X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/blobdiff_plain/c11339aa6ecbc7cde5f08f6c738f522f984ecb4c..e48d60147ff49ec4df9948967e6f8ef7c85202e9:/paper.tex

diff --git a/paper.tex b/paper.tex
index 89416df..24a86ca 100644
--- a/paper.tex
+++ b/paper.tex
@@ -307,7 +307,7 @@ propose to use the logarithm and the exponential of a complex in order to comput
 \end{align}
 %%\end{equation}
 
-Using the logarithm (eq. \ref{deflncomplex}) and the exponential (eq. \ref{defexpcomplex}) operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations
+Using the logarithm (eq.~\ref{deflncomplex}) and the exponential (eq.~\ref{defexpcomplex}) operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations
 manipulate lower absolute values and the roots for large polynomial's degrees can be looked for successfully~\cite{Karimall98}.
 
 Applying this solution for the Aberth method we obtain the
@@ -442,7 +442,7 @@ read-only caches.
 
 
 \subsubsection{A sequential Aberth algorithm}
-The main steps of Aberth method are shown in Algorithm.\ref{alg1-seq} :
+The main steps of Aberth method are shown in Algorithm.~\ref{alg1-seq} :
   
 \begin{algorithm}[H]
 \label{alg1-seq}
@@ -488,9 +488,9 @@ With the the Gauss-seidel iteration, we have:
 H(i,z^{k+1})=\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}
 
-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} 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 $z^{k+1}_{i}$.
 
-The $4^{th}$ step of the algorithm checks the convergence condition using Equation.\ref{eq:Aberth-Conv-Cond}.
+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}
@@ -502,31 +502,35 @@ 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 $Total\_time$ can be given as:
+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}
-Total\_time=\left[n\left(T_{i}(n)+T_{j}\right)+O(n)\right].K
+\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 parallelise these steps  in order to reduce the global execution time. In the following,  we explain how we did parrallelize these steps on a GPU architecture using the CUDA platform.
 
-\subsubsection{Parallelize the steps on GPU }
-On the CPU Aberth algorithm both steps 3 and 4 contain the loop \verb=for=, it use one thread to execute all the instruction in the loop N times. Here we explain how the GPU architecture can compute this loop and reduce the execution time.
-The GPU architecture assign the execution of this loop to a groups of parallel threads organized as a grid of blocks each block contain a number of threads. All threads within a block are executed concurrently in parallel. The instruction are executed as a kernel.
+\subsubsection{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 nbr\_thread be the number of threads executed in parallel, so you can easily transform the (18)formula like this: 
+Let N be the number of threads executed in parallel,  Equation.~\ref{eq:T-global} becomes then : 
 
 \begin{equation}
-Total\_time_{exe}=\left[\frac{N}{nbr\_thread}\left(T_{i}(N)+T_{j}\right)+O(n)\right].Nbr\_iter.
+T=\left[\frac{n}{N}\left(T_{i}(n)+T_{j}\right)+O(n)\right].K.
 \end{equation}
 
-In theory, the $Total\_time_{exe}$ on GPU is speed up nbr\_thread times as a $Total\_time_{exe}$ on CPU. We show more details in the experiment part. 
+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 platform, All the instruction of the loop \verb=for= are executed by the GPU as a kernel form. A kernel is a procedure written in CUDA and defined by a heading \verb=__global__=, which means that it is to be executed by the GPU. The following algorithm see the Aberth algorithm on 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 ``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.
 
 \begin{algorithm}[H]
+\label{alg2-cuda}
 %\LinesNumbered
-\caption{Algorithm to find root polynomial with Aberth method}
+\caption{CUDA Algorithm to find roots with the Aberth method}
 
 \KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error
 tolerance threshold),P(Polynomial to solve)}
@@ -535,24 +539,25 @@ tolerance threshold),P(Polynomial to solve)}
 
 \BlankLine
 
-Initialization of the parameter of the polynomial to solve\;
+Initialization of the coeffcients of the polynomial to solve\;
 Initialization of the solution vector $Z^{0}$\;
-Allocate and fill the data in the global memory GPU\;
+Allocate and copy initial data to the GPU global memory\;
 
 \While {$\Delta z_{max}\succ \epsilon$}{
  Let $\Delta z_{max}=0$\;
-$ kernel\_save(d\_Z^{k-1})$\;
+$ kernel\_save(d\_z^{k-1})$\;
 $ kernel\_update(d\_z^{k})$\;
-$kernel\_testConverge (d_?z_{max},d_Z^{k},d_Z^{k-1})$\;
+$kernel\_testConverge(\Delta z_{max},d_z^{k},d_z^{k-1})$\;
 }
 \end{algorithm}
 ~\\ 
 
-After the initialization step, all data of the root finding problem to be solved must be copied from the CPU memory to the GPU global memory, because the GPUs only work on the data filled in their memories. Next, all the data-parallel arithmetic operations inside the main loop \verb=(do ... while(...))= are executed as kernels by the GPU. The first kernel \textit{save} in line( 6, Algorithm 2) consist to save the vector of polynomial's root found at the previous time step on GPU memory, in order to test the convergence of the root at each iteration in line (8, Algorithm 2).
+After the initialisation step, all data of the root finding problem to be solved must be copied from the CPU memory to the GPU global memory, because the GPUs only access data already present in their memories. Next, all the data-parallel arithmetic operations inside the main loop \verb=(do ... while(...))= are executed as kernels by the GPU. The first kernel \textit{save} in line 6 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}).
 
-The second kernel executes the iterative function and update Z(k),as formula (), we notice that the kernel update are called in two forms,  separated with  the value of \emph{R} which determines the radius beyond which we apply the logarithm formula like this: 
+The second kernel executes the iterative function $H$ and updates $z^{k}$, according to Algorithm~\ref{alg3-update}. We notice that the update kernel is called in two forms, separated with thevalue of \emph{R} which determines the radius beyond which we apply the logarithm computation of the power of a complex. 
 
 \begin{algorithm}[H]
+\label{alg3-update}
 %\LinesNumbered
 \caption{A global Algorithm for the iterative function}
 
@@ -563,9 +568,9 @@ $kernel\_update\_Log(d\_z^{k})$\;
 }
 \end{algorithm}
 
-The first form execute the formula(8) if all the module's $( |Z(k)|<= R)$, else the kernel execute the formulas(13,14).the radius R was computed like:
+The first form execute the formula (8) if the modulus is of the current complex is less than the radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes the formulas (13,14).the radius R was computed like:
 
-$$R = \exp( \log(DBL\_MAX) / (2*(double)P.degrePolynome) )$$
+$$R = \exp( \log(DBL\_MAX) / (2*n) )$$
 
 The last kernel verify the convergence of the root after each update of $Z^{(k)}$, as formula(), we used the function of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. 
 
@@ -650,6 +655,16 @@ We initially carried out the convergence of Aberth algorithm with various sizes
 		
 \end{table}
 
+
+\begin{figure}[htbp]
+\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}
+
+
+
 \paragraph{A comparative study between Aberth and Durand-kerner algorithm}
 \begin{table}[htbp]
 	\centering
@@ -665,7 +680,6 @@ We initially carried out the convergence of Aberth algorithm with various sizes
 \end{table}
 
 
-
 \bibliography{mybibfile}
 
 \end{document}