Quelque modification de la partie theorique

diff --git a/figures/EA_DK.txt b/figures/EA_DK.txt
index 187bb3a690d480a5583334caaece5f869a4b341a..6414285280e318b227be485d03d89ffd712d168d 100644 (file)
--- a/figures/EA_DK.txt
+++ b/figures/EA_DK.txt
@@ -7,11 +7,11 @@
 150000          28.67          17              156.63          33                      
 200000         40              23              330.456         43                      
 250000         93.76           20              518.342         47                      
 150000          28.67          17              156.63          33                      
 200000         40              23              330.456         43                      
 250000         93.76           20              518.342         47                      

-300000         138.94          21              912.078         50                      
-350000         159.65          18              1398.64         56                      
-400000         258.91          22              3112            20                      
+300000         138.94          21              313.697         21                      
+350000         159.65          18                              28                      
+400000         258.91          22              580.649         20                      

 450000         339.47          23              
 450000         339.47          23              

-500000         419.78          23
+500000         419.78          23              739.882         28

 550000         415.94          19
 600000         549.70          21
 650000         612.12          20
 550000         415.94          19
 600000         549.70          21
 650000         612.12          20

diff --git a/paper.tex b/paper.tex
index 9fee850fbfa1c35aff39f3254619f1b4267ae742..bcac7f4c9511b8c207889a0d437d3f4912157172 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -147,7 +147,7 @@ approximation of all the roots, starting with the Durand-Kerner (DK)
 method:
 %%\begin{center}
 \begin{equation}
 method:
 %%\begin{center}
 \begin{equation}

- z_i^{k+1}=z_{i}^k-\frac{P(z_i^k)}{\prod_{i\neq j}(z_i^k-z_j^k)}
+ z_i^{k+1}=z_{i}^{k}-\frac{P(z_i^{k})}{\prod_{i\neq j}(z_i^{k}-z_j^{k})},   i = 1, . . . , n,

 \end{equation}
 %%\end{center}
 where $z_i^k$ is the $i^{th}$ root of the polynomial $P$ at the
 \end{equation}
 %%\end{center}
 where $z_i^k$ is the $i^{th}$ root of the polynomial $P$ at the


@@ -164,7 +164,8 @@ in the following form by Ehrlich~\cite{Ehrlich67} and
 Aberth~\cite{Aberth73} uses a different iteration formula given as fellows :
 %%\begin{center}
 \begin{equation}
 Aberth~\cite{Aberth73} uses a different iteration formula given as fellows :
 %%\begin{center}
 \begin{equation}

- z_i^{k+1}=z_i^k-\frac{1}{{\frac {P'(z_i^k)} {P(z_i^k)}}-{\sum_{i\neq j}\frac{1}{(z_i^k-z_j^k)}}}.
+\label{Eq:EA}
+ z_i^{k+1}=z_i^{k}-\frac{1}{{\frac {P'(z_i^{k})} {P(z_i^{k})}}-{\sum_{i\neq j}\frac{1}{(z_i^{k}-z_j^{k})}}}, i = 1, . . . , n,

 \end{equation}
 %%\end{center}
 where $P'(z)$ is the polynomial derivative of $P$ evaluated in the
 \end{equation}
 %%\end{center}
 where $P'(z)$ is the polynomial derivative of $P$ evaluated in the


@@ -192,7 +193,7 @@ Freeman and Bane~\cite{Freemanall90}  considered asynchronous
 algorithms, in which each processor continues to update its
 approximations even though the latest values of other $z_i((k))$
 have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before making a new iteration.
 algorithms, in which each processor continues to update its
 approximations even though the latest values of other $z_i((k))$
 have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before making a new iteration.

-Couturier et al. ~\cite{Raphaelall01} proposed two methods of parallelisation for
+Couturier and al~\cite{Raphaelall01} proposed two methods of parallelisation for

 a shared memory architecture and for distributed memory one. They were able to
 compute the roots of polynomials of degree 10000 in 430 seconds with only 8
 personal computers and 2 communications per iteration. Comparing to the sequential implementation
 a shared memory architecture and for distributed memory one. They were able to
 compute the roots of polynomials of degree 10000 in 430 seconds with only 8
 personal computers and 2 communications per iteration. Comparing to the sequential implementation


@@ -206,7 +207,7 @@ hardware resources provided by GPU in order to offer a stronger
 computing ability to the massive data computing.
 
 
 computing ability to the massive data computing.
 
 

-Ghidouche et al. ~\cite{Kahinall14} proposed an implementation of the
+Ghidouche and al~\cite{Kahinall14} proposed an implementation of the

 Durand-Kerner method on GPU. Their main
 result showed that a parallel CUDA implementation is 10 times as fast as
 the sequential implementation on a single CPU for high degree
 Durand-Kerner method on GPU. Their main
 result showed that a parallel CUDA implementation is 10 times as fast as
 the sequential implementation on a single CPU for high degree


@@ -220,35 +221,33 @@ In Section \ref{sec5} we propose a parallel implementation of the Ehrlich-Aberth
 \section{The Sequential Aberth method}
 \label{sec1}
 A cubically convergent iteration method for finding zeros of
 \section{The Sequential Aberth method}
 \label{sec1}
 A cubically convergent iteration method for finding zeros of

-polynomials was proposed by O.Aberth~\cite{Aberth73}. The Aberth
-method is a purely algebraic derivation. To illustrate the
-derivation, we let $w_{i}(z)$ be the product of linear factors 
+polynomials was proposed by O. Aberth~\cite{Aberth73}. In the fellowing we present the main stages of the running of the 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 

 
 

-\begin{equation}
-w_{i}(z)=\prod_{j=1,j \neq i}^{n} (z-x_{j})
-\end{equation}
+%\begin{equation}
+%w_{i}(z)=\prod_{j=1,j \neq i}^{n} (z-x_{j})
+%\end{equation}

 
 

-And let a rational function $R_{i}(z)$ be the correction term of the
-Weistrass method~\cite{Weierstrass03}
+%And let a rational function $R_{i}(z)$ be the correction term of the
+%Weistrass method~\cite{Weierstrass03}

 
 

-\begin{equation}
-R_{i}(z)=\frac{p(z)}{w_{i}(z)} , i=1,2,...,n.
-\end{equation}
+%\begin{equation}
+%R_{i}(z)=\frac{p(z)}{w_{i}(z)} , i=1,2,...,n.
+%\end{equation}

 
 

-Differentiating the rational function $R_{i}(z)$ and applying the
-Newton method, we have:
+%Differentiating the rational function $R_{i}(z)$ and applying the
+%Newton method, we have:

 
 

-\begin{equation}
-\frac{R_{i}(z)}{R_{i}^{'}(z)}= \frac{p(z)}{p^{'}(z)-p(z)\frac{w_{i}(z)}{w_{i}^{'}(z)}}= \frac{p(z)}{p^{'}(z)-p(z) \sum _{j=1,j \neq i}^{n}\frac{1}{z-x_{j}}}, i=1,2,...,n
-\end{equation}
-where R_{i}^{'}(z)is the rational function derivative of F evaluated in the point z 
-Substituting $x_{j}$ for $z_{j}$ we obtain the Aberth iteration method.
+%\begin{equation}
+%\frac{R_{i}(z)}{R_{i}^{'}(z)}= \frac{p(z)}{p^{'}(z)-p(z)\frac{w_{i}(z)}{w_{i}^{'}(z)}}= \frac{p(z)}{p^{'}(z)-p(z) \sum _{j=1,j \neq i}^{n}\frac{1}{z-x_{j}}}, i=1,2,...,n
+%\end{equation}
+%where R_{i}^{'}(z)is the rational function derivative of F evaluated in the point z 
+%Substituting $x_{j}$ for $z_{j}$ we obtain the Aberth iteration method.% 

 
 

-In the fellowing we present the main stages of the running of the Aberth method.

 
 \subsection{Polynomials Initialization}
 
 \subsection{Polynomials Initialization}

-The initialization of a polynomial p(z) is done by setting each of the $n$ complex coefficients $a_{i}$
-:
+The initialization of a polynomial p(z) is done by setting each of the $n$ complex coefficients $a_{i}$:

 
 \begin{equation}
 \label{eq:SimplePolynome}
 
 \begin{equation}
 \label{eq:SimplePolynome}


@@ -282,14 +281,15 @@ v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
 
 \subsection{Iterative Function $H_{i}(z^{k})$}
 The operator used by the Aberth method is corresponding to the
 
 \subsection{Iterative Function $H_{i}(z^{k})$}
 The operator used by the Aberth method is corresponding to the

-following equation which will enable the convergence towards
+following equation~\ref{Eq:EA} which will enable the convergence towards

 polynomial solutions, provided all the roots are distinct.
 
 \begin{equation}
 polynomial solutions, provided all the roots are distinct.
 
 \begin{equation}

-H_{i}(z^{k+1})=z_{i}^{k}-\frac{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}}}}
+\label{Eq:Hi}
+H_{i}(z^{k+1})=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=0,. . . .,n

 \end{equation}
 \end{equation}

-
+we notice that the function iterative $H_{i}$ in Eq.~\ref{Eq:Hi} it the same those presented in Eq.~\ref{Eq:EA}, but we prefer used the last one seen the advantage of its use to improve the Aberth method. More detail in the section ~\ref{sec2}.    

 \subsection{Convergence Condition}
 The convergence condition determines the termination of the algorithm. It consists in stopping from running
 the iterative function $H_{i}(z)$ when the roots are sufficiently stable. We consider that the method
 \subsection{Convergence Condition}
 The convergence condition determines the termination of the algorithm. It consists in stopping from running
 the iterative function $H_{i}(z)$ when the roots are sufficiently stable. We consider that the method


@@ -510,13 +510,13 @@ In this sequential algorithm, one CPU thread  executes all the steps. Let us loo
 There exists two ways to execute the iterative function that we call a Jacobi one and a Gauss-Seidel one. With the Jacobi iteration, at iteration $k+1$ we need all the previous values $z^{(k)}_{i}$ to compute the new values $z^{(k+1)}_{i}$, that is :
 
 \begin{equation}
 There exists two ways to execute the iterative function that we call a Jacobi one and a Gauss-Seidel one. With the Jacobi iteration, at iteration $k+1$ we need all the previous values $z^{(k)}_{i}$ to compute the new values $z^{(k+1)}_{i}$, that is :
 
 \begin{equation}

-H_{i}(z^{k+1})=\frac{p(z^{k}_{i})}{p'(z^{k}_{i})-p(z^{k}_{i})\sum^{n}_{j=1 j\neq i}\frac{1}{z^{k}_{i}-z^{k}_{j}}}, i=1,...,n.
+z^{k+1}_{i}=\frac{p(z^{k}_{i})}{p'(z^{k}_{i})-p(z^{k}_{i})\sum^{n}_{j=1 j\neq i}\frac{1}{z^{k}_{i}-z^{k}_{j}}}, i=1,...,n.

 \end{equation}
 
 \end{equation}
 

-With the Gauss-seidel iteration, we have:
+With the Gauss-Seidel iteration, we have:

 \begin{equation}
 \label{eq:Aberth-H-GS}
 \begin{equation}
 \label{eq:Aberth-H-GS}

-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.
+z^{k+1}_{i}=\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}
 %%Here a finiched my revision %%
 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 $z^{k+1}_{i}$.
 \end{equation}
 %%Here a finiched my revision %%
 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 $z^{k+1}_{i}$.