Ehrlich-Aberth au lieu de Aberth

diff --git a/paper.tex b/paper.tex
index d9c332420842844c03ce34d20615eaa0123df349..f66bf99b07b42a69c4fbb1837ed1fd5c8e106b85 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -229,10 +229,10 @@ and experimental study results. Finally, Section\ref{sec7} 6 concludes
 this paper and gives some hints for future research directions in this
 topic.
 
 this paper and gives some hints for future research directions in this
 topic.
 

-\section{The Sequential Aberth method}
+\section{The Sequential Ehrlich-Aberth method}

 \label{sec1}
 A cubically convergent iteration method for finding zeros of
 \label{sec1}
 A cubically convergent iteration method for finding zeros of

-polynomials was proposed by O. Aberth~\cite{Aberth73}. In the fellowing we present the main stages of the running of the Aberth method.
+polynomials was proposed by O. Aberth~\cite{Aberth73}. In the fellowing we present the main stages of the running of the Ehrlich-Aberth method.

 %The Aberth method is a purely algebraic derivation. 
 %To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors 
 
 %The Aberth method is a purely algebraic derivation. 
 %To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors 
 


@@ -271,7 +271,7 @@ The initialization of a polynomial p(z) is done by setting each of the $n$ compl
 Like for any iterative method, we need to choose $n$ initial guess points $z^{(0)}_{i}, i = 1, . . . , n.$
 The initial guess is very important since the number of steps needed by the iterative method to reach
 a given approximation strongly depends on it.
 Like for any iterative method, we need to choose $n$ initial guess points $z^{(0)}_{i}, i = 1, . . . , n.$
 The initial guess is very important since the number of steps needed by the iterative method to reach
 a given approximation strongly depends on it.

-In~\cite{Aberth73} the Aberth iteration is started by selecting $n$
+In~\cite{Aberth73} the Ehrlich-Aberth iteration is started by selecting $n$

 equi-spaced points on a circle of center 0 and radius r, where r is
 an upper bound to the moduli of the zeros. Later, Bini and al.~\cite{Bini96}
 performed this choice by selecting complex numbers along different
 equi-spaced points on a circle of center 0 and radius r, where r is
 an upper bound to the moduli of the zeros. Later, Bini and al.~\cite{Bini96}
 performed this choice by selecting complex numbers along different


@@ -480,20 +480,20 @@ provides two read-only memory spaces, the constant space and the
 texture space, which reside in external DRAM, and are accessed via
 read-only caches.
 
 texture space, which reside in external DRAM, and are accessed via
 read-only caches.
 

-\section{ The implementation of Aberth method on GPU}
+\section{ The implementation of Ehrlich-Aberth method on GPU}

 \label{sec5}
 %%\subsection{A CUDA implementation of the Aberth's method }
 %%\subsection{A GPU implementation of the Aberth's method }
 
 
 
 \label{sec5}
 %%\subsection{A CUDA implementation of the Aberth's method }
 %%\subsection{A GPU implementation of the Aberth's method }
 
 
 

-\subsection{A sequential Aberth algorithm}
-The main steps of Aberth method are shown in Algorithm.~\ref{alg1-seq} :
+\subsection{A sequential Ehrlich-Aberth algorithm}
+The main steps of Ehrlich-Aberth method are shown in Algorithm.~\ref{alg1-seq} :

   
 \begin{algorithm}[H]
 \label{alg1-seq}
 %\LinesNumbered
   
 \begin{algorithm}[H]
 \label{alg1-seq}
 %\LinesNumbered

-\caption{A sequential algorithm to find roots with the Aberth method}
+\caption{A sequential algorithm to find roots with the Ehrlich-Aberth method}

 
 \KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error tolerance threshold),P(Polynomial to solve)}
 \KwOut {Z(The solution root's vector)}
 
 \KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error tolerance threshold),P(Polynomial to solve)}
 \KwOut {Z(The solution root's vector)}


@@ -544,12 +544,12 @@ In the GPU, the schduler assigns the execution of this loop to a group of thread
 
 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. 
 
 
 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.
+Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth algorithm using CUDA.

 
 \begin{algorithm}[H]
 \label{alg2-cuda}
 %\LinesNumbered
 
 \begin{algorithm}[H]
 \label{alg2-cuda}
 %\LinesNumbered

-\caption{CUDA Algorithm to find roots with the Aberth method}
+\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)}
 
 \KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error
 tolerance threshold),P(Polynomial to solve)}


@@ -677,7 +677,7 @@ In this experiment we report the performance of log.exp solution describe in ~\r
 \label{fig:01}
 \end{figure}
 
 \label{fig:01}
 \end{figure}
 

-The figure 3, show a comparison between the execution time of the Ehrlich-Aberth algorithm applying log-exp solution and the execution time of the Ehrlich-Aberth algorithm without applying log-exp solution, with full and sparse polynomials degrees. We can see that the execution time for the both algorithms are the same while the full polynomials degrees are less than 4000 and full polynomials are less than 150,000. After,we show clearly that the classical version of Ehrlich-Aberth algorithm (without applying log.exp) stop to converge and can not solving any polynomial sparse or full. In counterpart, the new version of Ehrlich-Aberth algorithm (applying log.exp solution) can solve very high and large full polynomial exceed 100,000 degrees.
+The figure 3, show a comparison between the execution time of the Ehrlich-Aberth algorithm applying exp.log solution and the execution time of the Ehrlich-Aberth algorithm without applying exp.log solution, with full and sparse polynomials degrees. We can see that the execution time for the both algorithms are the same while the full polynomials degrees are less than 4000 and full polynomials are less than 150,000. After,we show clearly that the classical version of Ehrlich-Aberth algorithm (without applying log.exp) stop to converge and can not solving any polynomial sparse or full. In counterpart, the new version of Ehrlich-Aberth algorithm (applying log.exp solution) can solve very high and large full polynomial exceed 100,000 degrees.

 
 in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R},this exceed the limited number in the mantissa of floating points representations and can not compute the iterative function given in ~\ref{eq:Aberth-H-GS} to obtain the root solution, who justify the divergence of the classical Ehrlich-Aberth algorithm. However, applying log.exp solution given in ~\ref{sec2} took into account the limit of floating using the iterative function in(Eq.~\ref{Log_H1},Eq.~\ref{Log_H2} and allows to solve a very large polynomials degrees . 
 
 
 in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R},this exceed the limited number in the mantissa of floating points representations and can not compute the iterative function given in ~\ref{eq:Aberth-H-GS} to obtain the root solution, who justify the divergence of the classical Ehrlich-Aberth algorithm. However, applying log.exp solution given in ~\ref{sec2} took into account the limit of floating using the iterative function in(Eq.~\ref{Log_H1},Eq.~\ref{Log_H2} and allows to solve a very large polynomials degrees .