating polynomial zeros, Intern. J. Computer -Math. 14, 45-58. +%% \bibitem [10] {10}Loizon, G. (1983), Higher-order iteration functions for +%% simultaneously approximating polynomial zeros, Intern. J. Computer +%% Math. 14, 45-58. -\bibitem [11]{11} E. Durand, Solutions num´eriques des ´equations alg´ebriques, -Tome 1: Equations du type F(X) = 0; Racines d’un polynˆome, -Masson, Paris 1960. +%% \bibitem [11]{11} E. Durand, Solutions numŽeriques des Žequations algŽebriques, +%% Tome 1: Equations du type F(X) = 0; Racines d’un polynˆome, +%% Masson, Paris 1960. -\bibitem [12] {12} Weierstrass, K. (1903), Neuer Beweis des Satzes, dass -jede ganze rationale function einer veranderlichen dagestellt -werden kann als ein product aus linearen functionen derselben -veranderlichen, Ges. Werke 3, 251-269. -\bibitem [13] {13} Freeman, T. L. (1989), Calculating polynomial zeros on a -local memory parallel computer, Parallel Computing 12, 351-358. +%% \bibitem [12] {12} Weierstrass, K. (1903), Neuer Beweis des Satzes, dass +%% jede ganze rationale function einer veranderlichen dagestellt +%% werden kann als ein product aus linearen functionen derselben +%% veranderlichen, Ges. Werke 3, 251-269. +%% \bibitem [13] {13} Freeman, T. L. (1989), Calculating polynomial zeros on a +%% local memory parallel computer, Parallel Computing 12, 351-358. -\bibitem [14] {14} Freeman, T. L., Brankin, R. K. (1990), Asynchronous -polynomial zero-finding algorithms, Parallel Computing 17, -673-681. +%% \bibitem [14] {14} Freeman, T. L., Brankin, R. K. (1990), Asynchronous +%% polynomial zero-finding algorithms, Parallel Computing 17, +%% 673-681. -\bibitem [15] {15} Raphaël,C. François,S. (2001), Extraction de racines dans des -polynômes creux de degré élevé. RSRCP (Réseaux et Systèmes -Répartis, Calculateurs Parallèles), Numéro thématique : -Algorithmes itératifs parallèles et distribués, 13(1):67--81. +%% \bibitem [15] {15} Raphaël,C. François,S. (2001), Extraction de racines dans des +%% polynômes creux de degré élevé. RSRCP (Réseaux et Systèmes +%% Répartis, Calculateurs Parallèles), Numéro thématique : +%% Algorithmes itératifs parallèles et distribués, 13(1):67--81. -\bibitem [16]{16} Kahina, G. Raphaël, C. Abderrahmane, S. A -parallel implementation of the Durand-Kerner algorithm for -polynomial root-finding on GPU. In INDS 2014, Int. Conf. on -advanced Networking, Distributed Systems and Applications, Bejaia, -Algeria, pages 53--57, June 2014. IEEE +%% \bibitem [16]{16} Kahina, G. Raphaël, C. Abderrahmane, S. A +%% parallel implementation of the Durand-Kerner algorithm for +%% polynomial root-finding on GPU. In INDS 2014, Int. Conf. on +%% advanced Networking, Distributed Systems and Applications, Bejaia, +%% Algeria, pages 53--57, June 2014. IEEE -\end{thebibliography} +%% \end{thebibliography} \end{document} diff --git a/biblio.bib b/biblio.bib new file mode 100644 index 0000000..aef5348 --- /dev/null +++ b/biblio.bib @@ -0,0 +1,11 @@ +@Article{Aberth73, + author = "O. Aberth", + title = "Iteration Methods for Finding all Zeros of a + Polynomial Simultaneously", + journal = "Mathematics of Computation", + volume = "27", + number = "122", + pages = "339--344", + year = "1973", + keywords = "RBBRS3, root finding,", +}x \ No newline at end of file -- 2.39.5