-
\begin{document}
%
% paper title
{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=1,\ldots,n
\end{equation}
-This method contains 4 steps. The first step consists in the
-initializing the polynomial.\LZK{Pas compris?? \RC{changé}}.
-The second step initializes the solution vector $Z$ using the
-Guggenheimer method~\cite{Gugg86} to ensure that initial roots are all
-distinct from each other. \LZK{Quelle est la différence entre la 1st
- step et la 2nd step? Que veut dire " to ensure the distinction of
- the initial vector roots"? \RC{reformulé}}
-In step 3, the iterative function based on the Newton's
-method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03} is
-applied. In our case, the Ehrlich-Aberth is applied as in (\ref{Eq:EA1}).
-Iterations of the EA method will converge to the roots of the
-considered polynomial.\LZK{On ne peut pas expliquer un peu plus
- comment? Donner des formules comment elle se base sur la méthode de
- Newton et de l'opérateur de Weiestrass? \RC{amélioré}}
-\LZK{Elle est où la 4th step??}
-\LZK{Conclusion: Méthode mal présentée et j'ai presque rien compris!
- \RC{après} }
-
-
-In order to stop the iterative function, a stop condition is applied,
-this is the 4th step. This condition checks that all the root modules
-are lower than a fixed value $\epsilon$.
+This method contains 4 steps. The first step consists in the initializing the polynomial. The second step initializes the solution vector $Z$ using the Guggenheimer method~\cite{Gugg86} to ensure that initial roots are all distinct from each other. In step 3, the iterative function based on the Newton's method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03} is applied. In our case, the Ehrlich-Aberth is applied as in~(\ref{Eq:EA1}). Iterations of the Ehrlich-Aberth method will converge to the roots of the considered polynomial. In order to stop the iterative function, a stop condition is applied, this is the 4th step. This condition checks that all the root modules are lower than a fixed value $\epsilon$.
\begin{equation}
\label{eq:Aberth-Conv-Cond}
\forall i\in[1,n],~\vert\frac{z_i^k-z_i^{k-1}}{z_i^k}\vert<\epsilon
\end{equation}
-\LZK{On ne dit pas plutôt "the relative errors" à la place de "root
- modules"? Raph nous confirmera quelle critère d'arrêt a
- utilisé. \RC{normalement c'est bon, l'erreur est calculée avec le
- module de chaque racine}}
-
\subsection{Improving Ehrlich-Aberth method}
With high degree polynomials, the Ehrlich-Aberth method suffers from floating point overflows due to the mantissa of floating points representations. This induces errors in the computation of $p(z)$ when $z$ is large.
%We propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent.
-Using the logarithm and the exponential operators, we can replace any
-multiplications and divisions with additions and
-subtractions. Consequently, computations manipulate lower values in absolute
-values~\cite{Karimall98}. \LZK{Je n'ai pas compris cette dernière
- phrase? \RC{changé : on veut dire on manipule des valeurs plus petites en valeur absolues}}
+Using the logarithm and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower values in absolute values~\cite{Karimall98}.
%This problem was discussed earlier in~\cite{Karimall98} for the Durand-Kerner method. The authors
%propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent. Using the logarithm and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower absolute values and the roots for large polynomial degrees can be looked for successfully~\cite{Karimall98}.
implementation of Ehrlich-Aberth method.
\LZK{Vaut mieux expliquer l'implémentation en faisant référence à l'algo séquentiel que de parler des différentes steps.}
+%\begin{algorithm}[htpb]
+%\label{alg1-cuda}
+%\LinesNumbered
+%\SetAlgoNoLine
+%\caption{CUDA Algorithm to find polynomial roots with the Ehrlich-Aberth method}
+%\KwIn{$Z^{0}$ (Initial vector of roots), $\epsilon$ (Error tolerance threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degree), $\Delta z_{max}$ (Maximum value of stop condition)}
+%\KwOut{$Z$ (Solution vector of roots)}
+
+%\BlankLine
+
+%Initialization of P\;
+%Initialization of Pu\;
+%Initialization of the solution vector $Z^{0}$\;
+%Allocate and copy initial data to the GPU global memory\;
+%\While {$\Delta z_{max} > \epsilon$}{
+% $ ZPres=kernel\_save(Z)$\;
+% $ Z=kernel\_update(Z,P,Pu)$\;
+% $\Delta z_{max}=kernel\_testConv(Z,ZPrec)$\;
+
+%}
+%Copy results from GPU memory to CPU memory\;
+%\end{algorithm}
+
\begin{algorithm}[htpb]
-\label{alg1-cuda}
\LinesNumbered
\SetAlgoNoLine
-\caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
+\caption{Finding roots of polynomials with the Ehrlich-Aberth method on a GPU}
+\KwIn{$n$ (polynomial's degree), $\epsilon$ (tolerance threshold)}
+\KwOut{$Z$ (solution vector of roots)}
+Initialize the polynomial $P$ and its derivative $P'$\;
+Set the initial values of vector $Z$\;
+Copy $P$, $P'$ and $Z$ from the CPU memory to the GPU memory\;
+\While{\emph{not convergence}}{
+ $Z_{prev}$ = Kernel\_Save($Z$)\;
+ $Z$ = Kernel\_Update($P,P',Z$)\;
+ Kernel\_Test\_Convergence($Z,Z_{prev},\epsilon$)\;
+}
+Copy $Z$ from the GPU memory to the CPU memory\;
+\label{alg1-cuda}
+\end{algorithm}
+
+
+
+
+
-\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
- threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degrees), $\Delta z_{max}$ (Maximum value of stop condition)}
-\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
-%\BlankLine
-Initialization of P\;
-Initialization of Pu\;
-Initialization of the solution vector $Z^{0}$\;
-Allocate and copy initial data to the GPU global memory\;
-\While {$\Delta z_{max} > \epsilon$}{
- $ ZPres=kernel\_save(Z)$\;
- $ Z=kernel\_update(Z,P,Pu)$\;
- $\Delta z_{max}=kernel\_testConv(Z,ZPrec)$\;
-}
-Copy results from GPU memory to CPU memory\;
-\end{algorithm}
\RC{Si l'algo vous convient, il faudrait le détailler précisément}
\section{The EA algorithm on Multiple GPUs}
