-
\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 roots with the Ehrlich-Aberth method}
-
-\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)}
+%\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)$\;
- $\Delta z_{max}=kernel\_testConv(Z,ZPrec)$\;
+%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]
+\LinesNumbered
+\SetAlgoNoLine
+\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 CPU to GPU\;
+\While{\emph{not convergence}}{
+ $Z^{prev}$ = KernelSave($Z,n$)\;
+ $Z$ = KernelUpdate($P,P',Z^{prev},n$)\;
+ $\Delta Z$ = KernelComputeError($Z,Z^{prev},n$)\;
+ $\Delta Z_{max}$ = CudaMaxFunction($\Delta Z,n$)\;
+ TestConvergence($\Delta Z_{max},\epsilon$)\;
}
-Copy results from GPU memory to CPU memory\;
+Copy $Z$ from GPU to CPU\;
+\label{alg1-cuda}
+\RC{Si l'algo vous convient, il faudrait le détailler précisément\LZK{J'ai modifié l'algo. Sinon, est ce qu'on doit mettre en paramètre $Z^{prev}$ ou $Z$ tout court (dans le cas où on exploite l'asynchronisme des threads cuda!) pour le Kernel\_Update? }}
\end{algorithm}
-\RC{Si l'algo vous convient, il faudrait le détailler précisément}
+
\section{The EA algorithm on Multiple GPUs}
\label{sec4}
%% roots sufficiently converge.
-\begin{algorithm}[h]
-\label{alg2-cuda-openmp}
+%% \begin{algorithm}[h]
+%% \label{alg2-cuda-openmp}
+%% \LinesNumbered
+%% \SetAlgoNoLine
+%% \caption{CUDA-OpenMP Algorithm to find roots with the Ehrlich-Aberth method}
+
+%% \KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
+%% threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degree), $\Delta z$ ( Vector of errors for stop condition), $num\_gpus$ (number of OpenMP threads/ Number of GPUs), $Size$ (number of roots)}
+
+%% \KwOut {$Z$ ( Root's vector), $ZPrec$ (Previous root's vector)}
+
+%% \BlankLine
+
+%% Initialization of P\;
+%% Initialization of Pu\;
+%% Initialization of the solution vector $Z^{0}$\;
+%% Start of a parallel part with OpenMP (Z, $\Delta z$, P are shared variables)\;
+%% gpu\_id=cudaGetDevice()\;
+%% Allocate memory on GPU\;
+%% Compute local size and offet according to gpu\_id\;
+%% \While {$error > \epsilon$}{
+%% copy Z from CPU to GPU\;
+%% $ ZPrec_{loc}=kernel\_save(Z_{loc})$\;
+%% $ Z_{loc}=kernel\_update(Z,P,Pu)$\;
+%% $\Delta z[gpu\_id] = kernel\_testConv(Z_{loc},ZPrec_{loc})$\;
+%% $ error= Max(\Delta z)$\;
+%% copy $Z_{loc}$ from GPU to Z in CPU
+%% }
+%%\end{algorithm}
+
+\begin{algorithm}[htpb]
\LinesNumbered
\SetAlgoNoLine
-\caption{CUDA-OpenMP Algorithm to find roots with the Ehrlich-Aberth method}
-
-\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
- threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degree), $\Delta z$ ( Vector of errors for stop condition), $num\_gpus$ (number of OpenMP threads/ Number of GPUs), $Size$ (number of roots)}
-
-\KwOut {$Z$ ( Root's vector), $ZPrec$ (Previous root's vector)}
-
-\BlankLine
-
-Initialization of P\;
-Initialization of Pu\;
-Initialization of the solution vector $Z^{0}$\;
-Start of a parallel part with OpenMP (Z, $\Delta z$, P are shared variables)\;
-gpu\_id=cudaGetDevice()\;
-Allocate memory on GPU\;
-Compute local size and offet according to gpu\_id\;
-\While {$error > \epsilon$}{
- copy Z from CPU to GPU\;
-$ ZPrec_{loc}=kernel\_save(Z_{loc})$\;
-$ Z_{loc}=kernel\_update(Z,P,Pu)$\;
-$\Delta z[gpu\_id] = kernel\_testConv(Z_{loc},ZPrec_{loc})$\;
-$ error= Max(\Delta z)$\;
- copy $Z_{loc}$ from GPU to Z in CPU
+\caption{Finding roots of polynomials with the Ehrlich-Aberth method on multiple GPUs using OpenMP}
+\KwIn{$n$ (polynomial's degree), $\epsilon$ (tolerance threshold), $ngpu$ (number of GPUs)}
+\KwOut{$Z$ (solution vector of roots)}
+Initialize the polynomial $P$ and its derivative $P'$\;
+Set the initial values of vector $Z$\;
+Start of a parallel part with OpenMP ($Z$, $\Delta Z$, $\Delta Z_{max}$, $P$ are shared variables)\;
+$id_{gpu}$ = cudaGetDevice()\;
+$n_{loc}$ = $n/ngpu$ (local size)\;
+%$idx$ = $id_{gpu}\times n_{loc}$ (local offset)\;
+Copy $P$, $P'$ from CPU to GPU\;
+\While{\emph{not convergence}}{
+ Copy $Z$ from CPU to GPU\;
+ $Z^{prev}$ = KernelSave($Z,n$)\;
+ $Z_{loc}$ = KernelUpdate($P,P',Z^{prev},n_{loc}$)\;
+ $\Delta Z_{loc}$ = KernelComputeError($Z_{loc},Z^{prev}_{loc},n_{loc}$)\;
+ $\Delta Z_{max}[id_{gpu}]$ = CudaMaxFunction($\Delta Z_{loc},n_{loc}$)\;
+ Copy $Z_{loc}$ from GPU to $Z$ in CPU\;
+ $max$ = MaxFunction($\Delta Z_{max},ngpu$)\;
+ TestConvergence($max,\epsilon$)\;
}
+\label{alg2-cuda-openmp}
+\LZK{J'ai modifié l'algo. Le $P$ est mis shared. Qu'en est-il pour $P'$?}
\end{algorithm}
+
+
\subsection{an MPI-CUDA approach}
%\begin{figure}[htbp]
%\centering
