From: Kahina Date: Thu, 5 Nov 2015 12:35:18 +0000 (+0100) Subject: new X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/commitdiff_plain/2012ce2acb409df5a2db763463252497a2f32663 new --- diff --git a/mybibfile.bib b/mybibfile.bib index 4759950..a108141 100644 --- a/mybibfile.bib +++ b/mybibfile.bib @@ -141,7 +141,7 @@ }x @Article{Ostrowski41, - title = " On a Theorem by {J. L. Walsh} Concerning the Moduli of Roots of Algebraic Equations,Bull. A.M.S.", + title = " On a Theorem by {J. L. Walsh} Concerning the Moduli of Roots of Algebraic Equations. A.M.S.", journal = " Algorithmes itératifs paralléles et distribués", volume = "1", number = "47", @@ -165,7 +165,7 @@ OPTannote = {•} } @Article{Kahinall14, - title = "Parallel implementation of the {D}urand-{K}erner algorithm for polynomial root-finding on GPU", + title = "Parallel implementation of the {D}urand-{K}erner algorithm for polynomial root-finding on {GPU}", journal = "IEEE. Conf. on advanced Networking, Distributed Systems and Applications", volume = "", number = "", @@ -231,7 +231,7 @@ OPTannote = {•} @InProceedings{Winogard72, title = "Parallel Iteration Methods", - author = "Shmuel Winograd", + author = "S. Winograd", bibdate = "2011-09-13", bibsource = "DBLP, http://dblp.uni-trier.de/db/conf/coco/cocc1972.html#Winograd72", diff --git a/paper.tex b/paper.tex index 8f03506..96e01fd 100644 --- a/paper.tex +++ b/paper.tex @@ -364,12 +364,11 @@ Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left( \end{equation} This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as : +\begin{equation} \label{R.EL} -\begin{center} -\begin{verbatim} R = exp(log(DBL_MAX)/(2*n) ); -\end{verbatim} -\end{center} +\end{equation} + %\begin{equation} @@ -583,7 +582,7 @@ quickly because, just as any Jacobi algorithm (for solving linear systems of equ %In CUDA programming, all the instructions of the \verb=for= loop are executed by the GPU as a kernel. A kernel is a function written in CUDA and defined by the \verb=__global__= qualifier added before a usual \verb=C= function, which instructs the compiler to generate appropriate code to pass it to the CUDA runtime in order to be executed on the GPU. -Algorithm~\ref{alg2-cuda} shows steps of the Ehrlich-Aberth algorithm using CUDA. +Algorithm~\ref{alg2-cuda} shows sketch of the Ehrlich-Aberth algorithm using CUDA. \begin{enumerate} \begin{algorithm}[H] @@ -620,11 +619,11 @@ After the initialization step, all data of the root finding problem must be copied from the CPU memory to the GPU global memory. Next, all the data-parallel arithmetic operations inside the main loop \verb=(while(...))= are executed as kernels by the GPU. The -first kernel named \textit{save} in line 6 of +first kernel named \textit{save} in line 7 of Algorithm~\ref{alg2-cuda} consists in saving the vector of polynomial's root found at the previous time-step in GPU memory, in order to check the convergence of the roots after each iteration (line -8, Algorithm~\ref{alg2-cuda}). +10, Algorithm~\ref{alg2-cuda}). The second kernel executes the iterative function and updates $Z$, according to Algorithm~\ref{alg3-update}. We notice that the @@ -648,12 +647,10 @@ The first form executes formula the EA function Eq.~\ref{Eq:Hi} if the modulus of the current complex is less than the a certain value called the radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes the EA.EL function Eq.~\ref{Log_H2} -(with Eq.~\ref{deflncomplex}, Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as in ~\ref{R.EL} : - -$$R = \exp( \log(DBL\_MAX) / (2*n) )$$ where $DBL\_MAX$ stands for the maximum representable double value. +(with Eq.~\ref{deflncomplex}, Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as in Eq.~\ref{R.EL}. The last kernel checks the convergence of the roots after each update -of $Z^{(k)}$, according to formula Eq.~\ref{eq:Aberth-Conv-Cond}. We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. +of $Z^{k}$, according to formula Eq.~\ref{eq:Aberth-Conv-Cond}. We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. The kernel terminates its computations when all the roots have converged. It should be noticed that, as blocks of threads are @@ -702,7 +699,7 @@ all the coefficients are not null. A full polynomial is defined by: %polynomials. The execution time remains the %element-key which justifies our work of parallelization. For our tests, a CPU Intel(R) Xeon(R) CPU -E5620@2.40GHz and a GPU K40 (with 6 Go of ram) is used. +E5620@2.40GHz and a GPU K40 (with 6 Go of ram) are used. %\subsection{Comparative study} @@ -737,7 +734,7 @@ of the methods are given in Section~\ref{sec:vec_initialization}. In Figure~\ref{fig:01}, we report the execution times of the Ehrlich-Aberth method on one core of a Quad-Core Xeon E5620 CPU, on four cores on the same machine with \textit{OpenMP} and on a Nvidia -Tesla K40c GPU. We chose different sparse polynomials with degrees +Tesla K40 GPU. We chose different sparse polynomials with degrees ranging from 100,000 to 1,000,000. We can see that the implementation on the GPU is faster than those implemented on the CPU. However, the execution time for the