Quelques MAJ

diff --git a/paper.tex b/paper.tex
index ea5090609179092b5699ad2462f0059f89cff895..9c918b8b574597c0813047c781eb5a8c109de591 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -597,7 +597,7 @@ CUDA (Compute Unified Device Architecture) is a parallel computing architecture
 %Here we give a second form of the iterative function used by the Ehrlich-Aberth method: 
 
 %\begin{equation}
 %Here we give a second form of the iterative function used by the Ehrlich-Aberth method: 
 
 %\begin{equation}

-%\label{Eq:EA1}
+%\label{Eq:EA-1}

 %EA: z^{k+1}_{i}=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=1,. . . .,n
 %\end{equation}
 %EA: z^{k+1}_{i}=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=1,. . . .,n
 %\end{equation}


@@ -606,7 +606,7 @@ CUDA (Compute Unified Device Architecture) is a parallel computing architecture
 %The convergence condition determines the termination of the algorithm. It consists in stopping the %iterative function  when the roots are sufficiently stable. We consider that the method converges %sufficiently when:
 
 %\begin{equation}
 %The convergence condition determines the termination of the algorithm. It consists in stopping the %iterative function  when the roots are sufficiently stable. We consider that the method converges %sufficiently when:
 
 %\begin{equation}

-%\label{eq:Aberth-Conv-Cond}
+%\label{eq:AAberth-Conv-Cond}

 %\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
 %\end{equation}
 
 %\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
 %\end{equation}
 


@@ -654,9 +654,36 @@ R = exp(log(DBL\_MAX)/(2*n) );
 %R = \exp( \log(DBL\_MAX) / (2*n) )
 %\end{equation}
  where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.
 %R = \exp( \log(DBL\_MAX) / (2*n) )
 %\end{equation}
  where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.

+ 
+In order to hold into account the limit of size of floats, we propose to modifying the iterative function and compute the logarithm of:
+
+\begin{equation}
+EA: z^{k+1}_{i}=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=1,. . . .,n
+\end{equation}
+
+This method allows, indeed, to exceed the computation of the polynomials of degree 100,000 and to reach a degree upper to 1,000,000. For that purpose, it is necessary to use the logarithm and the exponential of a complex. The iterative  function of Ehrlich-Aberth method with exponential and logarithm is given as following:
+
+\begin{equation}
+\label{Log_H2}
+EA.EL: 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),
+\end{equation}
+
+where:
+
+\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)i=1,...,n,
+\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 and the roots for large polynomial degrees can be looked for successfully~\cite{Karimall98}.

 
 

-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}.
+%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}
 We introduced three paradigms of parallel programming. Our objective consists in implementing a root finding polynomial algorithm on multiple GPUs. To this end, it is primordial to know how 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.
 
 \subsection{Ehrlich-Aberth parallel implementation on CUDA}
 We introduced three paradigms of parallel programming. Our objective consists in implementing a root finding polynomial algorithm on multiple GPUs. To this end, it is primordial to know how 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.