X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/blobdiff_plain/f3cfcb3a5c68d2bc48c1087c56d50165364c133e..c5e1141650462fe452663bbf018f1c136914de28:/paper.tex?ds=inline diff --git a/paper.tex b/paper.tex index c8a6698..02f98de 100644 --- a/paper.tex +++ b/paper.tex @@ -4,6 +4,7 @@ %%\usepackage[utf8]{inputenc} %%\usepackage[T1]{fontenc} %%\usepackage[french]{babel} +\usepackage{float} \usepackage{amsmath,amsfonts,amssymb} \usepackage[ruled,vlined]{algorithm2e} %\usepackage[french,boxed,linesnumbered]{algorithm2e} @@ -96,7 +97,7 @@ root finding of polynomials, high degree, iterative methods, Durant-Kerner, GPU, Polynomials are algebraic structures used in mathematics that capture physical phenomenons and that express the outcome in the form of a function of some unknown variable. Formally speaking, a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} and zeros $\alpha_{i},\textit{i=1,...,n}$ %%\begin{center} \begin{equation} - {\Large p(x)=\sum{a_{i}x^{i}}=a_{n}\prod(x-\alpha_{i}),a_{0} a_{n}\neq 0}. + {\Large p(x)=\sum_{i=0}^{n}{a_{i}x^{i}}=a_{n}\prod_{i=1}^{n}(x-\alpha_{i}), a_{0} a_{n}\neq 0}. \end{equation} %%\end{center} @@ -329,7 +330,9 @@ Q(z_{k})=\exp\left( \ln (p(z_{k}))-\ln(p(z_{k}^{'}))+\ln \left( \sum_{k\neq j}^{n}\frac{1}{z_{k}-z_{j}}\right)\right). \end{equation} -This solution is applied when it is necessary ??? When ??? (SIDER) +This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated as: + +$$R = \exp( \log(DBL\_MAX) / (2*n) )$$ where $DBL\_MAX$ stands for the maximum representable double value. \section{The implementation of simultaneous methods in a parallel computer} \label{secStateofArt} @@ -355,7 +358,7 @@ parallelism that can be suitably exploited by SIMD machines. Moreover, they have fast rate of convergence (quadratic for the Durand-Kerner and cubic for the Ehrlisch-Aberth). Various parallel algorithms reported for these methods can be found -in~\cite{Cosnard90, Freeman89,Freemanall90,,Jana99,Janall99}. +in~\cite{Cosnard90, Freeman89,Freemanall90,Jana99,Janall99}. Freeman and Bane~\cite{Freemanall90} presented two parallel algorithms on a local memory MIMD computer with the compute-to communication time ratio O(n). However, their algorithms require @@ -381,6 +384,8 @@ GPUs, which details are discussed in the sequel. \section {A CUDA parallel Ehrlisch-Aberth method} +In the following, we describe the parallel implementation of Ehrlisch-Aberth method on GPU +for solving high degree polynomials. First, the hardware and software of the GPUs are presented. Then, a CUDA parallel Ehrlisch-Aberth method are presented. \subsection{Background on the GPU architecture} A GPU is viewed as an accelerator for the data-parallel and @@ -583,33 +588,26 @@ or from GPU memory to CPU memory \verb=(cudaMemcpyDeviceToHost))=. \section{Experimental study} \subsection{Definition of the polynomial used} -We use two forms of polynomials: -\paragraph{sparse polynomial}: -in this following form, the roots are distributed on 2 distinct circles: +We study two forms of polynomials the sparse polynomials and the full polynomials: +\paragraph{Sparse polynomial}: in this following form, the roots are distributed on 2 distinct circles: \begin{equation} - \forall \alpha_{1} \alpha_{2} \in C,\forall n_{1},n_{2} \in N^{*}; P(z)= (z^{n^{1}}-\alpha_{1})(z^{n^{2}}-\alpha_{2}) + \forall \alpha_{1} \alpha_{2} \in C,\forall n_{1},n_{2} \in N^{*}; P(z)= (z^{n_{1}}-\alpha_{1})(z^{n_{2}}-\alpha_{2}) \end{equation} - This form makes it possible to associate roots having two different modules and thus to work on a polynomial constitute of four non zero terms. -\paragraph{Full polynomial}: - the second form used to obtain a full polynomial is: +\paragraph{Full polynomial}: the second form used to obtain a full polynomial is: %%\begin{equation} %%\forall \alpha_{i} \in C,\forall n_{i}\in N^{*}; P(z)= \sum^{n}_{i=1}(z^{n^{i}}.a_{i}) %%\end{equation} \begin{equation} - {\Large \forall a_{i} \in C, i\in N; p(x)=\sum^{n-1}_{i=1} a_{i}.x^{i}} + {\Large \forall a_{i} \in C, i\in N; p(x)=\sum^{n}_{i=0} a_{i}.x^{i}} \end{equation} with this form, we can have until \textit{n} non zero terms. \subsection{The study condition} -In order to have representative average values, for each -point of our curves we measured the roots finding of 10 -different polynomials. - The our experiences results concern two parameters which are the polynomial degree and the execution time of our program to converge on the solution. The polynomial degree allows us @@ -617,7 +615,7 @@ to validate that our algorithm is powerful with high degree polynomials. The execution time remains the element-key which justifies our work of parallelization. For our tests we used a CPU Intel(R) Xeon(R) CPU -E5620@2.40GHz and a GPU K40 (with 6 Go of ram) +E5620@2.40GHz and a GPU K40 (with 6 Go of ram). \subsection{Comparative study} @@ -642,7 +640,7 @@ We initially carried out the convergence of Aberth algorithm with various sizes % \label{tab:theConvergenceOfAberthAlgorithm} %\end{table} -\begin{figure}[htbp] +\begin{figure}[H] \centering \includegraphics[width=0.8\textwidth]{figures/Compar_EA_algorithm_CPU_GPU} \caption{Aberth algorithm on CPU and GPU} @@ -671,28 +669,30 @@ We initially carried out the convergence of Aberth algorithm with various sizes %\end{table} -\begin{figure}[htbp] +\begin{figure}[H] \centering \includegraphics[width=0.8\textwidth]{figures/influence_nb_threads} \caption{Influence of the number of threads on the execution times of different polynomials (sparse and full)} \label{fig:01} \end{figure} - +\subsubsection{The impact of exp-log solution to compute very high degrees of polynomial} +\begin{figure}[H] +\centering + \includegraphics[width=0.8\textwidth]{figures/log_exp} +\caption{The impact of exp-log solution to compute very high degrees of polynomial.} +\label{fig:01} +\end{figure} \subsubsection{A comparative study between Aberth and Durand-kerner algorithm} -\begin{table}[htbp] - \centering - \begin{tabular} {|R{2cm}|L{2.5cm}|L{2.5cm}|L{1.5cm}|L{1.5cm}|} - \hline Polynomial's degrees & Aberth $T_{exe}$ & D-Kerner $T_{exe}$ & Aberth iteration & D-Kerner iteration\\ - \hline 5000 & 0.40 & 3.42 & 17 & 138 \\ - \hline 50000 & 3.92 & 385.266 & 17 & 823\\ - \hline 500000 & 497.109 & 4677.36 & 24 & 214\\ - \hline - \end{tabular} - \caption{Aberth algorithm compare to Durand-Kerner algorithm} - \label{tab:AberthAlgorithCompareToDurandKernerAlgorithm} -\end{table} + + +\begin{figure}[H] +\centering + \includegraphics[width=0.8\textwidth]{figures/EA_DK} +\caption{Ehrlisch-Aberth and Durand-Kerner algorithm on GPU} +\label{fig:01} +\end{figure} \bibliography{mybibfile}