From: Kahina Date: Fri, 30 Oct 2015 16:49:18 +0000 (+0100) Subject: MAJ X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/commitdiff_plain/e6206cbc48d80bf6ff86c59ba812f861c2b1cb17 MAJ --- diff --git a/figures/EA_DK.pdf b/figures/EA_DK.pdf index f992927..2bfe028 100644 Binary files a/figures/EA_DK.pdf and b/figures/EA_DK.pdf differ diff --git a/figures/EA_DK.plot b/figures/EA_DK.plot index 33c2d80..e9568f1 100644 --- a/figures/EA_DK.plot +++ b/figures/EA_DK.plot @@ -3,9 +3,14 @@ set encoding iso_8859_1 set terminal x11 set size 1,0.5 set term postscript enhanced portrait "Helvetica" 12 +#set decimalsign "," +#set decimalsign locale "fr_FR" set ylabel "execution times (in s)" -set xlabel "polynomial's degree" +set xlabel "polynomial's degree" +#set format x "%6.4e" +#set format x "%2.0e" +#set format x "%2.0t{/Symbol \327}10^{%L}" #set logscale x #set logscale y diff --git a/figures/EA_DK.txt b/figures/EA_DK.txt index 6414285..a22a151 100644 --- a/figures/EA_DK.txt +++ b/figures/EA_DK.txt @@ -6,28 +6,28 @@ 100000 12.45 16 54.5215 30 150000 28.67 17 156.63 33 200000 40 23 330.456 43 -250000 93.76 20 518.342 47 -300000 138.94 21 313.697 21 -350000 159.65 18 28 +250000 93.76 20 #518.342 47 +300000 138.94 21 #313.697 21 +350000 159.65 18 # 28 400000 258.91 22 580.649 20 -450000 339.47 23 +450000 339.47 23 # 500000 419.78 23 739.882 28 -550000 415.94 19 -600000 549.70 21 -650000 612.12 20 -700000 864.21 24 -750000 940.87 23 -800000 1247.16 26 -850000 1702.12 32 -900000 1803.17 30 -950000 2280.07 34 -1000000 2400.51 30 +550000 415.94 19 # +600000 549.70 21 1419.76 27 +650000 612.12 20 # +700000 864.21 24 2149.02 48 +750000 940.87 23 # +800000 1247.16 26 2459.5 26 +850000 1702.12 32 # +900000 1803.17 30 3001.47 28 +950000 2280.07 34 # +1000000 2400.51 30 3321.03 27 # Second data block (index 1) #DK sparse full times nb iter times nb iter -5000 3.42 138 638.572 9597 +5000 3.42 138 622.617 9483 50000 385.266 823 9.27 19 100000 447.364 408 7.73 15 150000 1524.08 552 8.64 21 diff --git a/paper.tex b/paper.tex index bcf399a..c49a705 100644 --- a/paper.tex +++ b/paper.tex @@ -147,7 +147,8 @@ approximation of all the roots, starting with the Durand-Kerner (DK) method: %%\begin{center} \begin{equation} - z_i^{k+1}=z_{i}^{k}-\frac{P(z_i^{k})}{\prod_{i\neq j}(z_i^{k}-z_j^{k})}, i = 1, . . . , n, +\label{DK} + DK: z_i^{k+1}=z_{i}^{k}-\frac{P(z_i^{k})}{\prod_{i\neq j}(z_i^{k}-z_j^{k})}, i = 1, . . . , n, \end{equation} %%\end{center} where $z_i^k$ is the $i^{th}$ root of the polynomial $P$ at the @@ -161,11 +162,11 @@ Docev~\cite{Docev62}, Durand~\cite{Durand60}, Kerner~\cite{Kerner66}. Another method discovered by Borsch-Supan~\cite{ Borch-Supan63} and also described and brought in the following form by Ehrlich~\cite{Ehrlich67} and -Aberth~\cite{Aberth73} uses a different iteration formula given as fellows : +Aberth~\cite{Aberth73} uses a different iteration formula given as: %%\begin{center} \begin{equation} \label{Eq:EA} - z_i^{k+1}=z_i^{k}-\frac{1}{{\frac {P'(z_i^{k})} {P(z_i^{k})}}-{\sum_{i\neq j}\frac{1}{(z_i^{k}-z_j^{k})}}}, i = 1, . . . , n, + EA: z_i^{k+1}=z_i^{k}-\frac{1}{{\frac {P'(z_i^{k})} {P(z_i^{k})}}-{\sum_{i\neq j}\frac{1}{(z_i^{k}-z_j^{k})}}}, i = 1, . . . , n, \end{equation} %%\end{center} where $P'(z)$ is the polynomial derivative of $P$ evaluated in the @@ -279,21 +280,21 @@ u_{i}=2.|a_{i}|^{\frac{1}{i}}; v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}. \end{equation} -\subsection{Iterative Function $H_{i}(z^{k})$} -The operator used by the Aberth method is corresponding to the -following equation~\ref{Eq:EA} which will enable the convergence towards -polynomial solutions, provided all the roots are distinct. +\subsection{Iterative Function} +%The operator used by the Aberth method is corresponding to the +%following equation~\ref{Eq:EA} which will enable the convergence towards +%polynomial solutions, provided all the roots are distinct. + +Here we give a second form of the iterative function used by Ehrlich-Aberth method: \begin{equation} \label{Eq:Hi} -H_{i}(z^{k+1})=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}} +EA2: z^{k+1}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}} {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=0,. . . .,n \end{equation} -we notice that the function iterative $H_{i}$ in Eq.~\ref{Eq:Hi} it the same those presented in Eq.~\ref{Eq:EA}, but we prefer used the last one seen the advantage of its use to improve the Aberth method. More detail in the section ~\ref{sec2}. +we notice that the function iterative in Eq.~\ref{Eq:Hi} it the same those presented in Eq.~\ref{Eq:EA}, but we prefer used the last one seen the advantage of its use to improve the Ehrlich-Aberth method and resolve very high degrees polynomials. More detail in the section ~\ref{sec2}. \subsection{Convergence Condition} -The convergence condition determines the termination of the algorithm. It consists in stopping from running -the iterative function $H_{i}(z)$ when the roots are sufficiently stable. We consider that the method -converges sufficiently when : +The convergence condition determines the termination of the algorithm. It consists in stopping from running the iterative function when the roots are sufficiently stable. We consider that the method converges sufficiently when: \begin{equation} \label{eq:Aberth-Conv-Cond} @@ -302,7 +303,7 @@ converges sufficiently when : \end{equation} -\section{Improving the Ehrlich-Aberth Method} +\section{Improving the Ehrlich-Aberth Method for high degree polynomials with exp.log formulation} \label{sec2} The Ehrlich-Aberth method implementation suffers of overflow problems. This situation occurs, for instance, in the case where a polynomial @@ -336,7 +337,7 @@ iteration function with logarithm: %%$$ \exp \bigl( \ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'}))- \ln(1- \exp(\ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'})+\ln\sum_{i\neq j}^{n}\frac{1}{z_{k}-z_{j}})$$ \begin{equation} \label{Log_H2} -H_{i}(z^{k+1})=z_{i}^{k}-\exp \left(\ln \left( +EA.EL: z^{k+1}=z_{i}^{k}-\exp \left(\ln \left( p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln \left(1-Q(z^{k}_{i})\right)\right), \end{equation} @@ -360,7 +361,7 @@ R = exp(log(DBL_MAX)/(2*n) ); %\end{equation} where \verb=DBL_MAX= stands for the maximum representable \verb=double= value. -\section{The implementation of simultaneous methods in a parallel computer} +\section{Implementation of simultaneous methods in a parallel computer} \label{secStateofArt} The main problem of simultaneous methods is that the necessary time needed for convergence is increased when we increase @@ -476,7 +477,7 @@ read-only caches. -\subsubsection{A sequential Aberth algorithm} +\subsection{A sequential Aberth algorithm} The main steps of Aberth method are shown in Algorithm.~\ref{alg1-seq} : \begin{algorithm}[H] @@ -484,9 +485,7 @@ The main steps of Aberth method are shown in Algorithm.~\ref{alg1-seq} : %\LinesNumbered \caption{A sequential algorithm to find roots with the Aberth method} -\KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error -tolerance threshold),P(Polynomial to solve)} - +\KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error tolerance threshold),P(Polynomial to solve)} \KwOut {Z(The solution root's vector)} \BlankLine @@ -503,7 +502,7 @@ $Z\left[j\right]=H\left(j,Z\right)$\; \For{$i \gets 0 $ \KwTo $n-1$}{ $c=\frac{\left|Z\left[i\right]-ZPrec\left[i\right]\right|}{Z\left[i\right]}$\; -\If{$c\succ\Delta z_{max}$ }{ +\If{$c > \Delta z_{max}$ }{ $\Delta z_{max}$=c\;} } } @@ -514,51 +513,26 @@ In this sequential algorithm, one CPU thread executes all the steps. Let us loo There exists two ways to execute the iterative function that we call a Jacobi one and a Gauss-Seidel one. With the Jacobi iteration, at iteration $k+1$ we need all the previous values $z^{(k)}_{i}$ to compute the new values $z^{(k+1)}_{i}$, that is : \begin{equation} -z^{k+1}_{i}=\frac{p(z^{k}_{i})}{p'(z^{k}_{i})-p(z^{k}_{i})\sum^{n}_{j=1 j\neq i}\frac{1}{z^{k}_{i}-z^{k}_{j}}}, i=1,...,n. +EAJ: z^{k+1}_{i}=\frac{p(z^{k}_{i})}{p'(z^{k}_{i})-p(z^{k}_{i})\sum^{n}_{j=1 j\neq i}\frac{1}{z^{k}_{i}-z^{k}_{j}}}, i=1,...,n. \end{equation} With the Gauss-Seidel iteration, we have: \begin{equation} \label{eq:Aberth-H-GS} -z^{k+1}_{i}=\frac{p(z^{k}_{i})}{p'(z^{k}_{i})-p(z^{k}_{i})(\sum^{i-1}_{j=1}\frac{1}{z^{k}_{i}-z^{k+1}_{j}}+\sum^{n}_{j=i+1}\frac{1}{z^{k}_{i}-z^{k}_{j}})}, i=1,...,n. +EAGS: z^{k+1}_{i}=\frac{p(z^{k}_{i})}{p'(z^{k}_{i})-p(z^{k}_{i})(\sum^{i-1}_{j=1}\frac{1}{z^{k}_{i}-z^{k+1}_{j}}+\sum^{n}_{j=i+1}\frac{1}{z^{k}_{i}-z^{k}_{j}})}, i=1,...,n. \end{equation} %%Here a finiched my revision %% -Using Equation.~\ref{eq:Aberth-H-GS} for the update sub-step of $H(i,z^{k+1})$, we expect the Gauss-Seidel iteration to converge more quickly because, just as its ancestor (for solving linear systems of equations), it uses the most fresh computed roots $z^{k+1}_{i}$. +Using Equation.~\ref{eq:Aberth-H-GS} to update the vector solution \textit{Z}, we expect the Gauss-Seidel iteration to converge more quickly because, just as its ancestor (for solving linear systems of equations), it uses the most fresh computed roots $z^{k+1}_{i}$. The $4^{th}$ step of the algorithm checks the convergence condition using Equation.~\ref{eq:Aberth-Conv-Cond}. Both steps 3 and 4 use 1 thread to compute all the $n$ roots on CPU, which is very harmful for performance in case of the large degree polynomials. -\paragraph{The execution time} -Let $T_{i}(n)$ be the time to compute one new root value at step 3, $T_{i}$ depends on the polynomial's degree $n$. When $n$ increase $T_{i}(n)$ increases too. We need $n.T_{i}(n)$ to compute all the new values in one iteration at step 3. - -Let $T_{j}$ be the time needed to check the convergence of one root value at the step 4, so we need $n.T_{j}$ to compute global convergence condition in each iteration at step 4. - -Thus, the execution time for both steps 3 and 4 is: -\begin{equation} -T_{iter}=n(T_{i}(n)+T_{j})+O(n). -\end{equation} -Let $K$ be the number of iterations necessary to compute all the roots, so the total execution time $T$ can be given as: - -\begin{equation} -\label{eq:T-global} -T=\left[n\left(T_{i}(n)+T_{j}\right)+O(n)\right].K -\end{equation} -The execution time increases with the increasing of the polynomial degree, which justifies to parallelize these steps in order to reduce the global execution time. In the following, we explain how we did parallelize these steps on a GPU architecture using the CUDA platform. -\subsubsection{A Parallel implementation with CUDA } +\subsection{A Parallel implementation with CUDA } On the CPU, both steps 3 and 4 contain the loop \verb=for= and a single thread executes all the instructions in the loop $n$ times. In this subsection, we explain how the GPU architecture can compute this loop and reduce the execution time. In the GPU, the schduler assigns the execution of this loop to a group of threads organised as a grid of blocks with block containing a number of threads. All threads within a block are executed concurrently in parallel. The instructions run on the GPU are grouped in special function called kernels. It's up to the programmer, to describe the execution context, that is the size of the Grid, the number of blocks and the number of threads per block upon the call of a given kernel, according to a special syntax defined by CUDA. -Let N be the number of threads executed in parallel, Equation.~\ref{eq:T-global} becomes then : - -\begin{equation} -T=\left[\frac{n}{N}\left(T_{i}(n)+T_{j}\right)+O(n)\right].K. -\end{equation} - -In theory, total execution time $T$ on GPU is speed up $N$ times as $T$ on CPU. We will see at what extent this is true in the experimental study hereafter. -~\\ -~\\ -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 ``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. +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 a sketch of the Aberth algorithm usind CUDA. @@ -594,7 +568,7 @@ The second kernel executes the iterative function $H$ and updates $z^{k}$, accor \begin{algorithm}[H] \label{alg3-update} %\LinesNumbered -\caption{A global Algorithm for the iterative function} +\caption{Kernel update} \eIf{$(\left|Z^{(k)}\right|<= R)$}{ $kernel\_update(d\_z^{k})$\;} @@ -614,15 +588,15 @@ or from GPU memory to CPU memory \verb=(cudaMemcpyDeviceToHost))=. %%HIER END MY REVISIONS (SIDER) \section{Experimental study} \label{sec6} -\subsection{Definition of the used polynomials } +%\subsection{Definition of the used polynomials } We study two categories of polynomials : the sparse polynomials and the full polynomials. -\paragraph{A sparse polynomial}: is a polynomial for which only some coefficients are not null. We use in the following polonymial for which the roots are distributed on 2 distinct circles : +\paragraph{A sparse polynomial}:is a polynomial for which only some coefficients are not null. We use in the following polynomial for which 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}) \end{equation} -\paragraph{A full polynomial} is in contrast, a polynomial for which all the coefficients are not null. the second form used to obtain a full polynomial is: +\paragraph{A full polynomial}:is in contrast, a polynomial for which all the coefficients are not null. 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} @@ -632,7 +606,7 @@ We study two categories of polynomials : the sparse polynomials and the full pol \end{equation} With this form, we can have until \textit{n} non zero terms whereas the sparse ones have just two non zero terms. -\subsection{The study condition} +%\subsection{The study condition} 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 @@ -643,13 +617,13 @@ element-key which justifies our work of parallelization. E5620@2.40GHz and a GPU K40 (with 6 Go of ram). -\subsection{Comparative study} +%\subsection{Comparative study} In this section, we discuss the performance Ehrlich-Aberth method of root finding polynomials implemented on CPUs and on GPUs. We performed a set of experiments on the sequential and the parallel algorithms, for both sparse and full polynomials and different sizes. We took into account the execution time, the polynomial size and the number of threads per block performed by sum or each experiment on CPUs and on GPUs. All experimental results obtained from the simulations are made in double precision data, for a convergence tolerance of the methods set to $10^{-7}$. Since we were more interested in the comparison of the performance behaviors of Ehrlich-Aberth and Durand-Kerner methods on CPUs versus on GPUs. The initialization values of the vector solution of the Ehrlich-Aberth method are given in section 2.2. -\subsubsection{The execution time in seconds of Ehrlich-Aberth algorithm on CPU OpenMP (1 core, 4 cores) vs. on a Tesla GPU} +\subsection{The execution time in seconds of Ehrlich-Aberth algorithm on CPU OpenMP (1 core, 4 cores) vs. on a Tesla GPU} %\begin{figure}[H] @@ -670,7 +644,7 @@ We report respectively the execution time of the Ehrlich-Aberth method implement %We notice that the convergence precision is a round $10^{-7}$ for the both implementation on CPU and GPU. Consequently, we can conclude that Ehrlich-Aberth on GPU are faster and accurately then CPU implementation. -\subsubsection{Influence of the number of threads on the execution times of different polynomials (sparse and full)} +\subsection{Influence of the number of threads on the execution times of different polynomials (sparse and full)} To optimize the performances of an algorithm on a GPU, it is necessary to maximize the use of cores GPU (maximize the number of threads executed in parallel) and to optimize the use of the various memoirs GPU. In fact, it is interesting to see the influence of the number of threads per block on the execution time of Ehrlich-Aberth algorithm. For that, we notice that the maximum number of threads per block for the Nvidia Tesla K40 GPU is 1024, so we varied the number of threads per block from 8 to 1024. We took into account the execution time for both sparse and full of 10 different polynomials of size 50000 and 10 different polynomials of size 500000 degrees. @@ -683,7 +657,7 @@ For that, we notice that the maximum number of threads per block for the Nvidia The figure 2 show that, the best execution time for both sparse and full polynomial are given when the threads number varies between 64 and 256 threads per bloc. We notice that with small polynomials the best number of threads per block is 64, Whereas, the large polynomials the best number of threads per block is 256. However,In the following experiments we specify that the number of thread by block is 256. -\subsubsection{The impact of exp-log solution to compute very high degrees of polynomial} +\subsection{The impact of exp-log solution to compute very high degrees of polynomial} In this experiment we report the performance of log.exp solution describe in ~\ref{sec2} to compute very high degrees polynomials. \begin{figure}[H] @@ -708,7 +682,7 @@ in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R}, %we report the performances of the exp.log for the Ehrlich-Aberth algorithm for solving very high degree of polynomial. -\subsubsection{A comparative study between Ehrlich-Aberth algorithm and Durand-kerner algorithm} +\subsection{A comparative study between Ehrlich-Aberth algorithm and Durand-kerner algorithm} In this part, we are interesting to compare the simultaneous methods, Ehrlich-Aberth and Durand-Kerner in parallel computer using GPU. We took into account the execution time, the number of iteration and the polynomial's size. for the both sparse and full polynomials. \begin{figure}[H] @@ -735,6 +709,7 @@ This figure show the execution time of the both algorithm EA and DK with sparse \section{Conclusion and perspective} + \label{sec7} \bibliography{mybibfile}