X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/blobdiff_plain/c5f581ffe6e32e34c3fb1048707698d5d4b67c53..555354e3bf6dbb62e78bbfd56f42c17be87d7bec:/Root.tex

diff --git a/Root.tex b/Root.tex
index c2359b7..e233465 100644
--- a/Root.tex
+++ b/Root.tex
@@ -56,7 +56,7 @@ This formula is mentioned for the first time from Weiestrass [12]
 as part of the fundamental theorem of Algebra and is rediscovered
 from Ilieff [2], Docev [3], Durand [4], Kerner [5]. 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}