X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper2.git/blobdiff_plain/87920647de834836776958ae2479e1e91567ab76..ddc3794278a5e10aa4895f7c8582a07fb06f3e88:/paper.tex

diff --git a/paper.tex b/paper.tex
index a790787..41c5960 100644
--- a/paper.tex
+++ b/paper.tex
@@ -682,10 +682,7 @@ than a fixed value $\epsilon$.
 \LZK{On ne dit pas plutÃ´t "the relative errors" Ã  la place de "root modules"? Raph nous confirmera quelle critÃ¨re d'arrÃªt a utilisÃ©.}
 
 \subsection{Improving Ehrlich-Aberth method}
-With high degree polynomials, the Ehrlich-Aberth method suffers from
-floating point overflows due to the mantissa of floating points
-representations. This induces errors in the computation of $p(z)$ when
-$z$ is large.
+With high degree polynomials, the Ehrlich-Aberth method suffers from floating point overflows due to the mantissa of floating points representations. This induces errors in the computation of $p(z)$ when $z$ is large.
  
 %Experimentally, it is very difficult to solve polynomials with the Ehrlich-Aberth method and have roots which except the circle of unit, represented by the radius $r$ evaluated as: 
 
@@ -702,33 +699,28 @@ In order to solve this problem, we propose to modify the iterative
 function by using the logarithm and the exponential of a complex and
 we propose a new version of the Ehrlich-Aberth method.  This method
 allows us to exceed the computation of the polynomials of degree
-100,000 and to reach a degree up to more than 1,000,000. This new
-version of the Ehrlich-Aberth method with exponential and logarithm is
-defined as follows:
+100,000 and to reach a degree up to more than 1,000,000. The reformulation of the iteration~(\ref{Eq:EA1}) of the Ehrlich-Aberth method with exponential and logarithm is defined as follows, for $i=1,\dots,n$:
 
 \begin{equation}
 \label{Log_H2}
-z^{k+1}_{i}=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),
+z^{k+1}_i = z_i^k - \exp(\ln(p(z_i^k)) - \ln(p'(z^k_i)) - \ln(1-Q(z^k_i))),
 \end{equation}
 
 where:
 
-\begin{eqnarray}
+\begin{equation}
 \label{Log_H1}
-Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left(
-\sum_{i\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right) \nonumber  \\
-i=1,...,n  
-\end{eqnarray}
+Q(z^k_i) = \exp(\ln(p(z^k_i)) - \ln(p'(z^k_i)) + \ln(\sum_{i\neq j}^n\frac{1}{z^k_i-z^k_j})).
+\end{equation}
 
 
 %We propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent. 
-Using the logarithm  and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower absolute values~\cite{Karimall98}.
+Using the logarithm  and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower absolute values~\cite{Karimall98}. \LZK{Je n'ai pas compris cette derniÃ¨re phrase?}
 
 %This problem was discussed earlier in~\cite{Karimall98} for the Durand-Kerner method. The authors
 %propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent. Using the logarithm  and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower absolute values and the roots for large polynomial degrees can be looked for successfully~\cite{Karimall98}.
 
-\subsection{Ehrlich-Aberth parallel implementation on CUDA}
+\subsection{The Ehrlich-Aberth parallel implementation on CUDA}
 %We introduced three paradigms of parallel programming.
 
 Our objective consists in implementing a root finding polynomial
@@ -737,22 +729,12 @@ to manage CUDA contexts of different GPUs. A direct method for
 controlling the various GPUs is to use as many threads or processes as
 GPU devices. We can choose the GPU index based on the identifier of
 OpenMP thread or the rank of the MPI process. Both approaches will be
-investigated.
+investigated. \LZK{RÃ©pÃ©tition! Le mÃªme texte est dÃ©jÃ  Ã©crit comme intro dans la section II. Sinon ici on parle seulement de l'implÃ©mentation cuda sans mpi et openmp!}
 
 
 
 
-Like any parallel code, a GPU parallel implementation first requires
-to determine the sequential tasks and the parallelizable parts of the
-sequential version of the program/algorithm. In our case, all the
-operations that are easy to execute in parallel must be made by the
-GPU to accelerate the execution of the application, like the step 3
-and step 4. On the other hand, all the sequential operations and the
-operations that have data dependencies between threads or recursive
-computations must be executed by only one CUDA or CPU thread (step 1
-and step 2). Initially, we specify the organization of parallel
-threads, by specifying the dimension of the grid Dimgrid, the number
-of blocks per grid DimBlock and the number of threads per block.
+Like any parallel code, a GPU parallel implementation first requires to determine the sequential code and the data-parallel operations of a algorithm. In fact, all the operations that are easy to execute in parallel must be made by the GPU to accelerate the execution, like the steps 3 and 4. On the other hand, all the sequential operations and the operations that have data dependencies between CUDA threads or recursive computations must be executed by only one CUDA thread or a CPU thread (the steps 1 and 2).\LZK{La mÃ©thode est dÃ©jÃ  mal prÃ©sentÃ©e, dans ce cas c'est encore plus difficile de comprendre que reprÃ©sentent ces diffÃ©rentes Ã©tapes!} Initially, we specify the organization of parallel threads by specifying the dimension of the grid \verb+Dimgrid+, the number of blocks per grid \verb+DimBlock+ and the number of threads per block.
 
 The code is organized kernels which are part of code that are run on
 GPU devices. For step 3, there are two kernels, the first named
