-
\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 of the initial approximations of all the roots of the polynomial.\LZK{Pas compris??}
-The second step initializes the solution vector $Z$ using the Guggenheimer method~\cite{Gugg86} to ensure the distinction of the initial vector roots.\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"?}
-In step 3, the iterative function based on the Newton's method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03} is applied. With this step the computation of roots will converge, provided that all roots are different.\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?}
-\LZK{Elle est où la 4th step??}
-\LZK{Conclusion: Méthode mal présentée et j'ai presque rien compris!}
-
-
-In order to stop the iterative function, a stop condition is
-applied. 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.}
-
\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 absolute values~\cite{Karimall98}. \LZK{Je n'ai pas compris cette dernière phrase?}
+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}.
controlling the various GPUs is to use as many threads or processes as
GPU devices. We can choose the GPU index based on the identifier of
OpenMP thread or the rank of the MPI process. Both approaches will be
-investigated. \LZK{Répétition! Le même texte est déjà écrit comme intro dans la section II. Sinon ici on parle seulement de l'implémentation cuda sans mpi et openmp!}
+investigated. \LZK{Répétition! Le même texte est déjà écrit comme
+ intro dans la section II. Sinon ici on parle seulement de
+ l'implémentation cuda sans mpi et openmp! \RC{Je suis d'accord à
+ revoir après, quand les 2 parties suivantes seront plus stables}}
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 {$\Delta z_{max} > \epsilon$}{
- $ kernel\_save(ZPrec,Z)$\;
- $ kernel\_update(Z,P,Pu)$\;
- $\Delta z_{max}=kernel\_testConverge(Z,ZPrec)$\;
+%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}
\label{sec4}
\subsection{an OpenMP-CUDA approach}
Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid
-OpenMP and CUDA programming model. All the data
-are shared with OpenMP amoung all the OpenMP threads. The shared data
-are the solution vector $Z$, the polynomial to solve $P$, and the
-error vector $\Delta z$. The number of OpenMP threads is equal to the
-number of GPUs, each OpenMP thread binds to one GPU, and it controls a
-part of the shared memory. More precisely each OpenMP thread owns of
-the vector Z, that is $(n/num\_gpu)$ roots where $n$ is the
-polynomial's degree and $num\_gpu$ the total number of available
-GPUs. Then all GPUs will have a grid of computation organized
+OpenMP and CUDA programming model. All the data are shared with
+OpenMP amoung all the OpenMP threads. The shared data are the solution
+vector $Z$, the polynomial to solve $P$, and the error vector $\Delta
+z$. The number of OpenMP threads is equal to the number of GPUs, each
+OpenMP thread binds to one GPU, and it controls a part of the shared
+memory. More precisely each OpenMP thread will be responsible to
+update its owns part of the vector Z. This part is call $Z_{loc}$ in
+the following. Then all GPUs will have a grid of computation organized
according to the device performance and the size of data on which it
runs the computation kernels.
To compute one iteration of the EA method each GPU performs the
-followings steps. First roots are shared with OpenMP. Each thread
-starts by copying all the previous roots inside its GPU. Then each GPU
-will compute an iteration of the EA method on its own roots. For that
-all the other roots are used. At the end of an iteration, the updated
-roots are copied from the GPU to the CPU. The convergence is checked
-on the new roots. Finally each CPU will update its own roots in the
-shared memory arrays containing all the roots.
+followings steps. First roots are shared with OpenMP and the
+computation of the local size for each GPU is performed (lines 5-7 in
+Algo\ref{alg2-cuda-openmp}). Each thread starts by copying all the
+previous roots inside its GPU (line 9). Then each GPU will copy the
+previous roots (line 10) and it will compute an iteration of the EA
+method on its own roots (line 11). For that all the other roots are
+used. The convergence is checked on the new roots (line 12). At the end
+of an iteration, the updated roots are copied from the GPU to the
+CPU (line 14) by direcly updating its own roots in the shared memory
+arrays containing all the roots.
%In principle a grid is set by two parameter DimGrid, the number of block per grid, DimBloc: the number of threads per block. The following schema shows the architecture of (CUDA,OpenMP).
%% 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 $Z^{0}$\;
-omp\_set\_num\_threads(num\_gpus)\;
-\#pragma omp parallel shared(Z,$\Delta z$,P)\;
-\Indp
-{
-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($max,\epsilon$)\;
}
-\Indm}
-\RC{Est ce qu'on fait apparaitre le pragma? J'hésite...}
+\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
