X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/blobdiff_plain/365ac29c1fc7a4a90a993cfe892d6153beb8a460..d70ea05e424af6b2925d036c3ff4b7f2552da8fc:/paper.tex

diff --git a/paper.tex b/paper.tex
index 00d805d..e8ec4e3 100644
--- a/paper.tex
+++ b/paper.tex
@@ -385,7 +385,7 @@ Authors usually adopt one of the two following approaches to parallelize root
 finding algorithms. The first approach aims at reducing the total number of
 iterations as by Miranker
 ~\cite{Mirankar68,Mirankar71}, Schedler~\cite{Schedler72} and
-Winogard~\cite{Winogard72}. The second approach aims at reducing the
+Winograd~\cite{Winogard72}. The second approach aims at reducing the
 computation time per iteration, as reported
 in~\cite{Benall68,Jana06,Janall99,Riceall06}. 
 
@@ -582,9 +582,9 @@ Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth algorithm using C
 \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 (the derivative of P), $n$ (Polynomial's degrees), $\Delta z_{max}$ (maximum value of stop condition)}
+  threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial's degrees), $\Delta z_{max}$ (Maximum value of stop condition)}
 
-\KwOut {$Z$ (The solution root's vector), $ZPrec$ (the previous solution root's vector)}
+\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
 
 \BlankLine
 
@@ -605,11 +605,19 @@ Copy results from GPU memory to CPU memory\;
 \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 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 named \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}).
+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
+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 $H$ and updates
-$d\_Z$, according to Algorithm~\ref{alg3-update}. We notice that the
-update kernel is called in two forms, separated with the value of
+Z, according to Algorithm~\ref{alg3-update}. We notice that the
+update kernel is called in two forms, according to the value
 \emph{R} which determines the radius beyond which we apply the
 exponential logarithm algorithm. 
 
@@ -619,9 +627,9 @@ exponential logarithm algorithm.
 \caption{Kernel update}
 
 \eIf{$(\left|Z\right|<= R)$}{
-$kernel\_update((Z,P,Pu)$\;}
+$kernel\_update(Z,P,Pu)$\;}
 {
-$kernel\_update\_ExpoLog((Z,P,Pu))$\;
+$kernel\_update\_ExpoLog(Z,P,Pu)$\;
 }
 \end{algorithm}
 
@@ -835,7 +843,8 @@ numerical applications on GPU.
 
 In future works, we plan to investigate the possibility of using
 several multiple GPUs simultaneously, either with multi-GPU machine or
-with cluster of GPUs.
+with cluster of GPUs. It may also be interesting to study the
+implementation of other root finding polynomial methods on GPU.