changement de l'encodage, ajout du fichier biblio

diff --git a/Root.pdf b/Root.pdf
index 6a1e7490aea98498906db23209637796393edddb..df62371f7f3bbafb8caee5f281bfa837988bd51a 100644 (file)
Binary files a/Root.pdf and b/Root.pdf differ

diff --git a/Root.tex b/Root.tex
index c2359b76ae17d99f8997ad89d2235d5153a8dd13..e2334654d11ffc212bedf07b6033031331ba3f62 100644 (file)
--- 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\r
 from Ilieff [2], Docev [3], Durand [4], Kerner [5]. Another method\r
 discovered from Borsch-Supan [6] and also described and brought in\r
 as part of the fundamental theorem of Algebra and is rediscovered\r
 from Ilieff [2], Docev [3], Durand [4], Kerner [5]. Another method\r
 discovered from Borsch-Supan [6] and also described and brought in\r

-the following form from Ehrlich [7] and Aberth [8]\r
+the following form from Ehrlich [7] and Aberth~\cite{Aberth73}.\r

 \begin{center}\r
 \r
 $  Z_{i}=Z_{i}-\frac{1}{{\frac {P'(Z_{i})}\r
 \begin{center}\r
 \r
 $  Z_{i}=Z_{i}-\frac{1}{{\frac {P'(Z_{i})}\r


@@ -113,67 +113,68 @@ implementation.................To be continued..................
 \r
 \r
 \r
 \r
 \r
 \r

-\bibliographystyle{alpha}\r
-\begin{thebibliography}{2}\r
+\bibliographystyle{plain}\r
+\bibliography{biblio}\r
+%% \begin{thebibliography}{2}\r

 \r
 \r

-\bibitem [1] {1} O. Aberth, Iteration Methods for Finding\r
-all Zeros of a Polynomial Simultaneously, Math. Comput. 27, 122\r
-(1973) 339\96344.\r
+%% \bibitem [1] {1} O. Aberth, Iteration Methods for Finding\r
+%% all Zeros of a Polynomial Simultaneously, Math. Comput. 27, 122\r
+%% (1973) 339Â\96344.\r

 \r
 \r

-\bibitem [2] {2} Ilieff, L. (1948-50), On the approximations of Newton, Annual\r
-Sofia Univ. 46, 167-171.\r
+%% \bibitem [2] {2} Ilieff, L. (1948-50), On the approximations of Newton, Annual\r
+%% Sofia Univ. 46, 167-171.\r

 \r
 \r

-\bibitem [3] {3} Docev, K. (1962), An alternative method of Newton for\r
-simultaneous calculation of all the roots of a given algebraic\r
-equation, Phys. Math. J., Bulg. Acad. Sci. 5, 136-139.\r
+%% \bibitem [3] {3} Docev, K. (1962), An alternative method of Newton for\r
+%% simultaneous calculation of all the roots of a given algebraic\r
+%% equation, Phys. Math. J., Bulg. Acad. Sci. 5, 136-139.\r

 \r
 \r

-\bibitem [4]{4} Durand, E. (1960), Solution Numerique des Equations\r
-Algebriques, Vol. 1, Equations du Type F(x)=0, Racines d'une\r
-Polynome. Masson, Paris.\r
+%% \bibitem [4]{4} Durand, E. (1960), Solution Numerique des Equations\r
+%% Algebriques, Vol. 1, Equations du Type F(x)=0, Racines d'une\r
+%% Polynome. Masson, Paris.\r

 \r
 \r

-\bibitem [4]  {4} Aberth, O. (1973), Iterative methods for finding all zeros of\r
-a polynomial simultaneously, Math. Comp. 27, 339-344.\r
+%% \bibitem [4]  {4} Aberth, O. (1973), Iterative methods for finding all zeros of\r
+%% a polynomial simultaneously, Math. Comp. 27, 339-344.\r

 \r
 \r

-\bibitem [5] {5} Kerner, I.O. (1966), Ein Gesamtschritteverfahren zur\r
-Berechnung der Nullstellen von Polynomen, Numer. Math. 8, 290-294.\r
+%% \bibitem [5] {5} Kerner, I.O. (1966), Ein Gesamtschritteverfahren zur\r
+%% Berechnung der Nullstellen von Polynomen, Numer. Math. 8, 290-294.\r

 \r
 \r

-\bibitem [6]{6} Borch-Supan, W. (1963), A posteriori error for the zeros of\r
-polynomials, Numer. Math. 5, 380-398.\r
+%% \bibitem [6]{6} Borch-Supan, W. (1963), A posteriori error for the zeros of\r
+%% polynomials, Numer. Math. 5, 380-398.\r

 \r
 \r

-\bibitem [7] {7} Ehrlich, L. W. (1967), A modified Newton method for\r
-polynomials, Comm. Ass. Comput. Mach. 10, 107-108.\r
+%% \bibitem [7] {7} Ehrlich, L. W. (1967), A modified Newton method for\r
+%% polynomials, Comm. Ass. Comput. Mach. 10, 107-108.\r

 \r
 \r
 \r
 \r
 \r
 \r

-\bibitem [10] {10}Loizon, G. (1983), Higher-order iteration functions for\r
-simultaneously approximating polynomial zeros, Intern. J. Computer\r
-Math. 14, 45-58.\r
+%% \bibitem [10] {10}Loizon, G. (1983), Higher-order iteration functions for\r
+%% simultaneously approximating polynomial zeros, Intern. J. Computer\r
+%% Math. 14, 45-58.\r

 \r
 \r

-\bibitem [11]{11} E. Durand, Solutions num´eriques des ´equations alg´ebriques,\r
-Tome 1: Equations du type F(X) = 0; Racines d\92un polyn\88ome,\r
-Masson, Paris 1960.\r
+%% \bibitem [11]{11} E. Durand, Solutions numŽeriques des Žequations algŽebriques,\r
+%% Tome 1: Equations du type F(X) = 0; Racines dÂ\92un polynÂ\88ome,\r
+%% Masson, Paris 1960.\r

 \r
 \r

-\bibitem [12] {12} Weierstrass, K. (1903), Neuer Beweis des Satzes, dass\r
-jede ganze rationale function einer veranderlichen dagestellt\r
-werden kann als ein product aus linearen functionen derselben\r
-veranderlichen, Ges. Werke 3, 251-269.\r
-\bibitem [13] {13} Freeman, T. L. (1989), Calculating polynomial zeros on a\r
-local memory parallel computer, Parallel Computing 12, 351-358.\r
+%% \bibitem [12] {12} Weierstrass, K. (1903), Neuer Beweis des Satzes, dass\r
+%% jede ganze rationale function einer veranderlichen dagestellt\r
+%% werden kann als ein product aus linearen functionen derselben\r
+%% veranderlichen, Ges. Werke 3, 251-269.\r
+%% \bibitem [13] {13} Freeman, T. L. (1989), Calculating polynomial zeros on a\r
+%% local memory parallel computer, Parallel Computing 12, 351-358.\r

 \r
 \r

-\bibitem [14] {14} Freeman, T. L., Brankin, R. K. (1990), Asynchronous\r
-polynomial zero-finding algorithms, Parallel Computing 17,\r
-673-681.\r
+%% \bibitem [14] {14} Freeman, T. L., Brankin, R. K. (1990), Asynchronous\r
+%% polynomial zero-finding algorithms, Parallel Computing 17,\r
+%% 673-681.\r

 \r
 \r

-\bibitem [15] {15} Raphaël,C. François,S. (2001), Extraction de racines dans des\r
-polynômes creux de degré élevé. RSRCP (Réseaux et Systèmes\r
-Répartis, Calculateurs Parallèles), Numéro thématique :\r
-Algorithmes itératifs parallèles et distribués, 13(1):67--81.\r
+%% \bibitem [15] {15} Raphaël,C. François,S. (2001), Extraction de racines dans des\r
+%% polynômes creux de degré élevé. RSRCP (Réseaux et Systèmes\r
+%% Répartis, Calculateurs Parallèles), Numéro thématique :\r
+%% Algorithmes itératifs parallèles et distribués, 13(1):67--81.\r

 \r
 \r

-\bibitem [16]{16} Kahina, G. Raphaël, C. Abderrahmane, S. A\r
-parallel implementation of the Durand-Kerner algorithm for\r
-polynomial root-finding on GPU. In INDS 2014, Int. Conf. on\r
-advanced Networking, Distributed Systems and Applications, Bejaia,\r
-Algeria, pages 53--57, June 2014. IEEE\r
+%% \bibitem [16]{16} Kahina, G. Raphaël, C. Abderrahmane, S. A\r
+%% parallel implementation of the Durand-Kerner algorithm for\r
+%% polynomial root-finding on GPU. In INDS 2014, Int. Conf. on\r
+%% advanced Networking, Distributed Systems and Applications, Bejaia,\r
+%% Algeria, pages 53--57, June 2014. IEEE\r

 \r
 \r

-\end{thebibliography}\r
+%% \end{thebibliography}\r

 \end{document}\r
 \end{document}\r

diff --git a/biblio.bib b/biblio.bib
new file mode 100644 (file)
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