From: couturie Date: Wed, 28 Oct 2015 02:19:14 +0000 (-0400) Subject: suite X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/commitdiff_plain/f3cbb035ec5c163d367699cff06a3c8ee10ea89d suite --- diff --git a/paper.tex b/paper.tex index 9e84151..9532afc 100644 --- a/paper.tex +++ b/paper.tex @@ -147,9 +147,12 @@ approximation of all the roots, starting with the Durand-Kerner (DK) method: %%\begin{center} \begin{equation} - Z_{i}=Z_{i}-\frac{P(Z_{i})}{\prod_{i\neq j}(z_{i}-z_{j})} + Z_i^{k+1}=Z_{i}^k-\frac{P(Z_i^k)}{\prod_{i\neq j}(Z_i^k-Z_j^k)} \end{equation} %%\end{center} +where $Z_i^k$ is the $i^{th}$ root of the polynomial $P$ at the +iteration $k$. + This formula is mentioned for the first time by Weiestrass~\cite{Weierstrass03} as part of the fundamental theorem @@ -161,9 +164,11 @@ in the following form by Ehrlich~\cite{Ehrlich67} and Aberth~\cite{Aberth73} uses a different iteration formula given as fellows : %%\begin{center} \begin{equation} - Z_{i}=Z_{i}-\frac{1}{{\frac {P'(Z_{i})} {P(Z_{i})}}-{\sum_{i\neq j}(z_{i}-z_{j})}}. + Z_i^{k+1}=Z_i^k-\frac{1}{{\frac {P'(Z_i^k)} {P(Z_i^k)}}-{\sum_{i\neq j}(Z_i^k-Z_j^k)}}. \end{equation} %%\end{center} +where $P'(Z)$ is the polynomial derivative of $P$ evaluated in the +point $Z$. Aberth, Ehrlich and Farmer-Loizou~\cite{Loizon83} have proved that the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of convergence.